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At the Rice Mathematics Tournament, 80% of contestants wear blue jeans, 70% wear tennis shoes, and 80% of those who wear blue jeans also wear tennis shoes. What fraction of people wearing tennis shoes are wearing blue jeans? | \frac{32}{35} | 0.875 |
A number of runners competed in a race. When Ammar finished, there were half as many runners who had finished before him compared to the number who finished behind him. Julia was the 10th runner to finish behind Ammar. There were twice as many runners who had finished before Julia compared to the number who finished behind her. How many runners were there in the race? | 31 | 0.875 |
A cube with side length $100cm$ is filled with water and has a hole through which the water drains into a cylinder of radius $100cm.$ If the water level in the cube is falling at a rate of $1 \frac{cm}{s} ,$ how fast is the water level in the cylinder rising? | \frac{1}{\pi} | 0.375 |
An equilateral triangle is inscribed inside of a circle of radius $R$ . Find the side length of the triangle | R\sqrt{3} | 0.75 |
If $0 < x < \frac{\pi}{2}$ and $\frac{\sin x}{1 + \cos x} = \frac{1}{3}$ , what is $\frac{\sin 2x}{1 + \cos 2x}$ ? | \frac{3}{4} | 0.75 |
Find the sum of all four-digit integers whose digits are a rearrangement of the digits $1$ , $2$ , $3$ , $4$ , such as $1234$ , $1432$ , or $3124$ . | 66660 | 0.75 |
Let $x_1, x_2,... , x_{84}$ be the roots of the equation $x^{84} + 7x - 6 = 0$ . Compute $\sum_{k=1}^{84} \frac{x_k}{x_k-1}$ . | \frac{77}{2} | 0.5 |
Juan wrote a natural number and Maria added a digit $ 1$ to the left and a digit $ 1$ to the right. Maria's number exceeds to the number of Juan in $14789$ . Find the number of Juan. | 532 | 0.875 |
The bisectors of the angles $A$ and $B$ of the triangle $ABC$ intersect the sides $BC$ and $AC$ at points $D$ and $E$ . It is known that $AE + BD = AB$ . Find the angle $\angle C$ . | 60^\circ | 0.875 |
A point $(x,y)$ in the first quadrant lies on a line with intercepts $(a,0)$ and $(0,b)$ , with $a,b > 0$ . Rectangle $M$ has vertices $(0,0)$ , $(x,0)$ , $(x,y)$ , and $(0,y)$ , while rectangle $N$ has vertices $(x,y)$ , $(x,b)$ , $(a,b)$ , and $(a,y)$ . What is the ratio of the area of $M$ to that of $N$ ?
*Proposed by Eugene Chen* | 1 | 0.75 |
Floyd looked at a standard $12$ hour analogue clock at $2\!:\!36$ . When Floyd next looked at the clock, the angles through which the hour hand and minute hand of the clock had moved added to $247$ degrees. How many minutes after $3\!:\!00$ was that? | 14 | 0.875 |
Real numbers are chosen at random from the interval $[0,1].$ If after choosing the $n$ -th number the sum of the numbers so chosen first exceeds $1$ , show that the expected value for $n$ is $e$ . | e | 0.875 |
Let $z_1$ and $z_2$ be the zeros of the polynomial $f(x) = x^2 + 6x + 11$ . Compute $(1 + z^2_1z_2)(1 + z_1z_2^2)$ . | 1266 | 0.625 |
Jerry's favorite number is $97$ . He knows all kinds of interesting facts about $97$ :
- $97$ is the largest two-digit prime.
- Reversing the order of its digits results in another prime.
- There is only one way in which $97$ can be written as a difference of two perfect squares.
- There is only one way in which $97$ can be written as a sum of two perfect squares.
- $\tfrac1{97}$ has exactly $96$ digits in the [smallest] repeating block of its decimal expansion.
- Jerry blames the sock gnomes for the theft of exactly $97$ of his socks.
A repunit is a natural number whose digits are all $1$ . For instance, \begin{align*}&1,&11,&111,&1111,&\vdots\end{align*} are the four smallest repunits. How many digits are there in the smallest repunit that is divisible by $97?$ | 96 | 0.875 |
Two circles $\mathcal{C}_1$ and $\mathcal{C}_2$ with centers $(1, 1)$ and $(4, 5)$ and radii $r_1 < r_2$ , respectively, are drawn on the coordinate plane. The product of the slopes of the two common external tangents of $\mathcal{C}_1$ and $\mathcal{C}_2$ is $3$ . If the value of $(r_2 - r_1)^2$ can be expressed as a common fraction in the form $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m + n$ . | 13 | 0.375 |
Show that there exists a composite number $n$ such that $a^n \equiv a \; \pmod{n}$ for all $a \in \mathbb{Z}$ . | n = 561 | 0.75 |
Tina writes four letters to her friends Silas, Jessica, Katie, and Lekan. She prepares an envelope for Silas, an envelope for Jessica, an envelope for Katie, and an envelope for Lekan. However, she puts each letter into a random envelope. What is the probability that no one receives the letter they are supposed to receive? | \frac{3}{8} | 0.875 |
For every positive integer $k$ let $a(k)$ be the largest integer such that $2^{a(k)}$ divides $k$ . For every positive integer $n$ determine $a(1)+a(2)+\cdots+a(2^n)$ . | 2^n - 1 | 0.875 |
What is the smallest possible natural number $n$ for which the equation $x^2 -nx + 2014 = 0$ has integer roots? | 91 | 0.875 |
Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$ . | 4648 | 0.25 |
Find the remainder when the number of positive divisors of the value $$ (3^{2020}+3^{2021})(3^{2021}+3^{2022})(3^{2022}+3^{2023})(3^{2023}+3^{2024}) $$ is divided by $1000$ .
*Proposed by pog* | 783 | 0.875 |
Suppose $xy-5x+2y=30$ , where $x$ and $y$ are positive integers. Find the sum of all possible values of $x$ | 31 | 0.875 |
Let $\triangle ABC$ have $\angle ABC=67^{\circ}$ . Point $X$ is chosen such that $AB = XC$ , $\angle{XAC}=32^\circ$ , and $\angle{XCA}=35^\circ$ . Compute $\angle{BAC}$ in degrees.
*Proposed by Raina Yang* | 81^\circ | 0.875 |
Paul fi lls in a $7\times7$ grid with the numbers $1$ through $49$ in a random arrangement. He then erases his work and does the same thing again, to obtain two diff erent random arrangements of the numbers in the grid. What is the expected number of pairs of numbers that occur in either the same row as each other or the same column as each other in both of the two arrangements? | 73.5 | 0.875 |
$\textbf{Problem 5.}$ Miguel has two clocks, one clock advances $1$ minute per day and the other one goes $15/10$ minutes per day.
If you put them at the same correct time, What is the least number of days that must pass for both to give the correct time simultaneously? | 1440 | 0.75 |
A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one roll of each type is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime integers, find $m+n$ . | 79 | 0.75 |
A group of women working together at the same rate can build a wall in $45$ hours. When the work started, all the women did not start working together. They joined the worked over a period of time, one by one, at equal intervals. Once at work, each one stayed till the work was complete. If the first woman worked 5 times as many hours as the last woman, for how many hours did the first woman work? | 75 | 0.875 |
Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks a sinusoidal path in search of home; that is, after $t$ minutes he is at position $(t,\sin t)$ .
Five minutes after he sets out, Alex enters the forest at the origin and sets out in search of Tim. He walks in such a way that after he has been in the forest for $m$ minutes, his position is $(m,\cos t)$ .
What is the greatest distance between Alex and Farmer Tim while they are walking in these paths? | 3\sqrt{3} | 0.5 |
Let $ a$ , $ b$ , $ c$ , $ x$ , $ y$ , and $ z$ be real numbers that satisfy the three equations
\begin{align*}
13x + by + cz &= 0
ax + 23y + cz &= 0
ax + by + 42z &= 0.
\end{align*}Suppose that $ a \ne 13$ and $ x \ne 0$ . What is the value of
\[ \frac{13}{a - 13} + \frac{23}{b - 23} + \frac{42}{c - 42} \, ?\] | -2 | 0.5 |
Let $T = TNFTPP$ . When properly sorted, $T - 35$ math books on a shelf are arranged in alphabetical order from left to right. An eager student checked out and read all of them. Unfortunately, the student did not realize how the books were sorted, and so after finishing the student put the books back on the shelf in a random order. If all arrangements are equally likely, the probability that exactly $6$ of the books were returned to their correct (original) position can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$ .
[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question. For those who wanted to try the problem separately, <details><summary>here's the value of T</summary>$T=44$</details>. | 2161 | 0.375 |
Eight football teams play matches against each other in such a way that no two teams meet twice and no three teams play all of the three possible matches.
What is the largest possible number of matches? | 16 | 0.875 |
At the Mountain School, Micchell is assigned a *submissiveness rating* of $3.0$ or $4.0$ for each class he takes. His *college potential* is then defined as the average of his submissiveness ratings over all classes taken. After taking 40 classes, Micchell has a college potential of $3.975$ . Unfortunately, he needs a college potential of at least $3.995$ to get into the [South Harmon Institute of Technology](http://en.wikipedia.org/wiki/Accepted#Plot). Otherwise, he becomes a rock. Assuming he receives a submissiveness rating of $4.0$ in every class he takes from now on, how many more classes does he need to take in order to get into the South Harmon Institute of Technology?
*Victor Wang* | 160 | 0.875 |
Let $m,n$ be natural numbers such that $\hspace{2cm} m+3n-5=2LCM(m,n)-11GCD(m,n).$ Find the maximum possible value of $m+n$ . | 70 | 0.75 |
Find value of $$ \frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx} $$ if $x$ , $y$ and $z$ are real numbers usch that $xyz=1$ | 1 | 0.625 |
**p1.** What is the maximum possible value of $m$ such that there exist $m$ integers $a_1, a_2, ..., a_m$ where all the decimal representations of $a_1!, a_2!, ..., a_m!$ end with the same amount of zeros?**p2.** Let $f : R \to R$ be a function such that $f(x) + f(y^2) = f(x^2 + y)$ , for all $x, y \in R$ . Find the sum of all possible $f(-2017)$ .**p3.** What is the sum of prime factors of $1000027$ ?**p4.** Let $$ \frac{1}{2!} +\frac{2}{3!} + ... +\frac{2016}{2017!} =\frac{n}{m}, $$ where $n, m$ are relatively prime. Find $(m - n)$ .**p5.** Determine the number of ordered pairs of real numbers $(x, y)$ such that $\sqrt[3]{3 - x^3 - y^3} =\sqrt{2 - x^2 - y^2}$ **p6.** Triangle $\vartriangle ABC$ has $\angle B = 120^o$ , $AB = 1$ . Find the largest real number $x$ such that $CA - CB > x$ for all possible triangles $\vartriangle ABC$ .**p7.**Jung and Remy are playing a game with an unfair coin. The coin has a probability of $p$ where its outcome is heads. Each round, Jung and Remy take turns to flip the coin, starting with Jung in round $ 1$ . Whoever gets heads first wins the game. Given that Jung has the probability of $8/15$ , what is the value of $p$ ?**p8.** Consider a circle with $7$ equally spaced points marked on it. Each point is $ 1$ unit distance away from its neighbors and labelled $0,1,2,...,6$ in that order counterclockwise. Feng is to jump around the circle, starting at the point $0$ and making six jumps counterclockwise with distinct lengths $a_1, a_2, ..., a_6$ in a way such that he will land on all other six nonzero points afterwards. Let $s$ denote the maximum value of $a_i$ . What is the minimum possible value of $s$ ?**p9.**Justin has a $4 \times 4 \times 4$ colorless cube that is made of $64$ unit-cubes. He then colors $m$ unit-cubes such that none of them belong to the same column or row of the original cube. What is the largest possible value of $m$ ?**p10.** Yikai wants to know Liang’s secret code which is a $6$ -digit integer $x$ . Furthermore, let $d(n)$ denote the digital sum of a positive integer $n$ . For instance, $d(14) = 5$ and $d(3) = 3$ . It is given that $$ x + d(x) + d(d(x)) + d(d(d(x))) = 999868. $$ Please find $x$ .
PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309). | 5 | 0.125 |
Suppose that $N$ is a three digit number divisible by $7$ such that upon removing its middle digit, the remaining two digit number is also divisible by $7$ . What is the minimum possible value of $N$ ?
*2019 CCA Math Bonanza Lightning Round #3.1* | 154 | 0.875 |
Garfield and Odie are situated at $(0,0)$ and $(25,0)$ , respectively. Suddenly, Garfield and Odie dash in the direction of the point $(9, 12)$ at speeds of $7$ and $10$ units per minute, respectively. During this chase, the minimum distance between Garfield and Odie can be written as $\frac{m}{\sqrt{n}}$ for relatively prime positive integers $m$ and $n$ . Find $m+n$ .
*Proposed by **Th3Numb3rThr33*** | 159 | 0.75 |
In a bag are all natural numbers less than or equal to $999$ whose digits sum to $6$ . What is the probability of drawing a number from the bag that is divisible by $11$ ? | \frac{1}{7} | 0.875 |
Find all pairs of real numbers $(a,b)$ so that there exists a polynomial $P(x)$ with real coefficients and $P(P(x))=x^4-8x^3+ax^2+bx+40$ . | (28, -48), (2, 56) | 0.75 |
On a $3 \times 3$ board the numbers from $1$ to $9$ are written in some order and without repeating. We say that the arrangement obtained is *Isthmian* if the numbers in any two adjacent squares have different parity. Determine the number of different Isthmian arrangements.
Note: Two arrangements are considered equal if one can be obtained from the other by rotating the board. | 720 | 0.875 |
Al and Bob play Rock Paper Scissors until someone wins a game. What is the probability that this happens on the sixth game?
| \frac{2}{729} | 0.875 |
For positive integers $n$ and $k$ , let $\mho(n,k)$ be the number of distinct prime divisors of $n$ that are at least $k$ . For example, $\mho(90, 3)=2$ , since the only prime factors of $90$ that are at least $3$ are $3$ and $5$ . Find the closest integer to
\[\sum_{n=1}^\infty \sum_{k=1}^\infty \frac{\mho(n,k)}{3^{n+k-7}}.\]
*Proposed by Daniel Zhu.* | 167 | 0.625 |
Find the largest value of the expression $\frac{p}{R}\left( 1- \frac{r}{3R}\right)$ , where $p,R, r$ is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle. | \frac{5\sqrt{3}}{2} | 0.75 |
Two toads named Gamakichi and Gamatatsu are sitting at the points $(0,0)$ and $(2,0)$ respectively. Their goal is to reach $(5,5)$ and $(7,5)$ respectively by making one unit jumps in positive $x$ or $y$ direction at a time. How many ways can they do this while ensuring that there is no point on the plane where both Gamakichi And Gamatatsu land on? | 19152 | 0.625 |
Let $a_1 < a_2 < \cdots < a_k$ denote the sequence of all positive integers between $1$ and $91$ which are relatively prime to $91$ , and set $\omega = e^{2\pi i/91}$ . Define \[S = \prod_{1\leq q < p\leq k}\left(\omega^{a_p} - \omega^{a_q}\right).\] Given that $S$ is a positive integer, compute the number of positive divisors of $S$ . | 1054 | 0.375 |
Let $f(x) = x^3 - 3x + b$ and $g(x) = x^2 + bx -3$ , where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f(x)$ = 0 and $g(x) = 0$ have a common root? | 0 | 0.875 |
There is a positive integer s such that there are s solutions to the equation $64sin^2(2x)+tan^2(x)+cot^2(x)=46$ in the interval $(0,\frac{\pi}{2})$ all of the form $\frac{m_k}{n_k}\pi$ where $m_k$ and $n_k$ are relatively prime positive integers, for $k = 1, 2, 3, . . . , s$ . Find $(m_1 + n_1) + (m_2 + n_2) + (m_3 + n_3) + · · · + (m_s + n_s)$ . | 100 | 0.375 |
Let $ABC$ be an acute triangle. $PQRS$ is a rectangle with $P$ on $AB$ , $Q$ and $R$ on $BC$ , and $S$ on $AC$ such that $PQRS$ has the largest area among all rectangles $TUVW$ with $T$ on $AB$ , $U$ and $V$ on $BC$ , and $W$ on $AC$ . If $D$ is the point on $BC$ such that $AD\perp BC$ , then $PQ$ is the harmonic mean of $\frac{AD}{DB}$ and $\frac{AD}{DC}$ . What is $BC$ ?
Note: The harmonic mean of two numbers $a$ and $b$ is the reciprocal of the arithmetic mean of the reciprocals of $a$ and $b$ .
*2017 CCA Math Bonanza Lightning Round #4.4* | 4 | 0.75 |
A binary string is a string consisting of only 0’s and 1’s (for instance, 001010, 101, etc.). What is the probability that a randomly chosen binary string of length 10 has 2 consecutive 0’s? Express your answer as a fraction. | \frac{55}{64} | 0.875 |
Find all positive integers $n$ , such that $n$ is a perfect number and $\varphi (n)$ is power of $2$ .
*Note:a positive integer $n$ , is called perfect if the sum of all its positive divisors is equal to $2n$ .* | n = 6 | 0.625 |
For a real number $x,$ let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x,$ and let $\{x\} = x -\lfloor x\rfloor$ denote the fractional part of $x.$ The sum of all real numbers $\alpha$ that satisfy the equation $$ \alpha^2+\{\alpha\}=21 $$ can be expressed in the form $$ \frac{\sqrt{a}-\sqrt{b}}{c}-d $$ where $a, b, c,$ and $d$ are positive integers, and $a$ and $b$ are not divisible by the square of any prime. Compute $a + b + c + d.$ | 169 | 0.75 |
Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $\log_2 n.$ | 742 | 0.75 |
Suppose that $(a_n)$ is a sequence of real numbers such that the series $$ \sum_{n=1}^\infty\frac{a_n}n $$ is convergent. Show that the sequence $$ b_n=\frac1n\sum^n_{j=1}a_j $$ is convergent and find its limit. | 0 | 0.875 |
What is the smallest positive integer $n$ such that there exists a choice of signs for which \[1^2\pm2^2\pm3^2\ldots\pm n^2=0\] is true?
*2019 CCA Math Bonanza Team Round #5* | 7 | 0.875 |
A triangle has sides of length $48$ , $55$ , and $73$ . Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the length of the shortest altitude of the triangle. Find the value of $a+b$ . | 2713 | 0.875 |
Two cubes $A$ and $B$ have different side lengths, such that the volume of cube $A$ is numerically equal to the surface area of cube $B$ . If the surface area of cube $A$ is numerically equal to six times the side length of cube $B$ , what is the ratio of the surface area of cube $A$ to the volume of cube $B$ ? | 7776 | 0.875 |
Jamie, Linda, and Don bought bundles of roses at a flower shop, each paying the same price for each
bundle. Then Jamie, Linda, and Don took their bundles of roses to a fair where they tried selling their
bundles for a fixed price which was higher than the price that the flower shop charged. At the end of the
fair, Jamie, Linda, and Don donated their unsold bundles of roses to the fair organizers. Jamie had bought
20 bundles of roses, sold 15 bundles of roses, and made $60$ profit. Linda had bought 34 bundles of roses,
sold 24 bundles of roses, and made $69 profit. Don had bought 40 bundles of roses and sold 36 bundles of
roses. How many dollars profit did Don make?
| 252 | 0.875 |
$$ \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\dots+\frac{1}{2014\cdot2015}=\frac{m}{n}, $$ where $\frac{m}{n}$ is irreducible.
a) Find $m+n.$ b) Find the remainder of division of $(m+3)^{1444}$ to $n{}$ . | 16 | 0.625 |
Evin and Jerry are playing a game with a pile of marbles. On each players' turn, they can remove $2$ , $3$ , $7$ , or $8$ marbles. If they can’t make a move, because there's $0$ or $1$ marble left, they lose the game. Given that Evin goes first and both players play optimally, for how many values of $n$ from $1$ to $1434$ does Evin lose the game?
*Proposed by Evin Liang*
<details><summary>Solution</summary>*Solution.* $\boxed{573}$ Observe that no matter how many marbles a one of them removes, the next player can always remove marbles such
that the total number of marbles removed is $10$ . Thus, when the number of marbles is a multiple of $10$ , the first player loses the game. We analyse this game based on the number of marbles modulo $10$ :
If the number of marbles is $0$ modulo $10$ , the first player loses the game
If the number of marbles is $2$ , $3$ , $7$ , or $8$ modulo $10$ , the first player wins the game by moving to $0$ modulo 10
If the number of marbles is $5$ modulo $10$ , the first player loses the game because every move leads to $2$ , $3$ , $7$ , or $8$ modulo $10$ In summary, the first player loses if it is $0$ mod 5, and wins if it is $2$ or $3$ mod $5$ . Now we solve the remaining cases by induction. The first player loses when it is $1$ modulo $5$ and wins when it is $4$ modulo $5$ . The base case is when there is $1$ marble, where the first player loses because there is no move. When it is $4$ modulo $5$ , then the first player can always remove $3$ marbles and win by the inductive hypothesis. When it is $1$ modulo $5$ , every move results in $3$ or $4$ modulo $5$ , which allows the other player to win by the inductive hypothesis.
Thus, Evin loses the game if n is $0$ or $1$ modulo $5$ . There are $\boxed{573}$ such values of $n$ from $1$ to $1434$ .</details> | 573 | 0.75 |
Find the largest possible value in the real numbers of the term $$ \frac{3x^2 + 16xy + 15y^2}{x^2 + y^2} $$ with $x^2 + y^2 \ne 0$ . | 19 | 0.625 |
The diagram shows a semicircle with diameter $20$ and the circle with greatest diameter that fits inside the semicircle. The area of the shaded region is $N\pi$ , where $N$ is a positive integer. Find $N$ . | N = 25 | 0.875 |
A square paper of side $n$ is divided into $n^2$ unit square cells. A maze is drawn on the paper with unit walls between some cells in such a way that one can reach every cell from every other cell not crossing any wall. Find, in terms of $n$ , the largest possible total length of the walls. | (n-1)^2 | 0.875 |
Let $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$ . Compute the prime $p$ satisfying $f(p) = 418{,}195{,}493$ .
*Proposed by Eugene Chen* | 2 | 0.625 |
Let $ \alpha_1$ , $ \alpha_2$ , $ \ldots$ , $ \alpha_{2008}$ be real numbers. Find the maximum value of \[ \sin\alpha_1\cos\alpha_2 \plus{} \sin\alpha_2\cos\alpha_3 \plus{} \cdots \plus{} \sin\alpha_{2007}\cos\alpha_{2008} \plus{} \sin\alpha_{2008}\cos\alpha_1\] | 1004 | 0.875 |
Let $ABCD$ be a trapezoid of bases $AB$ and $CD$ . Let $O$ be the intersection point of the diagonals $AC$ and $BD$ . If the area of the triangle $ABC$ is $150$ and the area of the triangle $ACD$ is $120$ , calculate the area of the triangle $BCO$ . | \frac{200}{3} | 0.75 |
Determine the smallest positive number $a$ such that the number of all integers belonging to $(a, 2016a]$ is $2016$ . | \frac{2017}{2016} | 0.5 |
Arthur, Bob, and Carla each choose a three-digit number. They each multiply the digits of their own numbers. Arthur gets 64, Bob gets 35, and Carla gets 81. Then, they add corresponding digits of their numbers together. The total of the hundreds place is 24, that of the tens place is 12, and that of the ones place is 6. What is the difference between the largest and smallest of the three original numbers?
*Proposed by Jacob Weiner* | 182 | 0.875 |
A rectangular pool table has vertices at $(0, 0) (12, 0) (0, 10),$ and $(12, 10)$ . There are pockets only in the four corners. A ball is hit from $(0, 0)$ along the line $y = x$ and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bounces off of before entering a pocket. | 9 | 0.625 |
A $9 \times 9$ square consists of $81$ unit squares. Some of these unit squares are painted black, and the others are painted white, such that each $2 \times 3$ rectangle and each $3 \times 2$ rectangle contain exactly 2 black unit squares and 4 white unit squares. Determine the number of black unit squares. | 27 | 0.875 |
Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!"
Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I'd know for certain what it is?"
Cat says, "Nope! However, if I now told you the units digit of my favorite number, you'd know which one it is!"
Claire says, "Now I know your favorite number!" What is Cat's favorite number?
*Proposed by Andrew Wu* | 13 | 0.625 |
Let be given a triangle $ABC$ and points $D,M,N$ belong to $BC,AB,AC$ , respectively. Suppose that $MD$ is parallel to $AC$ and $ND$ is parallel to $AB$ . If $S_{\vartriangle BMD} = 9$ cm $^2, S_{\vartriangle DNC} = 25$ cm $^2$ , compute $S_{\vartriangle AMN}$ ?
| 15 \, \text{cm}^2 | 0.625 |
Let $P(x,\ y)\ (x>0,\ y>0)$ be a point on the curve $C: x^2-y^2=1$ . If $x=\frac{e^u+e^{-u}}{2}\ (u\geq 0)$ , then find the area bounded by the line $OP$ , the $x$ axis and the curve $C$ in terms of $u$ . | \frac{1}{2} u | 0.375 |
Let $T_1$ be an isosceles triangle with sides of length 8, 11, and 11. Let $T_2$ be an isosceles triangle with sides of length $b$ , 1, and 1. Suppose that the radius of the incircle of $T_1$ divided by the radius of the circumcircle of $T_1$ is equal to the radius of the incircle of $T_2$ divided by the radius of the circumcircle of $T_2$ . Determine the largest possible value of $b$ . | \frac{14}{11} | 0.75 |
Given that the polynomial $P(x) = x^5 - x^2 + 1$ has $5$ roots $r_1, r_2, r_3, r_4, r_5$ . Find the value of the product $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$ , where $Q(x) = x^2 + 1$ . | 5 | 0.625 |
In the rectangle $ABCD, M, N, P$ and $Q$ are the midpoints of the sides. If the area of the shaded triangle is $1$ , calculate the area of the rectangle $ABCD$ .
 | 8 | 0.125 |
Let $P$ be the portion of the graph of $$ y=\frac{6x+1}{32x+8} - \frac{2x-1}{32x-8} $$ located in the first quadrant (not including the $x$ and $y$ axes). Let the shortest possible distance between the origin and a point on $P$ be $d$ . Find $\lfloor 1000d \rfloor$ .
*Proposed by **Th3Numb3rThr33*** | 433 | 0.875 |
Let $a_1,a_2,\ldots$ be a sequence defined by $a_1=a_2=1$ and $a_{n+2}=a_{n+1}+a_n$ for $n\geq 1$ . Find \[\sum_{n=1}^\infty \dfrac{a_n}{4^{n+1}}.\] | \frac{1}{11} | 0.875 |
Let $n$ be an integer greater than two, and let $A_1,A_2, \cdots , A_{2n}$ be pairwise distinct subsets of $\{1, 2, ,n\}$ . Determine the maximum value of
\[\sum_{i=1}^{2n} \dfrac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}\]
Where $A_{2n+1}=A_1$ and $|X|$ denote the number of elements in $X.$ | n | 0.5 |
In how many different ways can 900 be expressed as the product of two (possibly equal) positive integers? Regard $m \cdot n$ and $n \cdot m$ as the same product. | 14 | 0.875 |
In equilateral triangle $ABC$ , the midpoint of $\overline{BC}$ is $M$ . If the circumcircle of triangle $MAB$ has area $36\pi$ , then find the perimeter of the triangle.
*Proposed by Isabella Grabski* | 36 | 0.5 |
Let $S$ be the set of natural numbers that cannot be written as the sum of three squares. Legendre's three-square theorem states that $S$ consists of precisely the integers of the form $4^a(8b+7)$ where $a$ and $b$ are nonnegative integers. Find the smallest $n\in\mathbb N$ such that $n$ and $n+1$ are both in $S$ . | 111 | 0.375 |
Suppose $a_1, a_2, a_3, \dots$ is an increasing arithmetic progression of positive integers. Given that $a_3 = 13$ , compute the maximum possible value of \[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \]*Proposed by Evan Chen* | 365 | 0.875 |
There are $ 2010 $ people sitting around a round table. First, we give one person $ x $ a candy. Next, we give candies to $1$ st person, $1+2$ th person, $ 1+2+3$ th person, $\cdots$ , and $1+2+\cdots + 2009 $ th person clockwise from $ x $ . Find the number of people who get at least one candy. | 408 | 0.625 |
Tatjana imagined a polynomial $P(x)$ with nonnegative integer coefficients. Danica is trying to guess the polynomial. In each step, she chooses an integer $k$ and Tatjana tells her the value of $P(k)$ . Find the smallest number of steps Danica needs in order to find the polynomial Tatjana imagined. | 2 | 0.875 |
Find the smallest natural number, which divides $2^{n}+15$ for some natural number $n$ and can be expressed in the form $3x^2-4xy+3y^2$ for some integers $x$ and $y$ . | 23 | 0.5 |
For a table $n \times 9$ ( $n$ rows and $9$ columns), determine the maximum of $n$ that we can write one number in the set $\left\{ {1,2,...,9} \right\}$ in each cell such that these conditions are satisfied:
1. Each row contains enough $9$ numbers of the set $\left\{ {1,2,...,9} \right\}$ .
2. Any two rows are distinct.
3. For any two rows, we can find at least one column such that the two intersecting cells between it and the two rows contain the same number. | 8! | 0.625 |
Circles $k_1$ and $k_2$ with radii $r_1=6$ and $r_2=3$ are externally tangent and touch a circle $k$ with radius $r=9$ from inside. A common external tangent of $k_1$ and $k_2$ intersects $k$ at $P$ and $Q$ . Determine the length of $PQ$ . | 4\sqrt{14} | 0.125 |
For a positive integer $n$ , let $d_n$ be the units digit of $1 + 2 + \dots + n$ . Find the remainder when
\[\sum_{n=1}^{2017} d_n\]
is divided by $1000$ . | 69 | 0.375 |
Inside of the square $ABCD$ the point $P$ is given such that $|PA|:|PB|:|PC|=1:2:3$ . Find $\angle APB$ . | 135^\circ | 0.75 |
Consider the set $S$ of $100$ numbers: $1; \frac{1}{2}; \frac{1}{3}; ... ; \frac{1}{100}$ .
Any two numbers, $a$ and $b$ , are eliminated in $S$ , and the number $a+b+ab$ is added. Now, there are $99$ numbers on $S$ .
After doing this operation $99$ times, there's only $1$ number on $S$ . What values can this number take? | 100 | 0.875 |
You roll three fair six-sided dice. Given that the highest number you rolled is a $5$ , the expected value of the sum of the three dice can be written as $\tfrac ab$ in simplest form. Find $a+b$ . | 706 | 0.875 |
Let $a,b$ be positive reals such that $\frac{1}{a}+\frac{1}{b}\leq2\sqrt2$ and $(a-b)^2=4(ab)^3$ . Find $\log_a b$ . | -1 | 0.875 |
Let $p$ be an odd prime number less than $10^5$ . Granite and Pomegranate play a game. First, Granite picks a integer $c \in \{2,3,\dots,p-1\}$ .
Pomegranate then picks two integers $d$ and $x$ , defines $f(t) = ct + d$ , and writes $x$ on a sheet of paper.
Next, Granite writes $f(x)$ on the paper, Pomegranate writes $f(f(x))$ , Granite writes $f(f(f(x)))$ , and so on, with the players taking turns writing.
The game ends when two numbers appear on the paper whose difference is a multiple of $p$ , and the player who wrote the most recent number wins. Find the sum of all $p$ for which Pomegranate has a winning strategy.
*Proposed by Yang Liu* | 65819 | 0.5 |
The number $2^{1997}$ has $m$ decimal digits, while the number $5^{1997}$ has $n$ digits. Evaluate $m+n$ . | 1998 | 0.75 |
The Fibonacci sequence is defined as follows: $F_0=0$ , $F_1=1$ , and $F_n=F_{n-1}+F_{n-2}$ for all integers $n\ge 2$ . Find the smallest positive integer $m$ such that $F_m\equiv 0 \pmod {127}$ and $F_{m+1}\equiv 1\pmod {127}$ . | 256 | 0.5 |
Consider the $10$ -digit number $M=9876543210$ . We obtain a new $10$ -digit number from $M$ according to the following rule: we can choose one or more disjoint pairs of adjacent digits in $M$ and interchange the digits in these chosen pairs, keeping the remaining digits in their own places. For example, from $M=9\underline{87}6 \underline{54} 3210$ by interchanging the $2$ underlined pairs, and keeping the others in their places, we get $M_{1}=9786453210$ . Note that any number of (disjoint) pairs can be interchanged. Find the number of new numbers that can be so obtained from $M$ . | 88 | 0.5 |
Find the sum of all primes that can be written both as a sum of two primes and as a difference of two primes.
*Anonymous Proposal* | 5 | 0.625 |
Ethan Song and Bryan Guo are playing an unfair game of rock-paper-scissors. In any game, Ethan has a 2/5 chance to win, 2/5 chance to tie, and 1/5 chance to lose. How many games is Ethan expected to win before losing?
*2022 CCA Math Bonanza Lightning Round 4.3* | 2 | 0.625 |
Let $M\subset \Bbb{N}^*$ such that $|M|=2004.$ If no element of $M$ is equal to the sum of any two elements of $M,$ find the least value that the greatest element of $M$ can take. | 4007 | 0.375 |
The sequence of real numbers $\{a_n\}$ , $n \in \mathbb{N}$ satisfies the following condition: $a_{n+1}=a_n(a_n+2)$ for any $n \in \mathbb{N}$ . Find all possible values for $a_{2004}$ . | [-1, \infty) | 0.5 |
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