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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) |
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 |
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} |
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 |
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} |
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 |
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 |
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 |
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 |
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 |
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} |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
$a_1=-1$ , $a_2=2$ , and $a_n=\frac {a_{n-1}}{a_{n-2}}$ for $n\geq 3$ . What is $a_{2006}$ ? $
\textbf{(A)}\ -2
\qquad\textbf{(B)}\ -1
\qquad\textbf{(C)}\ -\frac 12
\qquad\textbf{(D)}\ \frac 12
\qquad\textbf{(E)}\ 2
$ | 2 |
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 |
$$ \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\dots+\frac{1}{2014\cdot2015}=\frac{m}{n}, $$ where $\frac{m}{n}$ is irreducible.
a) Find $m+n.$ b) Find the remainder of division of $(m+3)^{1444}$ to $n{}$ . | 16 |
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 |
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 |
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 |
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 |
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 |
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 |
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} |
Find the number of integer $n$ from the set $\{2000,2001,...,2010\}$ such that $2^{2n} + 2^n + 5$ is divisible by $7$ (A): $0$ , (B): $1$ , (C): $2$ , (D): $3$ , (E) None of the above. | 4 |
Determine the smallest positive number $a$ such that the number of all integers belonging to $(a, 2016a]$ is $2016$ . | \frac{2017}{2016} |
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 |
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 |
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 |
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 |
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 |
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 |
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} |
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 |
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 |
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 |
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} |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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! |
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} |
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 |
Inside of the square $ABCD$ the point $P$ is given such that $|PA|:|PB|:|PC|=1:2:3$ . Find $\angle APB$ . | 135^\circ |
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 |
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 |
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 |
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 |
The number $2^{1997}$ has $m$ decimal digits, while the number $5^{1997}$ has $n$ digits. Evaluate $m+n$ . | 1998 |
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 |
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 |
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 |
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 |
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 |
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) |
$ABC$ is a triangle with $AB = 33$ , $AC = 21$ and $BC = m$ , an integer. There are points $D$ , $E$ on the sides $AB$ , $AC$ respectively such that $AD = DE = EC = n$ , an integer. Find $m$ .
| 30 |
Find the number of permutations of the letters $ABCDE$ where the letters $A$ and $B$ are not adjacent and the letters $C$ and $D$ are not adjacent. For example, count the permutations $ACBDE$ and $DEBCA$ but not $ABCED$ or $EDCBA$ . | 48 |
Find the least positive integer $k$ so that $k + 25973$ is a palindrome (a number which reads the same forward and backwards). | 89 |
Let $ p > 2$ be a prime number. Find the least positive number $ a$ which can be represented as
\[ a \equal{} (X \minus{} 1)f(X) \plus{} (X^{p \minus{} 1} \plus{} X^{p \minus{} 2} \plus{} \cdots \plus{} X \plus{} 1)g(X),
\]
where $ f(X)$ and $ g(X)$ are integer polynomials.
*Mircea Becheanu*. | p |
A function $f$ is defined for all real numbers and satisfies \[f(2 + x) = f(2 - x)\qquad\text{and}\qquad f(7 + x) = f(7 - x)\] for all real $x$ . If $x = 0$ is a root of $f(x) = 0$ , what is the least number of roots $f(x) = 0$ must have in the interval $-1000 \le x \le 1000$ ? | 401 |
For each positive integer $ k$ , find the smallest number $ n_{k}$ for which there exist real $ n_{k}\times n_{k}$ matrices $ A_{1}, A_{2}, \ldots, A_{k}$ such that all of the following conditions hold:
(1) $ A_{1}^{2}= A_{2}^{2}= \ldots = A_{k}^{2}= 0$ ,
(2) $ A_{i}A_{j}= A_{j}A_{i}$ for all $ 1 \le i, j \le k$ , and
(3) $ A_{1}A_{2}\ldots A_{k}\ne 0$ . | n_k = 2^k |
There is a pile with $15$ coins on a table. At each step, Pedro choses one of the piles in the table with $a>1$ coins and divides it in two piles with $b\geq1$ and $c\geq1$ coins and writes in the board the product $abc$ . He continues until there are $15$ piles with $1$ coin each. Determine all possible values that the final sum of the numbers in the board can have. | 1120 |
In right triangle $ ABC$ with right angle at $ C$ , $ \angle BAC < 45$ degrees and $ AB \equal{} 4$ . Point $ P$ on $ AB$ is chosen such that $ \angle APC \equal{} 2\angle ACP$ and $ CP \equal{} 1$ . The ratio $ \frac{AP}{BP}$ can be represented in the form $ p \plus{} q\sqrt{r}$ , where $ p,q,r$ are positive integers and $ r$ is not divisible by the square of any prime. Find $ p\plus{}q\plus{}r$ . | 7 |
Suppose $a$ is a real number such that $\sin(\pi \cdot \cos a) = \cos(\pi \cdot \sin a)$ . Evaluate $35 \sin^2(2a) + 84 \cos^2(4a)$ . | 21 |
Given that the base- $17$ integer $\overline{8323a02421_{17}}$ (where a is a base- $17$ digit) is divisible by $\overline{16_{10}}$ , find $a$ . Express your answer in base $10$ .
*Proposed by Jonathan Liu* | 7 |
Wendy randomly chooses a positive integer less than or equal to $2020$ . The probability that the digits in Wendy's number add up to $10$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ . | 107 |
Suppose $ 2015= a_1 <a_2 < a_3<\cdots <a_k $ be a finite sequence of positive integers, and for all $ m, n \in \mathbb{N} $ and $1\le m,n \le k $ , $$ a_m+a_n\ge a_{m+n}+|m-n| $$ Determine the largest possible value $ k $ can obtain.
| 2016 |
Let A and B be fixed points in the plane with distance AB = 1. An ant walks on a straight
line from point A to some point C in the plane and notices that the distance from itself to B
always decreases at any time during this walk. Compute the area of the region in the plane
containing all points where point C could possibly be located. | \frac{\pi}{4} |
A circle of radius $4$ is inscribed in a triangle $ABC$ . We call $D$ the touchpoint between the circle and side BC. Let $CD =8$ , $DB= 10$ . What is the length of the sides $AB$ and $AC$ ? | 12.5 |
The sequence $ (a_n)$ satisfies $ a_1 \equal{} 1$ and $ \displaystyle 5^{(a_{n\plus{}1}\minus{}a_n)} \minus{} 1 \equal{} \frac{1}{n\plus{}\frac{2}{3}}$ for $ n \geq 1$ . Let $ k$ be the least integer greater than $ 1$ for which $ a_k$ is an integer. Find $ k$ . | 41 |
The four faces of a tetrahedral die are labelled $0, 1, 2,$ and $3,$ and the die has the property that, when it is rolled, the die promptly vanishes, and a number of copies of itself appear equal to the number on the face the die landed on. For example, if it lands on the face labelled $0,$ it disappears. If it lands on the face labelled $1,$ nothing happens. If it lands on the face labelled $2$ or $3,$ there will then be $2$ or $3$ copies of the die, respectively (including the original). Suppose the die and all its copies are continually rolled, and let $p$ be the probability that they will all eventually disappear. Find $\left\lfloor \frac{10}{p} \right\rfloor$ . | 24 |
Find the natural numbers $ n\ge 2 $ which have the property that the ring of integers modulo $ n $ has exactly an element that is not a sum of two squares. | 4 |
Find the number of pairs of integers $x, y$ with different parities such that $\frac{1}{x}+\frac{1}{y} = \frac{1}{2520}$ . | 90 |
Let $ \theta_1, \theta_2,\ldots , \theta_{2008}$ be real numbers. Find the maximum value of
$ \sin\theta_1\cos\theta_2 \plus{} \sin\theta_2\cos\theta_3 \plus{} \ldots \plus{} \sin\theta_{2007}\cos\theta_{2008} \plus{} \sin\theta_{2008}\cos\theta_1$ | 1004 |
Let $n$ and $k$ be integers satisfying $\binom{2k}{2} + n = 60$ . It is known that $n$ days before Evan's 16th birthday, something happened. Compute $60-n$ .
*Proposed by Evan Chen* | 45 |
Let be a natural number $ n\ge 3. $ Find $$ \inf_{\stackrel{ x_1,x_2,\ldots ,x_n\in\mathbb{R}_{>0}}{1=P\left( x_1,x_2,\ldots ,x_n\right)}}\sum_{i=1}^n\left( \frac{1}{x_i} -x_i \right) , $$ where $ P\left( x_1,x_2,\ldots ,x_n\right) :=\sum_{i=1}^n \frac{1}{x_i+n-1} , $ and find in which circumstances this infimum is attained. | 0 |
Max has a light bulb and a defective switch. The light bulb is initially off, and on the $n$ th time the switch is flipped, the light bulb has a $\tfrac 1{2(n+1)^2}$ chance of changing its state (i.e. on $\to$ off or off $\to$ on). If Max flips the switch 100 times, find the probability the light is on at the end.
*Proposed by Connor Gordon* | \frac{25}{101} |
In the city built are $2019$ metro stations. Some pairs of stations are connected. tunnels, and from any station through the tunnels you can reach any other. The mayor ordered to organize several metro lines: each line should include several different stations connected in series by tunnels (several lines can pass through the same tunnel), and in each station must lie at least on one line. To save money no more than $k$ lines should be made. It turned out that the order of the mayor is not feasible. What is the largest $k$ it could to happen? | 1008 |
Let $k$ be an integer. If the equation $(x-1)|x+1|=x+\frac{k}{2020}$ has three distinct real roots, how many different possible values of $k$ are there? | 4544 |
Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$ .
| 440 |
Let $ABC$ be a triangle and $I$ its incenter. Suppose $AI=\sqrt{2}$ , $BI=\sqrt{5}$ , $CI=\sqrt{10}$ and the inradius is $1$ . Let $A'$ be the reflection of $I$ across $BC$ , $B'$ the reflection across $AC$ , and $C'$ the reflection across $AB$ . Compute the area of triangle $A'B'C'$ . | \frac{24}{5} |
The sequences $(a_{n})$ , $(b_{n})$ are defined by $a_{1} = \alpha$ , $b_{1} = \beta$ , $a_{n+1} = \alpha a_{n} - \beta b_{n}$ , $b_{n+1} = \beta a_{n} + \alpha b_{n}$ for all $n > 0.$ How many pairs $(\alpha, \beta)$ of real numbers are there such that $a_{1997} = b_{1}$ and $b_{1997} = a_{1}$ ? | 1999 |
Let $n$ be a natural number. Find the least natural number $k$ for which there exist $k$ sequences of $0$ and $1$ of length $2n+2$ with the following property: any sequence of $0$ and $1$ of length $2n+2$ coincides with some of these $k$ sequences in at least $n+2$ positions. | k = 4 |
In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$ . Determine the length of the hypotenuse. | 994010 |
Let $S_{n}=\{1,n,n^{2},n^{3}, \cdots \}$ , where $n$ is an integer greater than $1$ . Find the smallest number $k=k(n)$ such that there is a number which may be expressed as a sum of $k$ (possibly repeated) elements in $S_{n}$ in more than one way. (Rearrangements are considered the same.) | k(n) = n + 1 |
The sum
\[ \sum_{k=0}^{\infty} \frac{2^{k}}{5^{2^{k}}+1}\]
can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$ . | 5 |
Compute the number of two digit positive integers that are divisible by both of their digits. For example, $36$ is one of these two digit positive integers because it is divisible by both $3$ and $6$ .
*2021 CCA Math Bonanza Lightning Round #2.4* | 14 |
A permutation of a finite set $S$ is a one-to-one function from $S$ to $S$ . A permutation $P$ of the set $\{ 1, 2, 3, 4, 5 \}$ is called a W-permutation if $P(1) > P(2) < P(3) > P(4) < P(5)$ . A permutation of the set $\{1, 2, 3, 4, 5 \}$ is selected at random. Compute the probability that it is a W-permutation. | \frac{2}{15} |
In a game, Jimmy and Jacob each randomly choose to either roll a fair six-sided die or to automatically roll a $1$ on their die. If the product of the two numbers face up on their dice is even, Jimmy wins the game. Otherwise, Jacob wins. The probability Jimmy wins $3$ games before Jacob wins $3$ games can be written as $\tfrac{p}{2^q}$ , where $p$ and $q$ are positive integers, and $p$ is odd. Find the remainder when $p+q$ is divided by $1000$ .
*Proposed by firebolt360* | 360 |
Let $\alpha$ and $\beta$ be positive integers such that $$ \frac{16}{37}<\frac{\alpha}{\beta}<\frac{7}{16} . $$ Find the smallest possible value of $\beta$ . | 23 |
The set $M= \{1;2;3;\ldots ; 29;30\}$ is divided in $k$ subsets such that if $a+b=n^2, (a,b \in M, a\neq b, n$ is an
integer number $)$ , then $a$ and $b$ belong different subsets.
Determine the minimum value of $k$ . | 3 |
Subsets and Splits