Olympiad problems

2019 MOAA, 10

Let $S$ be the set of all four digit palindromes (a palindrome is a number that reads the same forwards and backwards). The average value of $|m - n|$ over all ordered pairs $(m, n)$, where $m$ and $n$ are (not necessarily distinct) elements of $S$, is equal to $p/q$ , for relatively prime positive integers $p$ and $q$. Find $p + q$.

MOAA Team Rounds, 2019.3

Tags: number theory , team , 2019

For how many ordered pairs of positive integers $(a, b)$ such that $a \le 50$ is it true that $x^2 - ax + b$ has integer roots?

MOAA Team Rounds, 2019.8

Tags: algebra , team , 2019

Suppose that $$\frac{(\sqrt2)^5 + 1}{\sqrt2 + 1} \times \frac{2^5 + 1}{2 + 1} \times \frac{4^5 + 1}{4 + 1} \times \frac{16^5 + 1}{16 + 1} =\frac{m}{7 + 3\sqrt2}$$ for some integer $m$. How many $0$’s are in the binary representation of $m$? (For example, the number $20 = 10100_2$ has three $0$’s in its binary representation.)

2019 ASDAN Math Tournament, 2

Tags: Geometry Tiebreaker , 2019

Consider a triangle $\vartriangle ABC$ with $AB = 5$ and $BC = 4$. Let $G$ be the centroid of the triangle, and let $P$ lie on line $AG$ such that $AG = GP$ and $P\ne A$. Suppose that $P$ lies on the circumcircle of $\vartriangle ABC$. Compute $CA$.

MOAA Team Rounds, 2019.4

Tags: combinatorics , team , 2019

Brandon wants to split his orchestra of $20$ violins, $15$ violas, $10$ cellos, and $5$ basses into three distinguishable groups, where all of the players of each instrument are indistinguishable. He wants each group to have at least one of each instrument and for each group to have more violins than violas, more violas than cellos, and more cellos than basses. How many ways are there for Brandon to split his orchestra following these conditions?

2019 CMIMC, 14

Tags: 2019 , team , function

Consider the following function. $\textbf{procedure }\textsc{M}(x)$ $\qquad\textbf{if }0\leq x\leq 1$ $\qquad\qquad\textbf{return }x$ $\qquad\textbf{return }\textsc{M}(x^2\bmod 2^{32})$ Let $f:\mathbb N\to\mathbb N$ be defined such that $f(x) = 0$ if $\textsc{M}(x)$ does not terminate, and otherwise $f(x)$ equals the number of calls made to $\textsc{M}$ during the running of $\textsc{M}(x)$, not including the initial call. For example, $f(1) = 0$ and $f(2^{31}) = 1$. Compute the number of ones in the binary expansion of \[ f(0) + f(1) + f(2) + \cdots + f(2^{32} - 1). \]

2019 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities , algebra , 2019 , JBMO , romania

Find the maximum value of: $E(a,b)=\frac{a+b}{(4a^2+3)(4b^2+3)}$ For $a,b$ real numbers.

2019 CMIMC, 1

Tags: 2019 , team

David recently bought a large supply of letter tiles. One day he arrives back to his dorm to find that some of the tiles have been arranged to read $\textsc{Central Michigan University}$. What is the smallest number of tiles David must remove and/or replace so that he can rearrange them to read $\textsc{Carnegie Mellon University}$?

2019 Purple Comet Problems, 21

Tags: combinatorics , 2019

Each of the $48$ faces of eight $1\times 1\times 1$ cubes is randomly painted either blue or green. The probability that these eight cubes can then be assembled into a $2\times 2\times 2$ cube in a way so that its surface is solid green can be written $\frac{p^m}{q^n}$ , where $p$ and $q$ are prime numbers and $m$ and $n$ are positive integers. Find $p + q + m + n$.

2019 CMIMC, 13

Tags: 2019 , team

Points $A$, $B$, and $C$ lie in the plane such that $AB=13$, $BC=14$, and $CA=15$. A peculiar laser is fired from $A$ perpendicular to $\overline{BC}$. After bouncing off $BC$, it travels in a direction perpendicular to $CA$. When it hits $CA$, it travels in a direction perpendicular to $AB$, and after hitting $AB$ its new direction is perpendicular to $BC$ again. If this process is continued indefinitely, the laser path will eventually approach some finite polygonal shape $T_\infty$. What is the ratio of the perimeter of $T_\infty$ to the perimeter of $\triangle ABC$?

2019 CMIMC, 7

Tags: 2019 , algebra , number theory

For all positive integers $n$, let \[f(n) = \sum_{k=1}^n\varphi(k)\left\lfloor\frac nk\right\rfloor^2.\] Compute $f(2019) - f(2018)$. Here $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$.

2019 ASDAN Math Tournament, 4

Tags: geometry , 2019

Suppose $Z, Y$ , and $W$ are points on a circle such that lengths $ZY = Y W$. Extend $ZY$ and let $X$ be a point on $ZY$ where $ZY = Y X$. If $XW$ is a tangent of the circle, what is $\angle W XY$ ?

2019 CMIMC, 2

Tags: number theory , greatest common divisor , 2019 , team

Determine the number of ordered pairs of positive integers $(m,n)$ with $1\leq m\leq 100$ and $1\leq n\leq 100$ such that \[ \gcd(m+1,n+1) = 10\gcd(m,n). \]

2019 CMIMC, 9

Tags: 2019 , team

Let $f:\mathbb{N}\to \mathbb{N}$ be a bijection satisfying $f(ab)=f(a)f(b)$ for all $a,b\in \mathbb{N}$. Determine the minimum possible value of $f(n)/n$, taken over all possible $f$ and all $n\leq 2019$.

2019 JBMO Shortlist, A4

Tags: algebra , Junior , JBMO , 2019 , Balkan , inequalities

Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that $a^4 - 2019a = b^4 - 2019b = c$. Prove that $- \sqrt{c} < ab < 0$.

MOAA Team Rounds, 2019.9

Tags: number theory , team , 2019

Jonathan finds all ordered triples $(a, b, c)$ of positive integers such that $abc = 720$. For each ordered triple, he writes their sum $a + b + c$ on the board. (Numbers may appear more than once.) What is the sum of all the numbers written on the board?

MOAA Team Rounds, 2019.6

Tags: floor function , ceiling function , algebra , team , 2019

Let $f(x, y) = \left\lfloor \frac{5x}{2y} \right\rfloor + \left\lceil \frac{5y}{2x} \right\rceil$. Suppose $x, y$ are chosen independently uniformly at random from the interval $(0, 1]$. Let $p$ be the probability that $f(x, y) < 6$. If $p$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$, compute $m + n$. (Note: $\lfloor x\rfloor $ is defined as the greatest integer less than or equal to $x$ and $\lceil x \rceil$ is defined as the least integer greater than or equal to$ x$.)

2019 CMIMC, 6

Tags: 2019 , team , inequalities

Across all $x \in \mathbb{R}$, find the maximum value of the expression $$\sin x + \sin 3x + \sin 5x.$$

2019 ISI Entrance Examination, 2

Tags: isi , Indian Statistical Institute , 2019 , calculus

Let $f:(0,\infty)\to\mathbb{R}$ be defined by $$f(x)=\lim_{n\to\infty}\cos^n\bigg(\frac{1}{n^x}\bigg)$$ [b](a)[/b] Show that $f$ has exactly one point of discontinuity. [b](b)[/b] Evaluate $f$ at its point of discontinuity.

2019 MOAA, 7

Tags: number theory , team , 2019

Suppose $ABC$ is a triangle inscribed in circle $\omega$ . Let $A'$ be the point on $\omega$ so that $AA'$ is a diameter, and let $G$ be the centroid of $ABC$. Given that $AB = 13$, $BC = 14$, and $CA = 15$, let $x$ be the area of triangle $AGA'$ . If $x$ can be expressed in the form $m/n$ , where m and n are relatively prime positive integers, compute $100n + m$.

2019 India National OIympiad, 5

Tags: geometry , INMO , 2019 , P5

Let $AB$ be the diameter of a circle $\Gamma$ and let $C$ be a point on $\Gamma$ different from $A$ and $B$. Let $D$ be the foot of perpendicular from $C$ on to $AB$.Let $K$ be a point on the segment $CD$ such that $AC$ is equal to the semi perimeter of $ADK$.Show that the excircle of $ADK$ opposite $A$ is tangent to $\Gamma$.

2019 Junior Balkan MO, 4

Tags: combinatorics , Junior , Balkan , JBMO , 2019

A $5 \times 100$ table is divided into $500$ unit square cells, where $n$ of them are coloured black and the rest are coloured white. Two unit square cells are called [i]adjacent[/i] if they share a common side. Each of the unit square cells has at most two adjacent black unit square cells. Find the largest possible value of $n$.

2019 MOAA, 5

Tags: geometry , team , 2019

Let $ABC$ be a triangle with $AB = AC = 10$ and $BC = 12$. Define $\ell_A$ as the line through $A$ perpendicular to $\overline{AB}$. Similarly, $\ell_B$ is the line through $B$ perpendicular to $\overline{BC}$ and $\ell_C$ is the line through $C$ perpendicular to $\overline{CA}$. These three lines $\ell_A, \ell_B, \ell_C$ form a triangle with perimeter $m/n$ for relatively prime positive integers $m$ and $n$. Find $m + n$.

2019 CMIMC, 7

Tags: 2019 , team

Suppose you start at $0$, a friend starts at $6$, and another friend starts at $8$ on the number line. Every second, the leftmost person moves left with probability $\tfrac14$, the middle person with probability $\tfrac13$, and the rightmost person with probability $\tfrac12$. If a person does not move left, they move right, and if two people are on the same spot, they are randomly assigned which one of the positions they are. Determine the expected time until you all meet in one point.

2019 Iranian Geometry Olympiad, 3

Tags: IGO , Iran , geometry , projective geometry , cyclic quadrilateral , 2019

Circles $\omega_1$ and $\omega_2$ have centres $O_1$ and $O_2$, respectively. These two circles intersect at points $X$ and $Y$. $AB$ is common tangent line of these two circles such that $A$ lies on $\omega_1$ and $B$ lies on $\omega_2$. Let tangents to $\omega_1$ and $\omega_2$ at $X$ intersect $O_1O_2$ at points $K$ and $L$, respectively. Suppose that line $BL$ intersects $\omega_2$ for the second time at $M$ and line $AK$ intersects $\omega_1$ for the second time at $N$. Prove that lines $AM, BN$ and $O_1O_2$ concur. [i]Proposed by Dominik Burek - Poland[/i]