AFFFPupu/Maths_competition_questions · Datasets at Hugging Face

ID int64 1 120	Difficulty stringclasses 3 values	Label stringclasses 4 values	Question stringlengths 64 1.4k
1	Intermediate	Combinatorics	Convex polygons $P_1$ and $P_2$ are drawn in the same plane with $n_1$ and $n_2$ sides, respectively, $n_1 \leq n_2$. If $P_1$ and $P_2$ do not have any line segment in common, then the maximum number of intersections of $P_1$ and $P_2$ is?
2	Intermediate	Combinatorics	Suppose that $n$ people each know exactly one piece of information, and all $n$ pieces are different. Every time person $A$ phones person $B, A$ tells $B$ everything that $A$ knows, while $B$ tells $A$ nothing. What is the minimum number of phone calls between pairs of people needed for everyone to know everything? Prove your answer is a minimum.
3	Intermediate	Combinatorics	The numbers 1447,1005 and 1231 have something in common: each is a 4-digit number beginning with 1 that has exactly two identical digits. How many such numbers are there?
4	Intermediate	Combinatorics	For $\{1,2,3, \ldots, n\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1,2,3,6,9\}$ is $9-6+3-2+1=5$ and for $\{5\}$ it is simply 5 . Find the sum of all such alternating sums for $n=7$.
5	Intermediate	Combinatorics	For $\{1,2,3, \ldots, n\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1,2,3,6,9\}$ is $9-6+3-2+1=5$ and for $\{5\}$ it is simply 5 . Find the sum of all such alternating sums for $n=7$.
6	Intermediate	Combinatorics	Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
7	Intermediate	Combinatorics	What is the largest 2-digit prime factor of the integer $n=\left(\begin{array}{l}200 \\ 100\end{array}\right)$ ?
8	Intermediate	Combinatorics	A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac{m}{n}$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$.
9	Intermediate	Combinatorics	Let $A, B, C$ and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p=\frac{n}{729}$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.
10	Intermediate	Combinatorics	In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned $\frac{1}{2}$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?
11	Introductory	Combinatorics	The number of diagonals that can be drawn in a polygon of 100 sides is:
12	Introductory	Combinatorics	Six straight lines are drawn in a plane with no two parallel and no three concurrent. The number of regions into which they divide the plane is:
13	Introductory	Combinatorics	One thousand unit cubes are fastened together to form a large cube with edge length 10 units; this is painted and then separated into the original cubes. The number of these unit cubes which have at least one face painted is
14	Introductory	Combinatorics	A fair die is rolled six times. The probability of rolling at least a five at least five times is
15	Introductory	Combinatorics	Assume every 7-digit whole number is a possible telephone number except those that begin with 0 or 1 . What fraction of telephone numbers begin with 9 and end with 0 ?
16	Introductory	Combinatorics	The 600 students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of three lunch groups. The probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group is approximately
17	Introductory	Combinatorics	How many rearrangements of $a b c d$ are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either $a b$ or $b a$.
18	Introductory	Combinatorics	How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?
19	Introductory	Combinatorics	How many ways can a student schedule 3 mathematics courses - algebra, geometry, and number theory -- in a 6 -period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
20	Introductory	Combinatorics	The eighth grade class at Lincoln Middle School has 93 students. Each student takes a math class or a foreign language class or both. There are 70 eighth graders taking a math class, and there are 54 eighth graders taking a foreign language class. How many eighth graders take only a math class and not a foreign language class?
21	Olympiad	Combinatorics	Carina has three pins, labeled $A, B$, and $C$, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance 1 away. What is the least number of moves that Carina can make in order for triangle $A B C$ to have area $2021 ?$
22	Olympiad	Combinatorics	There is an integer $n>1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The $k$ cable cars of $A$ have $k$ different starting points and $k$ different finishing points, and a cable car that starts higher also finishes higher. The same conditions hold for $B$. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer $k$ for which one can guarantee that there are two stations that are linked by both companies.
23	Olympiad	Combinatorics	There are $4 n$ pebbles of weights $1,2,3, \ldots, 4 n$. Each pebble is colored in one of $n$ colors and there are four pebbles of each color. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied: - The total weights of both piles are the same. - Each pile contains two pebbles of each color.
24	Olympiad	Combinatorics	Let $n$ be a positive integer. Eric and a squid play a turn-based game on an infinite grid of unit squares. Eric's goal is to capture the squid by moving onto the same square as it. Initially, all the squares are colored white. The squid begins on an arbitrary square in the grid, and Eric chooses a different square to start on. On the squid's turn, it performs the following action exactly 2020 times: it chooses an adjacent unit square that is white, moves onto it, and sprays the previous unit square either black or gray. Once the squid has performed this action 2020 times, all squares colored gray are automatically colored white again, and the squid's turn ends. If the squid is ever unable to move, then Eric automatically wins. Moreover, the squid is claustrophobic, so at no point in time is it ever surrounded by a closed loop of black or gray squares. On Eric's turn, he performs the following action at most $n$ times: he chooses an adjacent unit square that is white and moves onto it. Note that the squid can trap Eric in a closed region, so that Eric can never win. Eric wins if he ever occupies the same square as the squid. Suppose the squid has the first turn, and both Eric and the squid play optimally. Both Eric and the squid always know each other's location and the colors of all the squares. Find all positive integers $n$ such that Eric can win in finitely many moves.
25	Olympiad	Combinatorics	Let $k$ be the number of students in a circle. Then let $m$ be the number of ways they can rearrange ourselves so that each of them is in the same spot or within one spot of where they started, and no two people are ever on the same spot. If $m$ leaves a remainder of 1 when divided by 5 , how many possible values are there of $k$, where $k$ is at least 3 and at most $2008 ?$
26	Olympiad	Combinatorics	A square of side $n$ is formed from $n^2$ unit squares, each colored in red, yellow or green. Find minimal $n$, such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or column).
27	Olympiad	Combinatorics	An $(n, k)$-tournament is a contest with $n$ players held in $k$ rounds such that: (i) Each player plays in each round, and every two players meet at most once. (ii) If player $A$ meets player $B$ in round $i$, player $C$ meets player $D$ on round $i$, and player $A$ meets player $C$ in round $j$, then player $B$ meets player $D$ in round $j$. Determine all pairs $(n, k)$ for which there exists an $(n, k)$-tournament.
28	Olympiad	Combinatorics	For a given positive integer $n>2$, let $C_1, C_2, C_3$ be the boundaries of three convex $n$-gons in the plane such that $C_1 \cap C_2, C_2 \cap C_3$, $C_3 \cap C_1$ are finite. Find the maximum number of points in the set $C_1 \cap C_2 \cap C_3$.
29	Olympiad	Combinatorics	The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer $n \geq 4$, find the least number of edges of a graph that can be obtained by repeated applications of this operation from a complete graph on $n$ vertices (where each pair of vertices are joined by an edge).
30	Olympiad	Combinatorics	The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer $n \geq 4$, find the least number of edges of a graph that can be obtained by repeated applications of this operation from a complete graph on $n$ vertices (where each pair of vertices are joined by an edge).
31	Introductory	Geometry	Triangle $A B C$ is equilateral with side length 6 . Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A, O$, and $C$ ?
32	Introductory	Geometry	The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle?
33	Introductory	Geometry	Let $A B C$ be a triangle. The bisector of $\angle A B C$ intersects $\overline{A C}$ at $E$, and the bisector of $\angle A C B$ intersects $\overline{A B}$ at $F$. If $B F=1, C E=2$, and $B C=3$, then the perimeter of $\triangle A B C$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
34	Introductory	Geometry	The circle having $(0,0)$ and $(8,6)$ as the endpoints of a diameter intersects the $x$-axis at a second point. What is the $x$-coordinate of this point?
35	Introductory	Geometry	A trapezoid has side lengths $3,5,7$, and 11 . The sum of all the possible areas of the trapezoid can be written in the form of $r_1 \sqrt{n_1}+r_2 \sqrt{n_2}+r_3$, where $r_1, r_2$, and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to $r_1+r_2+r_3+n_1+n_2$ ?
36	Introductory	Geometry	Square $P Q R S$ lies in the first quadrant. Points $(3,0),(5,0),(7,0)$, and $(13,0)$ lie on lines $S P, R Q, P Q$, and $S R$, respectively. What is the sum of the coordinates of the center of the square $P Q R S$ ?
37	Introductory	Geometry	Triangle $A B C$ has $A B=27, A C=26$, and $B C=25$. Let $I$ be the intersection of the internal angle bisectors of $\triangle A B C$. What is $B I$ ?
38	Introductory	Geometry	In rectangle $A B C D, A B=6, A D=30$, and $G$ is the midpoint of $\overline{A D}$. Segment $A B$ is extended 2 units beyond $B$ to point $E$, and $F$ is the intersection of $\overline{E D}$ and $\overline{B C}$. What is the area of quadrilateral $B F D G$ ?
39	Introductory	Geometry	In $\triangle A B C, A B=86$, and $A C=97$. A circle with center $A$ and radius $A B$ intersects $\overline{B C}$ at points $B$ and $X$. Moreover $\overline{B X}$ and $\overline{C X}$ have integer lengths. What is $B C$ ?
40	Introductory	Geometry	Two points on the circumference of a circle of radius $r$ are selected independently and at random. From each point a chord of length $r$ is drawn in a clockwise direction. What is the probability that the two chords intersect?
41	Intermediate	Geometry	Square $A B C D$ is inscribed in a circle. Point $P$ is on this circle such that $A P \cdot C P=56$, and $B P \cdot D P=90$. What is the area of the square?
42	Intermediate	Geometry	In triangle $A B C$ we have $A B=7, A C=8, B C=9$. Point $D$ is on the circumscribed circle of the triangle so that $A D$ bisects angle $B A C$ What is the value of $A D / C D$ ?
43	Intermediate	Geometry	A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\overline{A B}$, has length 31 . Find the sum of the lengths of the three diagonals that can be drawn from $A$.
44	Intermediate	Geometry	A hexagon with sides of lengths $2,2,7,7,11$, and 11 is inscribed in a circle. Find the diameter of the circle
45	Intermediate	Geometry	In a regular heptagon $A B C D E F G$, prove that: $\frac{1}{A B}=\frac{1}{A C}+\frac{1}{A E}$.
46	Intermediate	Geometry	$\triangle A B C$ has point $D$ on $A B$, point $E$ on $B C$, and point $F$ on $A C . A E, C D$, and $B F$ intersect at point $G$. The ratio $A D: D B$ is $3: 5$ and the ratio $C E: E B$ is $8: 3$. Find the ratio of $F G: G B$
47	Intermediate	Geometry	Consider a triangle $A B C$ with its three medians drawn, with the intersection points being $D, E, F$, corresponding to $A B, B C$, and $A C$ respectively. Thus, if we label point $A$ with a weight of $1, B$ must also have a weight of 1 since $A$ and $B$ are equidistant from $D$. By the same process, we find $C$ must also have a weight of 1 . Now, since $A$ and $B$ both have a weight of $1, D$ must have a weight of 2 (as is true for $E$ and $F$ ). Thus, if we label the centroid $P$, we can deduce that $D P: P C$ is $1: 2$ - the inverse ratio of their weights.
48	Intermediate	Geometry	In rectangle $A B C D, A B=6$ and $B C=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $B E=E F=F C$. Segments $\overline{A E}$ and $\overline{A F}$ intersect $\overline{B D}$ at $P$ and $Q$, respectively. The ratio $B P: P Q: Q D$ can be written as $r: s: t$ where the greatest common factor of $r, s$, and $t$ is 1 . What is $r+s+t$ ?
49	Intermediate	Geometry	Triangle $A B C$ has $A C=450$ and $B C=300$. Points $K$ and $L$ are located on $\overline{A C}$ and $\overline{A B}$ respectively so that $A K=C K$, and $\overline{C L}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\overline{B K}$ and $\overline{C L}$, and let $M$ be the point on line $B K$ for which $K$ is the midpoint of $\overline{P M}$. If $A M=180$, find $L P$.
50	Intermediate	Geometry	In parallelogram $A B C D$, point $M$ is on $\overline{A B}$ so that $\frac{A M}{A B}=\frac{17}{1000}$ and point $N$ is on $\overline{A D}$ so that $\frac{A N}{A D}=\frac{17}{2009}$. Let $P$ be the point of intersection of $\overline{A C}$ and $\overline{M N}$. Find $\frac{A C}{A P}$.
51	Olympiad	Geometry	Let $\triangle A B C$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $\overline{B C}, \overline{C A}$, and $\overline{A B}$ respectively. One of the intersection points of the line $\overline{E F}$ and the circumcircle is $P$. The lines $\overline{B P}$ and $\overline{D F}$ meet at point $Q$. Prove that $\|A P\|=\|A Q\|$. (IMO Shortlist $2010 \mathrm{G} 1$ )
52	Olympiad	Geometry	Four circles $\omega, \omega_A, \omega_B$, and $\omega_C$ with the same radius are drawn in the interior of triangle $A B C$ such that $\omega_A$ is tangent to sides $A B$ and $A C, \omega_B$ to $B C$ and $B A, \omega_C$ to $C A$ and $C B$, and $\omega$ is externally tangent to $\omega_A, \omega_B$, and $\omega_C$. If the sides of triangle $A B C$ are 13,14, and 15 , the radius of $\omega$ can be represented in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
53	Olympiad	Geometry	The following problems are constructions involving only a straightedge (no compass). 1. Construct the fourth harmonic line to three given lines through a point. 2. Construct the fourth harmonic point to three points on a line. 3. If a given right angle and a given arbitrary angle have their vertex and one side in common, double the given arbitrary angle. 4. Draw a parallel through a given point $P$ to two given parallel lines $l_1$ and $l_2$.
54	Olympiad	Geometry	Given two similar right triangles $A B C$ and $A^{\prime} B^{\prime} C, k=\frac{A C}{B C}$, $\angle A C B=90^{\circ}, D=A A^{\prime} \cap B B^{\prime}$. Find $\angle A D B$ and $\frac{A A^{\prime}}{B B^{\prime}}$.
55	Olympiad	Geometry	Let $\triangle A B C$ be an isosceles right triangle $(A C=B C)$. Let $S$ be a point on a circle with diameter $B C$. The line $\ell$ is symmetrical to $S C$ with respect to $A B$ and intersects $B C$ at $D$. Prove that $A S \perp D S$.
56	Olympiad	Geometry	$\triangle A B F \sim \triangle B C D \sim \triangle C A E$. Points $D, E, F$ are outside $\triangle A B C$. Prove that the centroids of triangles $\triangle A B C$ and $\triangle D E F$ are coinsite.
57	Olympiad	Geometry	Let triangle $\triangle A B C$ and point $A^{\prime}$ on sideline $B C$ be given. Construct $\triangle A^{\prime} B^{\prime} C^{\prime} \sim \triangle A B C$ where $B^{\prime}$ lies on sideline $A C$ and $C^{\prime}$ lies on sideline $A B$.
58	Olympiad	Geometry	Let triangle $\triangle A B C$ and point $C^{\prime}\left(C^{\prime} \neq C, C^{\prime} \neq B\right)$ on sideline $B C$ be given. $\triangle A^{\prime} B^{\prime} C^{\prime} \sim \triangle A B C$ where $B^{\prime}$ lies on sideline $A B$ and $A^{\prime}$ lies on sideline $A C$. The spiral symilarity $T$ maps $\triangle A B C$ into $\triangle A^{\prime} B^{\prime} C^{\prime}$. Prove a) $\angle A B^{\prime} A^{\prime}=\angle B C^{\prime} B^{\prime}=\angle C A^{\prime} C^{\prime}$. b) Center of $T$ is the First Brocard point of triangles $\triangle A B C$ and $\triangle A^{\prime} B^{\prime} C^{\prime}$.
59	Olympiad	Geometry	Let $\triangle A B C$ and point $A^{\prime}$ on sideline $B C$ be given. $\triangle A^{\prime} B^{\prime} C^{\prime} \sim \triangle A B C$ where $B^{\prime}$ lies on sideline $A C$ and $C^{\prime}$ lies on sideline $A B$. Denote $D=B B^{\prime} \cap C C^{\prime}, E=A A^{\prime} \cap C C^{\prime}, F=B B^{\prime} \cap A A^{\prime}$. Prove that circumcircles of triangles $\triangle A B F, \triangle A^{\prime} B^{\prime} F, \triangle B C D$, $\triangle B^{\prime} C^{\prime} D, \triangle A C E, \triangle A^{\prime} C^{\prime} E$ have the common point.
60	Olympiad	Geometry	Let triangle $\triangle A B C$ be given. The triangle $\triangle A C E$ is constructed using a spiral similarity of $\triangle A B C$ with center $A$, angle of rotation $\angle B A C$ and coefficient $\frac{A C}{A B}$ A point $D$ is centrally symmetrical to a point $B$ with respect to $C$. Prove that the spiral similarity with center $E$, angle of rotation $\angle A C B$ and coefficient $\frac{B C}{A C}$ taking $\triangle A C E$ to $\triangle C D E$.
61	Introductory	Algebra	In a jar there are blue jelly beans and green jelly beans. Then, $15 \%$ of the blue jelly beans are removed and $40 \%$ of the green jelly beans are removed. If afterwards the total number of jelly beans is $80 \%$ of the original number of jelly beans, then determine the percent of the remaining jelly beans that are blue.
62	Introductory	Algebra	Find all values of $B$ that have the property that if $(x, y)$ lies on the hyperbola $2 y^2-x^2=1$, then so does the point $(3 x+4 y, 2 x+B y)$.
63	Introductory	Algebra	Joe has a collection of 23 coins, consisting of 5 -cent coins, 10 -cent coins, and 25 -cent coins. He has 3 more 10 -cent coins than 5 -cent coins, and the total value of his collection is 320 cents. How many more 25 -cent coins does Joe have than 5 -cent coins?
64	Introductory	Algebra	Suppose that real number $x$ satisfies \begin{align} \sqrt{49-x^2}-\sqrt{25-x^2}=3 \end{align} What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}$ ?
65	Introductory	Algebra	Points $(\sqrt{\pi}, a)$ and $(\sqrt{\pi}, b)$ are distinct points on the graph of $y^2+x^4=2 x^2 y+1$. What is $\|a-b\|$ ?
66	Introductory	Algebra	Consider the set of all fractions $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by 1 , the value of the fraction is increased by $10 \%$ ?
67	Introductory	Algebra	Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20 \%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5 \%$ of the difference between $A$ and the cost of his movie ticket. To the nearest whole percent, what fraction of $A$ did Roger pay for his movie ticket and soda?
68	Introductory	Algebra	Suppose $a, b$, and $c$ are nonzero real numbers, and $a+b+c=0$. What are the possible value(s) for $\frac{a}{\|a\|}+\frac{b}{\|b\|}+\frac{c}{\|c\|}+\frac{a b c}{\|a b c\|}$ ?
69	Introductory	Algebra	A lattice point in an $x y$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y=m x+2$ passes through no lattice point with $0<x \leq 100$ for all $m$ such that $\frac{1}{2}<m<a$. What is the maximum possible value of $a$ ?
70	Introductory	Algebra	The first four terms of an arithmetic sequence are $p, 9,3 p-q$, and $3 p+q$. What is the $2010^{\text {th }}$ term of this sequence?
71	Intermediate	Algebra	The product $N$ of three positive integers is 6 times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N$.
72	Intermediate	Algebra	It is given that $\log _6 a+\log _6 b+\log _6 c=6$, where $a, b$, and $c$ are positive integers that form an increasing geometric sequence and $b-a$ is the square of an integer. Find $a+b+c$.
73	Intermediate	Algebra	For $t=1,2,3,4$, define $S_t=\sum_{i=1}^{350} a_i^t$ where $a_i \in\{1,2,3,4\}$. If $S_1=513$ and $S_4=4745$, find the minimum possible value for $S_2$.
74	Intermediate	Algebra	Determine the value of $a$ so that the following fraction reduces to a quotient of two linear expressions: \begin{align} \frac{x^3+(a-10) x^2-x+(a-6)}{x^3+(a-6) x^2-x+(a-10)} \end{align}
75	Intermediate	Algebra	For certain pairs $(m, n)$ of positive integers with $m \geq n$ there are exactly 50 distinct positive integers $k$ such that $\|\log m-\log k\|<\log n$. Find the sum of all possible values of the product $m n$.
76	Intermediate	Algebra	Suppose that $a, b$, and $c$ are positive real numbers such that $a^{\log _3 7}=27, b^{\log _7 11}=49$, and $c^{\log _{11} 25}=\sqrt{11}$. Find \begin{align} a^{\left(\log _3 7\right)^2}+b^{\left(\log _7 11\right)^2}+c^{\left(\log _{11} 25\right)^2} \end{align}
77	Intermediate	Algebra	Let $P(x)$ be a quadratic polynomial with real coefficients satisfying $x^2-2 x+2 \leq P(x) \leq 2 x^2-4 x+3$ for all real numbers $x$, and suppose $P(11)=181$. Find $P(16)$.
78	Intermediate	Algebra	Find the smallest positive integer $n$ with the property that the polynomial $x^4-n x+63$ can be written as a product of two nonconstant polynomials with integer coefficients.
79	Intermediate	Algebra	Let $z_1, z_2, z_3, \ldots, z_{12}$ be the 12 zeroes of the polynomial $z^{12}-2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $i z_j$. Then the maximum possible value of the real part of $\sum_{i=1}^{12} w_j$ can be written as $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m+n$.
80	Intermediate	Algebra	Let $f(x)=\left(x^2+3 x+2\right)^{\cos (\pi x)}$. Find the sum of all positive integers $n$ for which \begin{align} \left\|\sum_{k=1}^n \log _{10} f(k)\right\|=1 \end{align}
81	Olympiad	Algebra	Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for positive integers $a$ and $b$, \begin{align} f\left(a^2+b^2\right)=f(a) f(b) \text { and } f\left(a^2\right)=f(a)^2 . \end{align}
82	Olympiad	Algebra	The real numbers $a, b, c, d$ are such that $a \geq b \geq c \geq d>0$ and $a+b+c+d=1$. Prove that \begin{align} (a+2 b+3 c+4 d) a^a b^b c^c d^d<1 \end{align}
83	Olympiad	Algebra	Let $f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ (meaning $f$ takes positive real numbers to positive real numbers) be a nonconstant function such that for any positive real numbers $x$ and $y$, \begin{align} f(x) f(y) f(x+y)=f(x)+f(y)-f(x+y) . \end{align} Prove that there is a constant $a>1$ such that \begin{align} f(x)=\frac{a^x-1}{a^x+1} \end{align} for all positive real numbers $x$.
84	Olympiad	Algebra	Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$, \begin{align} (f(x)+x y) \cdot f(x-3 y)+(f(y)+x y) \cdot f(3 x-y)=(f(x+y))^2 \end{align}
85	Olympiad	Algebra	Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that \begin{align} f(x)+f(t)=f(y)+f(z) \end{align} for all rational numbers $x<y<z<t$ that form an arithmetic progression. ( $\mathbb{Q}$ is the set of all rational numbers.)
86	Olympiad	Algebra	Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the equation $f(x+f(x+y))+f(x y)=x+f(x+y)+y f(x)$ for all real numbers $x$ and $y$.
87	Olympiad	Algebra	Let $\mathbb{Z}$ be the set of integers. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that \begin{align} x f(2 f(y)-x)+y^2 f(2 x-f(y))=\frac{f(x)^2}{x}+f(y f(y)) \end{align} for all $x, y \in \mathbb{Z}$ with $x \neq 0$.
88	Olympiad	Algebra	Find all functions $f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$(where $\mathbb{Z}^{+}$is the set of positive integers) such that $f(n !)=f(n)$ ! for all positive integers $n$ and such that $m-n$ divides $f(m)-f(n)$ for all distinct positive integers $m, n$
89	Olympiad	Algebra	Let $a_1, a_2, \ldots, a_n$ be distinct positive integers and let $M$ be a set of $n-1$ positive integers not containing $s=a_1+a_2+\ldots+a_n$. A grasshopper is to jump along the real axis, starting at the point 0 and making $n$ jumps to the right with lengths $a_1, a_2, \ldots, a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $M$.
90	Olympiad	Algebra	Find the lowest possible values from the function \begin{align} f(x)=x^{2008}-2 x^{2007}+3 x^{2006}-4 x^{2005}+5 x^{2004}-\cdots-2006 x^3+2007 x^2-2008 x+2009 \end{align} for any real numbers $x$.
91	Introductory	Number Theory	The least common multiple of a positive integer $n$ and 18 is 180 , and the greatest common divisor of $n$ and 45 is 15 . What is the sum of the digits of $n$ ?
92	Introductory	Number Theory	Let $n$ be the least positive integer greater than 1000 for which \begin{align} \operatorname{gcd}(63, n+120)=21 \text { and } \operatorname{gcd}(n+63,120)=60 . \end{align} What is the sum of the digits of $n$ ?
93	Introductory	Number Theory	The sum of three consecutive integers is 54 . What is the smallest of the three integers?
94	Introductory	Number Theory	In the equation below, $A$ and $B$ are consecutive positive integers, and $A, B$, and $A+B$ represent number bases: \begin{align} 132_A+43_B=69_{A+B} \text {. } \end{align} What is $A+B$ ?
95	Introductory	Number Theory	Let $N$ be the greatest integer multiple of 36 all of whose digits are even and no two of whose digits are the same. Find the remainder when $N$ is divided by 1000 .
96	Introductory	Number Theory	How many positive two-digit integers are factors of $2^{24}-1$ ?
97	Introductory	Number Theory	Find $a+b+c$, where $a, b$, and $c$ are the hundreds, tens, and units digits of the six-digit integer $123 a b c$, which is a multiple of 990 .
98	Introductory	Number Theory	Let $a / b$ be the probability that a randomly selected divisor of 2007 is a multiple of 3 . If $a$ and $b$ are relatively prime positive integers, find $a+b$.
99	Introductory	Number Theory	Let $\phi(n)$ be the number of positive integers $k<n$ which are relatively prime to $n$. For how many distinct values of $n$ is $\phi(n)=12$ ?
100	Introductory	Number Theory	Call a number prime-looking if it is composite but not divisible by 2,3 , or 5 . The three smallest prime-looking numbers are 49,77 , and 91 . There are 168 prime numbers less than 1000 . How many prime-looking numbers are there less than $1000 ?$

1

Intermediate

Combinatorics

Convex polygons $P_1$ and $P_2$ are drawn in the same plane with $n_1$ and $n_2$ sides, respectively, $n_1 \leq n_2$. If $P_1$ and $P_2$ do not have any line segment in common, then the maximum number of intersections of $P_1$ and $P_2$ is?

2

Intermediate

Combinatorics

Suppose that $n$ people each know exactly one piece of information, and all $n$ pieces are different. Every time person $A$ phones person $B, A$ tells $B$ everything that $A$ knows, while $B$ tells $A$ nothing. What is the minimum number of phone calls between pairs of people needed for everyone to know everything? Prove your answer is a minimum.

3

Intermediate

Combinatorics

The numbers 1447,1005 and 1231 have something in common: each is a 4-digit number beginning with 1 that has exactly two identical digits. How many such numbers are there?

4

Intermediate

Combinatorics

For $\{1,2,3, \ldots, n\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1,2,3,6,9\}$ is $9-6+3-2+1=5$ and for $\{5\}$ it is simply 5 . Find the sum of all such alternating sums for $n=7$.

5

Intermediate

Combinatorics

For $\{1,2,3, \ldots, n\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1,2,3,6,9\}$ is $9-6+3-2+1=5$ and for $\{5\}$ it is simply 5 . Find the sum of all such alternating sums for $n=7$.

6

Intermediate

Combinatorics

Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

7

Intermediate

Combinatorics

What is the largest 2-digit prime factor of the integer $n=\left(\begin{array}{l}200 \\ 100\end{array}\right)$ ?

8

Intermediate

Combinatorics

A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac{m}{n}$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$.

9

Intermediate

Combinatorics

Let $A, B, C$ and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p=\frac{n}{729}$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.

10

Intermediate

Combinatorics

In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned $\frac{1}{2}$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?

11

Introductory

Combinatorics

The number of diagonals that can be drawn in a polygon of 100 sides is:

12

Introductory

Combinatorics

Six straight lines are drawn in a plane with no two parallel and no three concurrent. The number of regions into which they divide the plane is:

13

Introductory

Combinatorics

One thousand unit cubes are fastened together to form a large cube with edge length 10 units; this is painted and then separated into the original cubes. The number of these unit cubes which have at least one face painted is

14

Introductory

Combinatorics

A fair die is rolled six times. The probability of rolling at least a five at least five times is

15

Introductory

Combinatorics

Assume every 7-digit whole number is a possible telephone number except those that begin with 0 or 1 . What fraction of telephone numbers begin with 9 and end with 0 ?

16

Introductory

Combinatorics

The 600 students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of three lunch groups. The probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group is approximately

17

Introductory

Combinatorics

How many rearrangements of $a b c d$ are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either $a b$ or $b a$.

18

Introductory

Combinatorics

How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?

19

Introductory

Combinatorics

How many ways can a student schedule 3 mathematics courses - algebra, geometry, and number theory -- in a 6 -period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)

20

Introductory

Combinatorics

The eighth grade class at Lincoln Middle School has 93 students. Each student takes a math class or a foreign language class or both. There are 70 eighth graders taking a math class, and there are 54 eighth graders taking a foreign language class. How many eighth graders take only a math class and not a foreign language class?

21

Olympiad

Combinatorics

Carina has three pins, labeled $A, B$, and $C$, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance 1 away. What is the least number of moves that Carina can make in order for triangle $A B C$ to have area $2021 ?$

22

Olympiad

Combinatorics

There is an integer $n>1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The $k$ cable cars of $A$ have $k$ different starting points and $k$ different finishing points, and a cable car that starts higher also finishes higher. The same conditions hold for $B$. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer $k$ for which one can guarantee that there are two stations that are linked by both companies.

23

Olympiad

Combinatorics

There are $4 n$ pebbles of weights $1,2,3, \ldots, 4 n$. Each pebble is colored in one of $n$ colors and there are four pebbles of each color. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied: - The total weights of both piles are the same. - Each pile contains two pebbles of each color.

24

Olympiad

Combinatorics

Let $n$ be a positive integer. Eric and a squid play a turn-based game on an infinite grid of unit squares. Eric's goal is to capture the squid by moving onto the same square as it. Initially, all the squares are colored white. The squid begins on an arbitrary square in the grid, and Eric chooses a different square to start on. On the squid's turn, it performs the following action exactly 2020 times: it chooses an adjacent unit square that is white, moves onto it, and sprays the previous unit square either black or gray. Once the squid has performed this action 2020 times, all squares colored gray are automatically colored white again, and the squid's turn ends. If the squid is ever unable to move, then Eric automatically wins. Moreover, the squid is claustrophobic, so at no point in time is it ever surrounded by a closed loop of black or gray squares. On Eric's turn, he performs the following action at most $n$ times: he chooses an adjacent unit square that is white and moves onto it. Note that the squid can trap Eric in a closed region, so that Eric can never win. Eric wins if he ever occupies the same square as the squid. Suppose the squid has the first turn, and both Eric and the squid play optimally. Both Eric and the squid always know each other's location and the colors of all the squares. Find all positive integers $n$ such that Eric can win in finitely many moves.

25

Olympiad

Combinatorics

Let $k$ be the number of students in a circle. Then let $m$ be the number of ways they can rearrange ourselves so that each of them is in the same spot or within one spot of where they started, and no two people are ever on the same spot. If $m$ leaves a remainder of 1 when divided by 5 , how many possible values are there of $k$, where $k$ is at least 3 and at most $2008 ?$

26

Olympiad

Combinatorics

A square of side $n$ is formed from $n^2$ unit squares, each colored in red, yellow or green. Find minimal $n$, such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or column).

27

Olympiad

Combinatorics

An $(n, k)$-tournament is a contest with $n$ players held in $k$ rounds such that: (i) Each player plays in each round, and every two players meet at most once. (ii) If player $A$ meets player $B$ in round $i$, player $C$ meets player $D$ on round $i$, and player $A$ meets player $C$ in round $j$, then player $B$ meets player $D$ in round $j$. Determine all pairs $(n, k)$ for which there exists an $(n, k)$-tournament.

28

Olympiad

Combinatorics

For a given positive integer $n>2$, let $C_1, C_2, C_3$ be the boundaries of three convex $n$-gons in the plane such that $C_1 \cap C_2, C_2 \cap C_3$, $C_3 \cap C_1$ are finite. Find the maximum number of points in the set $C_1 \cap C_2 \cap C_3$.

29

Olympiad

Combinatorics

The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer $n \geq 4$, find the least number of edges of a graph that can be obtained by repeated applications of this operation from a complete graph on $n$ vertices (where each pair of vertices are joined by an edge).

30

Olympiad

Combinatorics

The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer $n \geq 4$, find the least number of edges of a graph that can be obtained by repeated applications of this operation from a complete graph on $n$ vertices (where each pair of vertices are joined by an edge).

31

Introductory

Geometry

Triangle $A B C$ is equilateral with side length 6 . Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A, O$, and $C$ ?

32

Introductory

Geometry

The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle?

33

Introductory

Geometry

Let $A B C$ be a triangle. The bisector of $\angle A B C$ intersects $\overline{A C}$ at $E$, and the bisector of $\angle A C B$ intersects $\overline{A B}$ at $F$. If $B F=1, C E=2$, and $B C=3$, then the perimeter of $\triangle A B C$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

34

Introductory

Geometry

The circle having $(0,0)$ and $(8,6)$ as the endpoints of a diameter intersects the $x$-axis at a second point. What is the $x$-coordinate of this point?

35

Introductory

Geometry

A trapezoid has side lengths $3,5,7$, and 11 . The sum of all the possible areas of the trapezoid can be written in the form of $r_1 \sqrt{n_1}+r_2 \sqrt{n_2}+r_3$, where $r_1, r_2$, and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to $r_1+r_2+r_3+n_1+n_2$ ?

36

Introductory

Geometry

Square $P Q R S$ lies in the first quadrant. Points $(3,0),(5,0),(7,0)$, and $(13,0)$ lie on lines $S P, R Q, P Q$, and $S R$, respectively. What is the sum of the coordinates of the center of the square $P Q R S$ ?

37

Introductory

Geometry

Triangle $A B C$ has $A B=27, A C=26$, and $B C=25$. Let $I$ be the intersection of the internal angle bisectors of $\triangle A B C$. What is $B I$ ?

38

Introductory

Geometry

In rectangle $A B C D, A B=6, A D=30$, and $G$ is the midpoint of $\overline{A D}$. Segment $A B$ is extended 2 units beyond $B$ to point $E$, and $F$ is the intersection of $\overline{E D}$ and $\overline{B C}$. What is the area of quadrilateral $B F D G$ ?

39

Introductory

Geometry

In $\triangle A B C, A B=86$, and $A C=97$. A circle with center $A$ and radius $A B$ intersects $\overline{B C}$ at points $B$ and $X$. Moreover $\overline{B X}$ and $\overline{C X}$ have integer lengths. What is $B C$ ?

40

Introductory

Geometry

Two points on the circumference of a circle of radius $r$ are selected independently and at random. From each point a chord of length $r$ is drawn in a clockwise direction. What is the probability that the two chords intersect?

41

Intermediate

Geometry

Square $A B C D$ is inscribed in a circle. Point $P$ is on this circle such that $A P \cdot C P=56$, and $B P \cdot D P=90$. What is the area of the square?

42

Intermediate

Geometry

In triangle $A B C$ we have $A B=7, A C=8, B C=9$. Point $D$ is on the circumscribed circle of the triangle so that $A D$ bisects angle $B A C$ What is the value of $A D / C D$ ?

43

Intermediate

Geometry

A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\overline{A B}$, has length 31 . Find the sum of the lengths of the three diagonals that can be drawn from $A$.

44

Intermediate

Geometry

A hexagon with sides of lengths $2,2,7,7,11$, and 11 is inscribed in a circle. Find the diameter of the circle

45

Intermediate

Geometry

In a regular heptagon $A B C D E F G$, prove that: $\frac{1}{A B}=\frac{1}{A C}+\frac{1}{A E}$.

46

Intermediate

Geometry

$\triangle A B C$ has point $D$ on $A B$, point $E$ on $B C$, and point $F$ on $A C . A E, C D$, and $B F$ intersect at point $G$. The ratio $A D: D B$ is $3: 5$ and the ratio $C E: E B$ is $8: 3$. Find the ratio of $F G: G B$

47

Intermediate

Geometry

Consider a triangle $A B C$ with its three medians drawn, with the intersection points being $D, E, F$, corresponding to $A B, B C$, and $A C$ respectively. Thus, if we label point $A$ with a weight of $1, B$ must also have a weight of 1 since $A$ and $B$ are equidistant from $D$. By the same process, we find $C$ must also have a weight of 1 . Now, since $A$ and $B$ both have a weight of $1, D$ must have a weight of 2 (as is true for $E$ and $F$ ). Thus, if we label the centroid $P$, we can deduce that $D P: P C$ is $1: 2$ - the inverse ratio of their weights.

48

Intermediate

Geometry

In rectangle $A B C D, A B=6$ and $B C=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $B E=E F=F C$. Segments $\overline{A E}$ and $\overline{A F}$ intersect $\overline{B D}$ at $P$ and $Q$, respectively. The ratio $B P: P Q: Q D$ can be written as $r: s: t$ where the greatest common factor of $r, s$, and $t$ is 1 . What is $r+s+t$ ?

49

Intermediate

Geometry

Triangle $A B C$ has $A C=450$ and $B C=300$. Points $K$ and $L$ are located on $\overline{A C}$ and $\overline{A B}$ respectively so that $A K=C K$, and $\overline{C L}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\overline{B K}$ and $\overline{C L}$, and let $M$ be the point on line $B K$ for which $K$ is the midpoint of $\overline{P M}$. If $A M=180$, find $L P$.

50

Intermediate

Geometry

In parallelogram $A B C D$, point $M$ is on $\overline{A B}$ so that $\frac{A M}{A B}=\frac{17}{1000}$ and point $N$ is on $\overline{A D}$ so that $\frac{A N}{A D}=\frac{17}{2009}$. Let $P$ be the point of intersection of $\overline{A C}$ and $\overline{M N}$. Find $\frac{A C}{A P}$.

51

Olympiad

Geometry

Let $\triangle A B C$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $\overline{B C}, \overline{C A}$, and $\overline{A B}$ respectively. One of the intersection points of the line $\overline{E F}$ and the circumcircle is $P$. The lines $\overline{B P}$ and $\overline{D F}$ meet at point $Q$. Prove that $|A P|=|A Q|$. (IMO Shortlist $2010 \mathrm{G} 1$ )

52

Olympiad

Geometry

Four circles $\omega, \omega_A, \omega_B$, and $\omega_C$ with the same radius are drawn in the interior of triangle $A B C$ such that $\omega_A$ is tangent to sides $A B$ and $A C, \omega_B$ to $B C$ and $B A, \omega_C$ to $C A$ and $C B$, and $\omega$ is externally tangent to $\omega_A, \omega_B$, and $\omega_C$. If the sides of triangle $A B C$ are 13,14, and 15 , the radius of $\omega$ can be represented in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

53

Olympiad

Geometry

The following problems are constructions involving only a straightedge (no compass). 1. Construct the fourth harmonic line to three given lines through a point. 2. Construct the fourth harmonic point to three points on a line. 3. If a given right angle and a given arbitrary angle have their vertex and one side in common, double the given arbitrary angle. 4. Draw a parallel through a given point $P$ to two given parallel lines $l_1$ and $l_2$.

54

Olympiad

Geometry

Given two similar right triangles $A B C$ and $A^{\prime} B^{\prime} C, k=\frac{A C}{B C}$, $\angle A C B=90^{\circ}, D=A A^{\prime} \cap B B^{\prime}$. Find $\angle A D B$ and $\frac{A A^{\prime}}{B B^{\prime}}$.

55

Olympiad

Geometry

Let $\triangle A B C$ be an isosceles right triangle $(A C=B C)$. Let $S$ be a point on a circle with diameter $B C$. The line $\ell$ is symmetrical to $S C$ with respect to $A B$ and intersects $B C$ at $D$. Prove that $A S \perp D S$.

56

Olympiad

Geometry

$\triangle A B F \sim \triangle B C D \sim \triangle C A E$. Points $D, E, F$ are outside $\triangle A B C$. Prove that the centroids of triangles $\triangle A B C$ and $\triangle D E F$ are coinsite.

57

Olympiad

Geometry

Let triangle $\triangle A B C$ and point $A^{\prime}$ on sideline $B C$ be given. Construct $\triangle A^{\prime} B^{\prime} C^{\prime} \sim \triangle A B C$ where $B^{\prime}$ lies on sideline $A C$ and $C^{\prime}$ lies on sideline $A B$.

58

Olympiad

Geometry

Let triangle $\triangle A B C$ and point $C^{\prime}\left(C^{\prime} \neq C, C^{\prime} \neq B\right)$ on sideline $B C$ be given. $\triangle A^{\prime} B^{\prime} C^{\prime} \sim \triangle A B C$ where $B^{\prime}$ lies on sideline $A B$ and $A^{\prime}$ lies on sideline $A C$. The spiral symilarity $T$ maps $\triangle A B C$ into $\triangle A^{\prime} B^{\prime} C^{\prime}$. Prove a) $\angle A B^{\prime} A^{\prime}=\angle B C^{\prime} B^{\prime}=\angle C A^{\prime} C^{\prime}$. b) Center of $T$ is the First Brocard point of triangles $\triangle A B C$ and $\triangle A^{\prime} B^{\prime} C^{\prime}$.

59

Olympiad

Geometry

Let $\triangle A B C$ and point $A^{\prime}$ on sideline $B C$ be given. $\triangle A^{\prime} B^{\prime} C^{\prime} \sim \triangle A B C$ where $B^{\prime}$ lies on sideline $A C$ and $C^{\prime}$ lies on sideline $A B$. Denote $D=B B^{\prime} \cap C C^{\prime}, E=A A^{\prime} \cap C C^{\prime}, F=B B^{\prime} \cap A A^{\prime}$. Prove that circumcircles of triangles $\triangle A B F, \triangle A^{\prime} B^{\prime} F, \triangle B C D$, $\triangle B^{\prime} C^{\prime} D, \triangle A C E, \triangle A^{\prime} C^{\prime} E$ have the common point.

60

Olympiad

Geometry

Let triangle $\triangle A B C$ be given. The triangle $\triangle A C E$ is constructed using a spiral similarity of $\triangle A B C$ with center $A$, angle of rotation $\angle B A C$ and coefficient $\frac{A C}{A B}$ A point $D$ is centrally symmetrical to a point $B$ with respect to $C$. Prove that the spiral similarity with center $E$, angle of rotation $\angle A C B$ and coefficient $\frac{B C}{A C}$ taking $\triangle A C E$ to $\triangle C D E$.

61

Introductory

Algebra

In a jar there are blue jelly beans and green jelly beans. Then, $15 \%$ of the blue jelly beans are removed and $40 \%$ of the green jelly beans are removed. If afterwards the total number of jelly beans is $80 \%$ of the original number of jelly beans, then determine the percent of the remaining jelly beans that are blue.

62

Introductory

Algebra

Find all values of $B$ that have the property that if $(x, y)$ lies on the hyperbola $2 y^2-x^2=1$, then so does the point $(3 x+4 y, 2 x+B y)$.

63

Introductory

Algebra

Joe has a collection of 23 coins, consisting of 5 -cent coins, 10 -cent coins, and 25 -cent coins. He has 3 more 10 -cent coins than 5 -cent coins, and the total value of his collection is 320 cents. How many more 25 -cent coins does Joe have than 5 -cent coins?

64

Introductory

Algebra

Suppose that real number $x$ satisfies \begin{align*} \sqrt{49-x^2}-\sqrt{25-x^2}=3 \end{align*} What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}$ ?

65

Introductory

Algebra

Points $(\sqrt{\pi}, a)$ and $(\sqrt{\pi}, b)$ are distinct points on the graph of $y^2+x^4=2 x^2 y+1$. What is $|a-b|$ ?

66

Introductory

Algebra

Consider the set of all fractions $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by 1 , the value of the fraction is increased by $10 \%$ ?

67

Introductory

Algebra

Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20 \%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5 \%$ of the difference between $A$ and the cost of his movie ticket. To the nearest whole percent, what fraction of $A$ did Roger pay for his movie ticket and soda?

68

Introductory

Algebra

Suppose $a, b$, and $c$ are nonzero real numbers, and $a+b+c=0$. What are the possible value(s) for $\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{a b c}{|a b c|}$ ?

69

Introductory

Algebra

A lattice point in an $x y$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y=m x+2$ passes through no lattice point with $0<x \leq 100$ for all $m$ such that $\frac{1}{2}<m<a$. What is the maximum possible value of $a$ ?

70

Introductory

Algebra

The first four terms of an arithmetic sequence are $p, 9,3 p-q$, and $3 p+q$. What is the $2010^{\text {th }}$ term of this sequence?

71

Intermediate

Algebra

The product $N$ of three positive integers is 6 times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N$.

72

Intermediate

Algebra

It is given that $\log _6 a+\log _6 b+\log _6 c=6$, where $a, b$, and $c$ are positive integers that form an increasing geometric sequence and $b-a$ is the square of an integer. Find $a+b+c$.

73

Intermediate

Algebra

For $t=1,2,3,4$, define $S_t=\sum_{i=1}^{350} a_i^t$ where $a_i \in\{1,2,3,4\}$. If $S_1=513$ and $S_4=4745$, find the minimum possible value for $S_2$.

74

Intermediate

Algebra

Determine the value of $a$ so that the following fraction reduces to a quotient of two linear expressions: \begin{align*} \frac{x^3+(a-10) x^2-x+(a-6)}{x^3+(a-6) x^2-x+(a-10)} \end{align*}

75

Intermediate

Algebra

For certain pairs $(m, n)$ of positive integers with $m \geq n$ there are exactly 50 distinct positive integers $k$ such that $|\log m-\log k|<\log n$. Find the sum of all possible values of the product $m n$.

76

Intermediate

Algebra

Suppose that $a, b$, and $c$ are positive real numbers such that $a^{\log _3 7}=27, b^{\log _7 11}=49$, and $c^{\log _{11} 25}=\sqrt{11}$. Find \begin{align*} a^{\left(\log _3 7\right)^2}+b^{\left(\log _7 11\right)^2}+c^{\left(\log _{11} 25\right)^2} \end{align*}

77

Intermediate

Algebra

Let $P(x)$ be a quadratic polynomial with real coefficients satisfying $x^2-2 x+2 \leq P(x) \leq 2 x^2-4 x+3$ for all real numbers $x$, and suppose $P(11)=181$. Find $P(16)$.

78

Intermediate

Algebra

Find the smallest positive integer $n$ with the property that the polynomial $x^4-n x+63$ can be written as a product of two nonconstant polynomials with integer coefficients.

79

Intermediate

Algebra

Let $z_1, z_2, z_3, \ldots, z_{12}$ be the 12 zeroes of the polynomial $z^{12}-2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $i z_j$. Then the maximum possible value of the real part of $\sum_{i=1}^{12} w_j$ can be written as $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m+n$.

80

Intermediate

Algebra

Let $f(x)=\left(x^2+3 x+2\right)^{\cos (\pi x)}$. Find the sum of all positive integers $n$ for which \begin{align*} \left|\sum_{k=1}^n \log _{10} f(k)\right|=1 \end{align*}

81

Olympiad

Algebra

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for positive integers $a$ and $b$, \begin{align*} f\left(a^2+b^2\right)=f(a) f(b) \text { and } f\left(a^2\right)=f(a)^2 . \end{align*}

82

Olympiad

Algebra

The real numbers $a, b, c, d$ are such that $a \geq b \geq c \geq d>0$ and $a+b+c+d=1$. Prove that \begin{align*} (a+2 b+3 c+4 d) a^a b^b c^c d^d<1 \end{align*}

83

Olympiad

Algebra

Let $f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ (meaning $f$ takes positive real numbers to positive real numbers) be a nonconstant function such that for any positive real numbers $x$ and $y$, \begin{align*} f(x) f(y) f(x+y)=f(x)+f(y)-f(x+y) . \end{align*} Prove that there is a constant $a>1$ such that \begin{align*} f(x)=\frac{a^x-1}{a^x+1} \end{align*} for all positive real numbers $x$.

84

Olympiad

Algebra

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$, \begin{align*} (f(x)+x y) \cdot f(x-3 y)+(f(y)+x y) \cdot f(3 x-y)=(f(x+y))^2 \end{align*}

85

Olympiad

Algebra

Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that \begin{align*} f(x)+f(t)=f(y)+f(z) \end{align*} for all rational numbers $x<y<z<t$ that form an arithmetic progression. ( $\mathbb{Q}$ is the set of all rational numbers.)

86

Olympiad

Algebra

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the equation $f(x+f(x+y))+f(x y)=x+f(x+y)+y f(x)$ for all real numbers $x$ and $y$.

87

Olympiad

Algebra

Let $\mathbb{Z}$ be the set of integers. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that \begin{align*} x f(2 f(y)-x)+y^2 f(2 x-f(y))=\frac{f(x)^2}{x}+f(y f(y)) \end{align*} for all $x, y \in \mathbb{Z}$ with $x \neq 0$.

88

Olympiad

Algebra

Find all functions $f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$(where $\mathbb{Z}^{+}$is the set of positive integers) such that $f(n !)=f(n)$ ! for all positive integers $n$ and such that $m-n$ divides $f(m)-f(n)$ for all distinct positive integers $m, n$

89

Olympiad

Algebra

Let $a_1, a_2, \ldots, a_n$ be distinct positive integers and let $M$ be a set of $n-1$ positive integers not containing $s=a_1+a_2+\ldots+a_n$. A grasshopper is to jump along the real axis, starting at the point 0 and making $n$ jumps to the right with lengths $a_1, a_2, \ldots, a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $M$.

90

Olympiad

Algebra

Find the lowest possible values from the function \begin{align*} f(x)=x^{2008}-2 x^{2007}+3 x^{2006}-4 x^{2005}+5 x^{2004}-\cdots-2006 x^3+2007 x^2-2008 x+2009 \end{align*} for any real numbers $x$.

91

Introductory

Number Theory

The least common multiple of a positive integer $n$ and 18 is 180 , and the greatest common divisor of $n$ and 45 is 15 . What is the sum of the digits of $n$ ?

92

Introductory

Number Theory

Let $n$ be the least positive integer greater than 1000 for which \begin{align*} \operatorname{gcd}(63, n+120)=21 \text { and } \operatorname{gcd}(n+63,120)=60 . \end{align*} What is the sum of the digits of $n$ ?

93

Introductory

Number Theory

The sum of three consecutive integers is 54 . What is the smallest of the three integers?

94

Introductory

Number Theory

In the equation below, $A$ and $B$ are consecutive positive integers, and $A, B$, and $A+B$ represent number bases: \begin{align*} 132_A+43_B=69_{A+B} \text {. } \end{align*} What is $A+B$ ?

95

Introductory

Number Theory

Let $N$ be the greatest integer multiple of 36 all of whose digits are even and no two of whose digits are the same. Find the remainder when $N$ is divided by 1000 .

96

Introductory

Number Theory

How many positive two-digit integers are factors of $2^{24}-1$ ?

97

Introductory

Number Theory

Find $a+b+c$, where $a, b$, and $c$ are the hundreds, tens, and units digits of the six-digit integer $123 a b c$, which is a multiple of 990 .

98

Introductory

Number Theory

Let $a / b$ be the probability that a randomly selected divisor of 2007 is a multiple of 3 . If $a$ and $b$ are relatively prime positive integers, find $a+b$.

99

Introductory

Number Theory

Let $\phi(n)$ be the number of positive integers $k<n$ which are relatively prime to $n$. For how many distinct values of $n$ is $\phi(n)=12$ ?

100

Introductory

Number Theory

Call a number prime-looking if it is composite but not divisible by 2,3 , or 5 . The three smallest prime-looking numbers are 49,77 , and 91 . There are 168 prime numbers less than 1000 . How many prime-looking numbers are there less than $1000 ?$