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Given that $A,B,$ and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB,AC,$ and $BC$ are integers, what is the smallest possible value of $AB?$ | 3 | 0.875 |
In a massive school which has $m$ students, and each student took at least one subject. Let $p$ be an odd prime. Given that:
(i) each student took at most $p+1$ subjects.
(ii) each subject is taken by at most $p$ students.
(iii) any pair of students has at least $1$ subject in common.
Find the maximum possible value of $m$ . | p^2 | 0.375 |
In a rectangular $57\times 57$ grid of cells, $k$ of the cells are colored black. What is the smallest positive integer $k$ such that there must exist a rectangle, with sides parallel to the edges of the grid, that has its four vertices at the center of distinct black cells?
[i]Proposed by James Lin | 457 | 0.125 |
Let $\alpha$ be a solution satisfying the equation $|x|=e^{-x}.$ Let $I_n=\int_0^{\alpha} (xe^{-nx}+\alpha x^{n-1})dx\ (n=1,\ 2,\ \cdots).$ Find $\lim_{n\to\infty} n^2I_n.$ | 1 | 0.625 |
Find the minimum positive value of $ 1*2*3*4*...*2020*2021*2022$ where you can replace $*$ as $+$ or $-$ | 1 | 0.875 |
Suppose $F$ is a polygon with lattice vertices and sides parralell to x-axis and y-axis.Suppose $S(F),P(F)$ are area and perimeter of $F$ .
Find the smallest k that:
$S(F) \leq k.P(F)^2$ | \frac{1}{16} | 0.5 |
Let $T = (a,b,c)$ be a triangle with sides $a,b$ and $c$ and area $\triangle$ . Denote by $T' = (a',b',c')$ the triangle whose sides are the altitudes of $T$ (i.e., $a' = h_a, b' = h_b, c' = h_c$ ) and denote its area by $\triangle '$ . Similarly, let $T'' = (a'',b'',c'')$ be the triangle formed from the altitudes of $T'$ , and denote its area by $\triangle ''$ . Given that $\triangle ' = 30$ and $\triangle '' = 20$ , find $\triangle$ . | 45 | 0.75 |
Let $k$ be a positive integer. Lexi has a dictionary $\mathbb{D}$ consisting of some $k$ -letter strings containing only the letters $A$ and $B$ . Lexi would like to write either the letter $A$ or the letter $B$ in each cell of a $k \times k$ grid so that each column contains a string from $\mathbb{D}$ when read from top-to-bottom and each row contains a string from $\mathbb{D}$ when read from left-to-right.
What is the smallest integer $m$ such that if $\mathbb{D}$ contains at least $m$ different strings, then Lexi can fill her grid in this manner, no matter what strings are in $\mathbb{D}$ ? | 2^{k-1} | 0.375 |
Define the sequence $x_1, x_2, ...$ inductively by $x_1 = \sqrt{5}$ and $x_{n+1} = x_n^2 - 2$ for each $n \geq 1$ . Compute $\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$ . | 1 | 0.75 |
Let $S$ be the set of triples $(a,b,c)$ of non-negative integers with $a+b+c$ even. The value of the sum
\[\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}\]
can be expressed as $\frac{m}{n}$ for relative prime positive integers $m$ and $n$ . Compute $m+n$ .
*Proposed by Nathan Xiong* | 37 | 0.75 |
What is the smallest positive integer $n$ such that $n=x^3+y^3$ for two different positive integer tuples $(x,y)$ ? | 1729 | 0.5 |
Find the number of $12$ -digit "words" that can be formed from the alphabet $\{0,1,2,3,4,5,6\}$ if neighboring digits must differ by exactly $2$ . | 882 | 0.125 |
Let $k$ and $a$ are positive constants. Denote by $V_1$ the volume of the solid generated by a rotation of the figure enclosed
by the curve $C: y=\frac{x}{x+k}\ (x\geq 0)$ , the line $x=a$ and the $x$ -axis around the $x$ -axis, and denote by $V_2$ that of
the solid by a rotation of the figure enclosed by the curve $C$ , the line $y=\frac{a}{a+k}$ and the $y$ -axis around the $y$ -axis.
Find the ratio $\frac{V_2}{V_1}.$ | k | 0.375 |
Find all triples $(a, b, c)$ of positive integers for which $$ \begin{cases} a + bc=2010 b + ca = 250\end{cases} $$ | (3, 223, 9) | 0.625 |
Let $a_1$ , $a_2$ , $\ldots\,$ , $a_{2019}$ be a sequence of real numbers. For every five indices $i$ , $j$ , $k$ , $\ell$ , and $m$ from 1 through 2019, at least two of the numbers $a_i$ , $a_j$ , $a_k$ , $a_\ell$ , and $a_m$ have the same absolute value. What is the greatest possible number of distinct real numbers in the given sequence? | 8 | 0.625 |
The equation $166\times 56 = 8590$ is valid in some base $b \ge 10$ (that is, $1, 6, 5, 8, 9, 0$ are digits in base $b$ in the above equation). Find the sum of all possible values of $b \ge 10$ satisfying the equation. | 12 | 0.5 |
In English class, you have discovered a mysterious phenomenon -- if you spend $n$ hours on an essay, your score on the essay will be $100\left( 1-4^{-n} \right)$ points if $2n$ is an integer, and $0$ otherwise. For example, if you spend $30$ minutes on an essay you will get a score of $50$ , but if you spend $35$ minutes on the essay you somehow do not earn any points.
It is 4AM, your English class starts at 8:05AM the same day, and you have four essays due at the start of class. If you can only work on one essay at a time, what is the maximum possible average of your essay scores?
*Proposed by Evan Chen* | 75 | 0.375 |
Find all triples $(a,b,c)$ of real numbers all different from zero that satisfies:
\begin{eqnarray} a^4+b^2c^2=16a\nonumber b^4+c^2a^2=16b \nonumber c^4+a^2b^2=16c \nonumber \end{eqnarray}
| (2, 2, 2) | 0.875 |
How many ordered pairs of real numbers $(x, y)$ are there such that $x^2+y^2 = 200$ and
\[\sqrt{(x-5)^2+(y-5)^2}+\sqrt{(x+5)^2+(y+5)^2}\]
is an integer? | 12 | 0.375 |
Define $\triangle ABC$ with incenter $I$ and $AB=5$ , $BC=12$ , $CA=13$ . A circle $\omega$ centered at $I$ intersects $ABC$ at $6$ points. The green marked angles sum to $180^\circ.$ Find $\omega$ 's area divided by $\pi.$ | \frac{16}{3} | 0.25 |
Compute the sum of all 2-digit prime numbers $p$ such that there exists a prime number $q$ for which $100q + p$ is a perfect square. | 179 | 0.625 |
Find the greatest possible value of $ sin(cos x) \plus{} cos(sin x)$ and determine all real
numbers x, for which this value is achieved. | \sin(1) + 1 | 0.875 |
Let $S$ be a set of $2020$ distinct points in the plane. Let
\[M=\{P:P\text{ is the midpoint of }XY\text{ for some distinct points }X,Y\text{ in }S\}.\]
Find the least possible value of the number of points in $M$ . | 4037 | 0.125 |
Find all pairs of integers $(c, d)$ , both greater than 1, such that the following holds:
For any monic polynomial $Q$ of degree $d$ with integer coefficients and for any prime $p > c(2c+1)$ , there exists a set $S$ of at most $\big(\tfrac{2c-1}{2c+1}\big)p$ integers, such that
\[\bigcup_{s \in S} \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \dots\}\]
contains a complete residue system modulo $p$ (i.e., intersects with every residue class modulo $p$ ). | (c, d) | 0.125 |
Let $c_i$ denote the $i$ th composite integer so that $\{c_i\}=4,6,8,9,...$ Compute
\[\prod_{i=1}^{\infty} \dfrac{c^{2}_{i}}{c_{i}^{2}-1}\]
(Hint: $\textstyle\sum^\infty_{n=1} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}$ ) | \frac{12}{\pi^2} | 0.5 |
Let $ABC$ be an equilateral triangle. $A $ point $P$ is chosen at random within this triangle. What is the probability that the sum of the distances from point $P$ to the sides of triangle $ABC$ are measures of the sides of a triangle? | \frac{1}{4} | 0.875 |
Decomposition of number $n$ is showing $n$ as a sum of positive integers (not neccessarily distinct). Order of addends is important. For every positive integer $n$ show that number of decompositions is $2^{n-1}$ | 2^{n-1} | 0.875 |
**a)** Solve in $ \mathbb{R} $ the equation $ 2^x=x+1. $ **b)** If a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ has the property that $$ (f\circ f)(x)=2^x-1,\quad\forall x\in\mathbb{R} , $$ then $ f(0)+f(1)=1. $ | f(0) + f(1) = 1 | 0.875 |
Consider \(n^2\) unit squares in the \(xy\) plane centered at point \((i,j)\) with integer coordinates, \(1 \leq i \leq n\), \(1 \leq j \leq n\). It is required to colour each unit square in such a way that whenever \(1 \leq i < j \leq n\) and \(1 \leq k < l \leq n\), the three squares with centres at \((i,k),(j,k),(j,l)\) have distinct colours. What is the least possible number of colours needed? | 2n-1 | 0.5 |
Integers a, b, c, d, and e satisfy the following three properties:
(i) $2 \le a < b <c <d <e <100$
(ii) $ \gcd (a,e) = 1 $ (iii) a, b, c, d, e form a geometric sequence.
What is the value of c? | 36 | 0.875 |
Let $a>1$ be an odd positive integer. Find the least positive integer $n$ such that $2^{2000}$ is a divisor of $a^n-1$ .
*Mircea Becheanu* | 2^{1998} | 0.25 |
Suppose $z_1, z_2 , \cdots z_n$ are $n$ complex numbers such that $min_{j \not= k} | z_{j} - z_{k} | \geq max_{1 \leq j \leq n} |z_j|$ . Find the maximum possible value of $n$ . Further characterise all such maximal configurations. | n = 7 | 0.5 |
Let $ABCD$ be a cyclic quadrilateral, and suppose that $BC = CD = 2$ . Let $I$ be the incenter of triangle $ABD$ . If $AI = 2$ as well, find the minimum value of the length of diagonal $BD$ . | 2\sqrt{3} | 0.125 |
$2000$ people are standing on a line. Each one of them is either a *liar*, who will always lie, or a *truth-teller*, who will always tell the truth. Each one of them says: "there are more liars to my left than truth-tellers to my right". Determine, if possible, how many people from each class are on the line. | 1000 | 0.25 |
Someones age is equal to the sum of the digits of his year of birth. How old is he and when was he born, if it is known that he is older than $11$ .
P.s. the current year in the problem is $2010$ . | 24 | 0.875 |
Alice writes differents real numbers in the board, if $a,b,c$ are three numbers in this board, least one of this numbers $a + b, b + c, a + c$ also is a number in the board. What's the largest quantity of numbers written in the board??? | 7 | 0.125 |
The difference between the maximal and the minimal diagonals of the regular $n$ -gon equals to its side ( $n > 5$ ). Find $n$ . | n = 9 | 0.625 |
Each of the spots in a $8\times 8$ chessboard is occupied by either a black or white “horse”. At most how many black horses can be on the chessboard so that none of the horses attack more than one black horse?**Remark:** A black horse could attack another black horse. | 16 | 0.125 |
Let $n,k$ , $1\le k\le n$ be fixed integers. Alice has $n$ cards in a row, where the card has position $i$ has the label $i+k$ (or $i+k-n$ if $i+k>n$ ). Alice starts by colouring each card either red or blue. Afterwards, she is allowed to make several moves, where each move consists of choosing two cards of different colours and swapping them. Find the minimum number of moves she has to make (given that she chooses the colouring optimally) to put the cards in order (i.e. card $i$ is at position $i$ ).
NOTE: edited from original phrasing, which was ambiguous. | n - \gcd(n, k) | 0.875 |
Given a random string of 33 bits (0 or 1), how many (they can overlap) occurrences of two consecutive 0's would you expect? (i.e. "100101" has 1 occurrence, "0001" has 2 occurrences) | 8 | 0.875 |
A rope of length 10 *m* is tied tautly from the top of a flagpole to the ground 6 *m* away from the base of the pole. An ant crawls up the rope and its shadow moves at a rate of 30 *cm/min*. How many meters above the ground is the ant after 5 minutes? (This takes place on the summer solstice on the Tropic of Cancer so that the sun is directly overhead.) | 2 | 0.625 |
Compute the area of the trapezoid $ABCD$ with right angles $BAD$ and $ADC$ and side lengths of $AB=3$ , $BC=5$ , and $CD=7$ . | 15 | 0.875 |
Let $m > n$ be positive integers such that $3(3mn - 2)^2 - 2(3m -3n)^2 = 2019$ . Find $3m + n$ .
| 46 | 0.875 |
Given an interger $n\geq 2$ , determine the maximum value the sum $\frac{a_1}{a_2}+\frac{a_2}{a_3}+...+\frac{a_{n-1}}{a_n}$ may achieve, and the points at which the maximum is achieved, as $a_1,a_2,...a_n$ run over all positive real numers subject to $a_k\geq a_1+a_2...+a_{k-1}$ , for $k=2,...n$ | \frac{n}{2} | 0.75 |
The graph of the equation $y = 5x + 24$ intersects the graph of the equation $y = x^2$ at two points. The two points are a distance $\sqrt{N}$ apart. Find $N$ . | 3146 | 0.875 |
Let $\Gamma_1$ be a circle with radius $\frac{5}{2}$ . $A$ , $B$ , and $C$ are points on $\Gamma_1$ such that $\overline{AB} = 3$ and $\overline{AC} = 5$ . Let $\Gamma_2$ be a circle such that $\Gamma_2$ is tangent to $AB$ and $BC$ at $Q$ and $R$ , and $\Gamma_2$ is also internally tangent to $\Gamma_1$ at $P$ . $\Gamma_2$ intersects $AC$ at $X$ and $Y$ . $[PXY]$ can be expressed as $\frac{a\sqrt{b}}{c}$ . Find $a+b+c$ .
*2022 CCA Math Bonanza Individual Round #5* | 19 | 0.125 |
Suppose $p < q < r < s$ are prime numbers such that $pqrs + 1 = 4^{p+q}$ . Find $r + s$ . | 274 | 0.875 |
In how many ways can one select eight integers $a_1,a_2, ... ,a_8$ , not necesarily distinct, such that $1 \le a_1 \le ... \le a_8 \le 8$ ? | \binom{15}{7} | 0.75 |
Graph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trinomial $y = x^2$ in points $A$ , $B$ . The intersection of tangents to $G_2$ in points $A$ , $B$ is point $C$ . If $C \in G_1$ , find all possible values of $p$ . | p = 2 | 0.75 |
Find, with explanation, the maximum value of $f(x)=x^3-3x$ on the set of all real numbers $x$ satisfying $x^4+36\leq 13x^2$ . | 18 | 0.875 |
In the picture there are six coins, each with radius 1cm. Each coin is tangent to exactly two other coins next to it (as in the picture). Between the coins, there is an empty area whose boundary is a star-like shape. What is the perimeter of this shape?
 | 4\pi \text{ cm} | 0.125 |
Let $(a_n)_{n \geq 0}$ be the sequence of integers defined recursively by $a_0 = 0, a_1 = 1, a_{n+2} = 4a_{n+1} + a_n$ for $n \geq 0.$ Find the common divisors of $a_{1986}$ and $a_{6891}.$ | 17 | 0.125 |
Determine the number of pairs of positive integers $x,y$ such that $x\le y$ , $\gcd (x,y)=5!$ and $\text{lcm}(x,y)=50!$ . | 16384 | 0.75 |
Determine the largest natural number $r$ with the property that among any five subsets with $500$ elements of the set $\{1,2,\ldots,1000\}$ there exist two of them which share at least $r$ elements. | r = 200 | 0.25 |
Let $ABC$ be a triangle with side lengths $AB=6, AC=7,$ and $BC=8.$ Let $H$ be the orthocenter of $\triangle ABC$ and $H'$ be the reflection of $H$ across the midpoint $M$ of $BC.$ $\tfrac{[ABH']}{[ACH']}$ can be expressed as $\frac{p}{q}$ . Find $p+q$ .
*2022 CCA Math Bonanza Individual Round #14* | 251 | 0.5 |
The sequence $ (a_n)$ is given by $ a_1\equal{}1,a_2\equal{}0$ and:
$ a_{2k\plus{}1}\equal{}a_k\plus{}a_{k\plus{}1}, a_{2k\plus{}2}\equal{}2a_{k\plus{}1}$ for $ k \in \mathbb{N}.$
Find $ a_m$ for $ m\equal{}2^{19}\plus{}91.$ | 91 | 0.5 |
Let $k$ be a fixed positive integer. The $n$ th derivative of $\tfrac{1}{x^k-1}$ has the form $\tfrac{P_n(x)}{(x^k-1)^{n+1}}$ , where $P_n(x)$ is a polynomial. Find $P_n(1)$ . | (-1)^n n! k^n | 0.875 |
Evaluate the sum \[ 11^2 - 1^1 + 12^2 - 2^2 + 13^2 - 3^2 + \cdots + 20^2 - 10^2. \] | 2100 | 0.625 |
Find $ \sum_{k \in A} \frac{1}{k-1}$ where $A= \{ m^n : m,n \in \mathbb{Z} m,n \geq 2 \} $ .
Problem was post earlier [here](http://www.artofproblemsolving.com/Forum/viewtopic.php?f=67&t=29456&hilit=silk+road) , but solution not gives and olympiad doesn't indicate, so I post it again :blush:
Official solution [here](http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659) | 1 | 0.625 |
A good approximation of $\pi$ is $3.14.$ Find the least positive integer $d$ such that if the area of a circle with diameter $d$ is calculated using the approximation $3.14,$ the error will exceed $1.$ | 51 | 0.875 |
You have 2 six-sided dice. One is a normal fair die, while the other has 2 ones, 2 threes, and 2 fives. You pick a die and roll it. Because of some secret magnetic attraction of the unfair die, you have a 75% chance of picking the unfair die and a 25% chance of picking the fair die. If you roll a three, what is the probability that you chose the fair die? | \frac{1}{7} | 0.875 |
Let $ d_n$ be the determinant of the $ n\times n$ matrix whose entries, from left to right and then from top to bottom, are $ \cos 1,\cos 2,\dots,\cos n^2.$ (For example, $ d_3 \equal{} \begin{vmatrix}\cos 1 & \cos2 & \cos3
\cos4 & \cos5 & \cos 6
\cos7 & \cos8 & \cos 9\end{vmatrix}.$ The argument of $ \cos$ is always in radians, not degrees.)
Evaluate $ \lim_{n\to\infty}d_n.$ | 0 | 0.875 |
Compute the sum of $x^2+y^2$ over all four ordered pairs $(x,y)$ of real numbers satisfying $x=y^2-20$ and $y=x^2+x-21$ .
*2021 CCA Math Bonanza Lightning Round #3.4* | 164 | 0.875 |
Suppose $\{ x_n \}_{n\geq 1}$ is a sequence of positive real numbers such that $x_1 \geq x_2 \geq x_3 \ldots \geq x_n \ldots$ , and for all $n$ \[ \frac{x_1}{1} + \frac{x_4}{2} + \frac{x_9}{3} + \ldots + \frac{x_{n^2}}{n} \leq 1 . \] Show that for all $k$ \[ \frac{x_1}{1} + \frac{x_2}{2} +\ldots + \frac{x_k}{k} \leq 3. \] | 3 | 0.75 |
Let $ P(x)$ be a nonzero polynomial such that, for all real numbers $ x$ , $ P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x)$ . Determine the maximum possible number of real roots of $ P(x)$ . | 4 | 0.25 |
Find all triples $ \left(p,x,y\right)$ such that $ p^x\equal{}y^4\plus{}4$ , where $ p$ is a prime and $ x$ and $ y$ are natural numbers. | (5, 1, 1) | 0.625 |
Let $ n>1$ and for $ 1 \leq k \leq n$ let $ p_k \equal{} p_k(a_1, a_2, . . . , a_n)$ be the sum of the products of all possible combinations of k of the numbers $ a_1,a_2,...,a_n$ . Furthermore let $ P \equal{} P(a_1, a_2, . . . , a_n)$ be the sum of all $ p_k$ with odd values of $ k$ less than or equal to $ n$ .
How many different values are taken by $ a_j$ if all the numbers $ a_j (1 \leq j \leq n)$ and $ P$ are prime? | 2 | 0.5 |
From the vertex $ A$ of the equilateral triangle $ ABC$ a line is drown that intercepts the segment $ [BC]$ in the point $ E$ . The point $ M \in (AE$ is such that $ M$ external to $ ABC$ , $ \angle AMB \equal{} 20 ^\circ$ and $ \angle AMC \equal{} 30 ^ \circ$ . What is the measure of the angle $ \angle MAB$ ? | 20^\circ | 0.25 |
Define the sequence $a_0,a_1,\dots$ inductively by $a_0=1$ , $a_1=\frac{1}{2}$ , and
\[a_{n+1}=\dfrac{n a_n^2}{1+(n+1)a_n}, \quad \forall n \ge 1.\]
Show that the series $\displaystyle \sum_{k=0}^\infty \dfrac{a_{k+1}}{a_k}$ converges and determine its value.
*Proposed by Christophe Debry, KU Leuven, Belgium.* | 1 | 0.75 |
An exam at a university consists of one question randomly selected from the $ n$ possible questions. A student knows only one question, but he can take the exam $n$ times. Express as a function of $n$ the probability $p_n$ that the student will pass the exam. Does $p_n$ increase or decrease as $n$ increases? Compute $lim_{n\to \infty}p_n$ . What is the largest lower bound of the probabilities $p_n$ ? | 1 - \frac{1}{e} | 0.75 |
You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1, the value of each blue card is equal to twice the number of red cards, and the value of each yellow card is equal to three times the number of blue cards. What is the maximum score you can get with fifteen cards? | 168 | 0.625 |
Find all $x \in R$ such that $$ x - \left[ \frac{x}{2016} \right]= 2016 $$ , where $[k]$ represents the largest smallest integer or equal to $k$ . | 2017 | 0.875 |
The Fahrenheit temperature ( $F$ ) is related to the Celsius temperature ( $C$ ) by $F = \tfrac{9}{5} \cdot C + 32$ . What is the temperature in Fahrenheit degrees that is one-fifth as large if measured in Celsius degrees? | F = -4 | 0.75 |
A computer generates all pairs of real numbers $x, y \in (0, 1)$ for which the numbers $a = x+my$ and $b = y+mx$ are both integers, where $m$ is a given positive integer. Finding one such pair $(x, y)$ takes $5$ seconds. Find $m$ if the computer needs $595$ seconds to find all possible ordered pairs $(x, y)$ . | m = 11 | 0.625 |
Jeffrey rolls fair three six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $m+n$ .
*Proposed by Nathan Xiong* | 101 | 0.25 |
In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by 30. Find the sum of the four terms. | 129 | 0.875 |
In the diagram below $ \angle CAB, \angle CBD$ , and $\angle CDE$ are all right angles with side lengths $AC = 3$ , $BC = 5$ , $BD = 12$ , and $DE = 84$ . The distance from point $E$ to the line $AB$ can be expressed as the ratio of two relatively prime positive integers, $m$ and $n$ . Find $m + n$ .
[asy]
size(300);
defaultpen(linewidth(0.8));
draw(origin--(3,0)--(0,4)--cycle^^(0,4)--(6,8)--(3,0)--(30,-4)--(6,8));
label(" $A$ ",origin,SW);
label(" $B$ ",(0,4),dir(160));
label(" $C$ ",(3,0),S);
label(" $D$ ",(6,8),dir(80));
label(" $E$ ",(30,-4),E);[/asy] | 5393 | 0.25 |
In square $ABCD$ , $\overline{AC}$ and $\overline{BD}$ meet at point $E$ .
Point $F$ is on $\overline{CD}$ and $\angle CAF = \angle FAD$ .
If $\overline{AF}$ meets $\overline{ED}$ at point $G$ , and if $\overline{EG} = 24$ cm, then find the length of $\overline{CF}$ . | 48 | 0.75 |
Points $A$ , $B$ , $C$ , and $D$ lie on a circle. Let $AC$ and $BD$ intersect at point $E$ inside the circle. If $[ABE]\cdot[CDE]=36$ , what is the value of $[ADE]\cdot[BCE]$ ? (Given a triangle $\triangle ABC$ , $[ABC]$ denotes its area.) | 36 | 0.875 |
Let $a$ and $b$ be real numbers such that $a+b=\log_2( \log_2 3)$ . What is the minimum value of $2^a + 3^b$ ? | \log_2 6 | 0.375 |
Find all positive integers $n$ that have 4 digits, all of them perfect squares, and such that $n$ is divisible by 2, 3, 5 and 7. | 4410 | 0.875 |
Let $ABCD$ be a quadrilateral with $\overline{AB}\parallel\overline{CD}$ , $AB=16$ , $CD=12$ , and $BC<AD$ . A circle with diameter $12$ is inside of $ABCD$ and tangent to all four sides. Find $BC$ . | 13 | 0.5 |
In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$ , $\beta$ , and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$ . If $\cos \alpha$ , which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator? | 49 | 0.875 |
Find the root that the following three polynomials have in common:
\begin{align*} & x^3+41x^2-49x-2009
& x^3 + 5x^2-49x-245
& x^3 + 39x^2 - 117x - 1435\end{align*} | 7 | 0.875 |
Find all complex-valued functions $f$ of a complex variable such that $$ f(z)+zf(1-z)=1+z $$ for all $z\in \mathbb{C}$ . | f(z) = 1 | 0.875 |
There are $800$ marbles in a bag. Each marble is colored with one of $100$ colors, and there are eight marbles of each color. Anna draws one marble at a time from the bag, without replacement, until she gets eight marbles of the same color, and then she immediately stops.
Suppose Anna has not stopped after drawing $699$ marbles. Compute the probability that she stops immediately after drawing the $700$ th marble. | \frac{99}{101} | 0.375 |
A set of 10 distinct integers $S$ is chosen. Let $M$ be the number of nonempty subsets of $S$ whose elements have an even sum. What is the minimum possible value of $M$ ?
<details><summary>Clarifications</summary>
- $S$ is the ``set of 10 distinct integers'' from the first sentence.
</details>
*Ray Li* | 511 | 0.25 |
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Proposed by Minseok Eli Park (wolfpack) | 47 | 0.75 |
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