This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 116

2018 CMIMC Combinatorics, 4
Tags: 2018 , combinatorics
At CMU, the A and the B buses arrive once every 20 and 18 minutes, respectively. Kevin prefers the A bus but does not want to wait for too long. He commits to the following waiting scheme: he will take the first A bus that arrives, but after waiting for five minutes he will take the next bus that comes, no matter what it is. Determine the probability that he ends up on an A bus.

2018 CMIMC Algebra, 10
Tags: 2018 , algebra
Define a sequence of polynomials $F_n(x)$ by $F_0(x)=0, F_1(x)=x-1$, and for $n\geq 1$, $$F_{n+1}(x)=2xF_n(x)-F_{n-1}(x)+2F_1(x).$$ For each $n$, $F_n(x)$ can be written in the form $$F_n(x)=c_nP_1(x)P_2(x)\cdots P_{g(n)}(x)$$ where $c_n$ is a constant and $P_1(x),P_2(x)\cdots, P_{g(n)}(x)$ are non-constant polynomials with integer coefficients and $g(n)$ is as large as possible. For all $2< n< 101$, let $t$ be the minimum possible value of $g(n)$ in the above expression; for how many $k$ in the specified range is $g(k)=t$?

2018 CMIMC CS, 2
Tags: 2018 , computer science
Consider the natural implementation of computing Fibonacci numbers: \begin{tabular}{l} 1: \textbf{FUNCTION} $\text{FIB}(n)$: \\ 2:$\qquad$ \textbf{IF} $n = 0$ \textbf{OR} $n = 1$ \textbf{RETURN} 1 \\ 3:$\qquad$ \textbf{RETURN} $\text{FIB}(n-1) + \text{FIB}(n-2)$ \end{tabular} When $\text{FIB}(10)$ is evaluated, how many recursive calls to $\text{FIB}$ occur?

2019 Philippine TST, 4
Tags: number theory , Arithmetic Function , IMO Shortlist , number of divisors , 2018 , divisor
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2018 ASDAN Math Tournament, 10
Tags: 2018 , Algebra Test
Compute the unique value of $\theta$, in degrees, where $0^\circ<\theta<90^\circ$, such that $$\csc\theta=\sum_{i=3}^{11}\csc(2^i)^\circ.$$

2018 CMIMC Algebra, 7
Tags: 2018 , algebra , trigonometry
Compute \[\sum_{k=0}^{2017}\dfrac{5+\cos\left(\frac{\pi k}{1009}\right)}{26+10\cos\left(\frac{\pi k}{1009}\right)}.\]

2018 CMIMC Combinatorics, 2
Tags: 2018 , combinatorics
Compute the number of ways to rearrange nine white cubes and eighteen black cubes into a $3\times 3\times 3$ cube such that each $1\times1\times3$ row or column contains exactly one white cube. Note that rotations are considered $\textit{distinct}$.

2018 ASDAN Math Tournament, 2
Tags: 2018 , Algebra Test
Given that $x$ is a real number, compute the minimum possible value of $(x-20)^2 + (x-18)^2$.

2018 ASDAN Math Tournament, 1
Tags: 2018 , Algebra Tiebreaker
Given that $x\ge0$, $y\ge0$, $x+2y\le6$, and $2x+y\le6$, compute the maximum possible value of $x+y$.

2018 ASDAN Math Tournament, 2
Tags: Geometry Test , 2018
The intersection of $2$ cubes of side length $5$ is a cube of side length $3$. Compute the surface area of the entire figure.

2018 CMIMC Algebra, 5
Tags: 2018 , algebra , system of equations
Suppose $a$, $b$, and $c$ are nonzero real numbers such that \[bc+\frac1a = ca+\frac2b = ab+\frac7c = \frac1{a+b+c}.\] Find $a+b+c$.

2018 CMIMC Algebra, 4
Tags: 2018 , algebra
2018 little ducklings numbered 1 through 2018 are standing in a line, with each holding a slip of paper with a nonnegative number on it; it is given that ducklings 1 and 2018 have the number zero. At some point, ducklings 2 through 2017 change their number to equal the average of the numbers of the ducklings to their left and right. Suppose the new numbers on the ducklings sum to 1000. What is the maximum possible sum of the original numbers on all 2018 slips?

2018 CMIMC Combinatorics, 6
Tags: 2018 , combinatorics
Richard rolls a fair six-sided die repeatedly until he rolls his twentieth prime number or his second even number. Compute the probability that his last roll is prime.

2018 ASDAN Math Tournament, 9
Tags: geometry , 2018
A circle inscribed within quadrilateral $ABCD$ is tangent to $AB$ at $E$, to $BC$ at $F$, to $CD$ at $G$, and to $DA$ at $H$. Suppose that $AE = 6$, $EB = 30$, $CG = 10$, and $GD = 2$. Compute $EF^2 + F G^2 + GH^2 + HE^2$. .

2018 ASDAN Math Tournament, 9
Tags: 2018 , Discrete Math Test
Alice starts at the top of Pascal’s triangle. Every move, she moves one layer below, choosing either the left or the right with equal probability. After making $6$ moves, what is the expected sum of the values she visited, including the starting and ending values? For example, in the path shown below, the sum of the values Alice visited is $1 + 1 + 1 + 3 + 6 + 10 + 20 = 42$. [center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMC84L2JjZDhiYjkzNjgyMTczMGQ0ZWIzZjE4NDVkOWIxODQxYzQxODdlLnBuZw==&rn=QS5wbmc=[/img][/center]

2019 Germany Team Selection Test, 1
Tags: number theory , Arithmetic Function , IMO Shortlist , number of divisors , 2018 , divisor
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2018 CMIMC CS, 8
Tags: 2018 , computer science
We consider a simple model for balanced parenthesis checking. Let $\mathcal R=\{\texttt{(())}\rightarrow \texttt{A},\texttt{(A)}\rightarrow\texttt{A},\texttt{AA}\rightarrow\texttt{A}\}$ be a set of rules for phrase reduction. Ideally, any given phrase is balanced if and only if the model is able to reduce the phrase to $\texttt{A}$ by some arbitrary sequence of rule applications. For example, to show $\texttt{((()))}$ is balanced we can perform the following sequence of reductions. \[\texttt{((()))}\rightarrow\texttt{(A)}\rightarrow\texttt{A}\qquad \checkmark\] Unfortunately, the above set of rules $\mathcal R$ is not complete, since there exist parenthetical phrases which are balanced but which are not balanced according to $\mathcal R$. Determine the number of such phrases of length $14$.

2018 ASDAN Math Tournament, 4
Tags: 2018 , Algebra Test
Given that $4^{x_1} = 5, 5^{x_2} = 6, \dots , 2047^{x_{2044}} = 2048$, compute the product $x_1 \dots x_{2044}$.

2018 MOAA, 5
Tags: algebra , team , 2018
Mr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves one problem every $3$ minutes. When he listens to rap music, however, he only solves one problem every $5$ minutes. Mr. DoBa listens to a playlist comprised of $60\%$ classical music and $40\%$ rap music. Each song is exactly $4$ minutes long. Suppose that the expected number of problems he solves in an hour does not depend on whether or not Mr. DoBa is listening to music at any given moment, and let $m$ the average number of problems Mr. DoBa solves per minute when he is not listening to music. Determine the value of $1000m$.

2018 CMI B.Sc. Entrance Exam, 1
Tags: CMI , Chennai Mathematical Institute , 2018 , calculus , number theory
Answer the following questions : $\textbf{(a)}~$ A natural number $k$ is called stable if there exist $k$ distinct natural numbers $a_1, a_2,\cdots, a_k$, each $a_i>1$, such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_k}=1$$ Show that if $k$ is stable, then $(k+1)$ is also stable. Using this or otherwise, find all stable numbers. $\textbf{(b)}$ Let $f$ be a differentiable function defined on a subset $A$ of the real numbers. Define $$f^*(y):=\max_{x\in A} \left\{yx-f(x)\right\}$$ whenever the above maximum is finite. For the function $f(x)=\ln x$, determine the set of points for which $f^*$ is defined and find an expression for $f^*(y)$ involving only $y$ and constants.

2018 CMIMC Combinatorics, 7
Tags: 2018 , combinatorics
Nine distinct light bulbs are placed in a circle, each of which is off. Determine the number of ways to turn on some of the light bulbs in the circle such that no four consecutive bulbs are all off.

MOAA Team Rounds, 2018.1
Tags: geometry , team , 2018
In $\vartriangle ABC$, $AB = 3$, $BC = 5$, and $CA = 6$. Points $D$ and $E$ are chosen such that $ACDE$ is a square which does not overlap with $\vartriangle ABC$. The length of $BD$ can be expressed in the form $\sqrt{m + n\sqrt{p}}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of a prime. Determine the value of $m + n + p$.

2018 ASDAN Math Tournament, 1
Tags: Geometry Tiebreaker , 2018
Point $X$ is placed on segment $AB$ of a regular hexagon $ABCDEF$ such that the ratio of the area of $AXEF$ to the area of $XBCDE$ is $\frac12$. If $AB = 2018$, find $AX$.

2018 ASDAN Math Tournament, 5
Tags: geometry , 2018
In pentagon $ABCDE$, $BE$ intersects $AC$ and $AD$ at $F$ and $G$, respectively. Suppose that $A[\vartriangle AF G] = A[\vartriangle BCF] = A[\vartriangle DEG] = 16$, where$ A[\vartriangle AF G]$ denotes the area of $\vartriangle AF G$. Next, suppose that $BF = 4$, $F G = 5$, and $GE = 6$. Compute $A[ABCDE]$.

2018 IMO, 1
Tags: geometry , IMO , imo 2018 , IMO P1 , 2018
Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line. [i]Proposed by Silouanos Brazitikos, Evangelos Psychas and Michael Sarantis, Greece[/i]