Olympiad problems

1975 AMC 12/AHSME, 14

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If the $ whatsis$ is $ so$ when the $ whosis$ is $ is$ and the $ so$ and $ so$ is $ is \cdot so$, what is the $ whosis \cdot whatsis$ when the $ whosis$ is $ so$, the $ so$ and $ so$ is $ so \cdot so$ and the $ is$ is two ($ whatsis$, $ whosis$, $ is$ and $ so$ are variables taking positive values)? $ \textbf{(A)}\ whosis \cdot is \cdot so \qquad \textbf{(B)}\ whosis \qquad \textbf{(C)}\ is \qquad \textbf{(D)}\ so \qquad \textbf{(E)}\ so \text{ and } so$

2019 IMEO, 5

Find all pairs of positive integers $(s, t)$, so that for any two different positive integers $a$ and $b$ there exists some positive integer $n$, for which $$a^s + b^t | a^n + b^{n+1}.$$ [i]Proposed by Oleksii Masalitin (Ukraine)[/i]

2021-2022 OMMC, 14

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The corners of a $2$-dimensional room in the shape of an isosceles right triangle are labeled $A$, $B$, $C$ where $AB = BC$. Walls $BC$ and $CA$ are mirrors. A laser is shot from $A$, hits off of each of the mirrors once and lands at a point $X$ on $AB$. Let $Y$ be the point where the laser hits off $AC$. If $\tfrac{AB}{AX} = 64$, $\tfrac{CA}{AY} = \tfrac pq$ for coprime positive integers $p$, $q$. Find $p + q$. [i]Proposed by Sid Doppalapudi[/i]

2005 Greece Junior Math Olympiad, 2

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If $f(n)=\frac{2n+1+\sqrt{n(n+1)}}{\sqrt{n+1}+\sqrt{n}}$ for all positive integers $n$, evaluate (a) $f(1)$, (b) the sum $A=f(1)+f(2)+...+f(400)$.

1983 AIME Problems, 13

Tags: probability , algebra , binomial theorem

For $\{1, 2, 3, \dots, n\}$ and each of its nonempty subsets a unique [b]alternating sum[/b] is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. (For example, the alternating sum for $\{1, 2, 4, 6, 9\}$ is $9 - 6 + 4 - 2 + 1 = 6$ and for $\{5\}$ it is simply 5.) Find the sum of all such alternating sums for $n = 7$.

2008 China Team Selection Test, 1

Tags: trigonometry , circumcircle , ratio , projective geometry , geometry

Let $ ABC$ be a triangle, let $ AB > AC$. Its incircle touches side $ BC$ at point $ E$. Point $ D$ is the second intersection of the incircle with segment $ AE$ (different from $ E$). Point $ F$ (different from $ E$) is taken on segment $ AE$ such that $ CE \equal{} CF$. The ray $ CF$ meets $ BD$ at point $ G$. Show that $ CF \equal{} FG$.

2016 Saint Petersburg Mathematical Olympiad, 1

Tags: product , number theory , divisor

Sasha multiplied all the divisors of the natural number $n$. Fedya increased each divider by $1$, and then multiplied the results. If the product found Fedya is divided by the product found by Sasha , what can $n$ be equal to ?

PEN P Problems, 24

Tags: additive number theory

Show that any integer can be expressed as the form $a^{2}+b^{2}-c^{2}$, where $a, b, c \in \mathbb{Z}$.

2000 Harvard-MIT Mathematics Tournament, 6

Tags: probability

$6$ people each have a hat. If they shuffle their hats and redistribute them, what is the probability that exactly one person gets their own hat back?

1999 AIME Problems, 5

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For any positive integer $x$, let $S(x)$ be the sum of the digits of $x$, and let $T(x)$ be $|S(x+2)-S(x)|.$ For example, $T(199)=|S(201)-S(199)|=|3-19|=16.$ How many values $T(x)$ do not exceed 1999?

2020 Brazil National Olympiad, 2

Tags: quadratic equation , quadratic , number theory

The following sentece is written on a board: [center]The equation $x^2-824x+\blacksquare 143=0$ has two integer solutions.[/center] Where $\blacksquare$ represents algarisms of a blurred number on the board. What are the possible equations originally on the board?

2017 AIME Problems, 8

Tags: algebra , polynomial

Find the number of positive integers $n$ less than $2017$ such that \[ 1+n+\frac{n^2}{2!}+\frac{n^3}{3!}+\frac{n^4}{4!}+\frac{n^5}{5!}+\frac{n^6}{6!} \] is an integer.

2016 JBMO TST - Turkey, 2

Tags: combinatorics

A and B plays a game on a pyramid whose base is a $2016$-gon. In each turn, a player colors a side (which was not colored before) of the pyramid using one of the $k$ colors such that none of the sides with a common vertex have the same color. If A starts the game, find the minimal value of $k$ for which $B$ can guarantee that all sides are colored.

2010 India IMO Training Camp, 11

Tags: function , algebra

Find all functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that $f(x+y)+xy=f(x)f(y)$ for all reals $x, y$

2021 Brazil National Olympiad, 5

Tags: number theory , equation , square , triplets

Find all triples of non-negative integers $(a, b, c)$ such that \[a^{2}+b^{2}+c^{2} = a b c+1.\]

2024 Assara - South Russian Girl's MO, 7

Tags: number theory

Find all positive integers $n$ for such the following condition holds: "If $a$, $b$ and $c$ are positive integers such are all numbers \[ a^2+2ab+b^2,\ b^2+2bc+c^2, \ c^2+2ca+a^2 \] are divisible by $n$, then $(a+b+c)^2$ is also divisible by $n$." [i]G.M.Sharafetdinova[/i]

1991 Tournament Of Towns, (291) 1

Tags: number theory , diophantine , diophantine equation

Find all natural numbers $n$, and all integers $x,y$ ($x\ne y$) for which the following equation is satisfied: $$x + x^2 + x^4 + ...+ x^{2^n} = y + y^2 + y^4 + ... + y^{2^n} .$$

2023 Canadian Mathematical Olympiad Qualification, 2

Tags: combinatorics

How many ways are there to fill a $3 \times 3$ grid with the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, and $9$, such that the set of three elements in every row and every column form an arithmetic progression in some order? (Each number must be used exactly once)

2007 Tournament Of Towns, 5

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The audience chooses two of five cards, numbered from $1$ to $5$ respectively. The assistant of a magician chooses two of the remaining three cards, and asks a member of the audience to take them to the magician, who is in another room. The two cards are presented to the magician in arbitrary order. By an arrangement with the assistant beforehand, the magician is able to deduce which two cards the audience has chosen only from the two cards he receives. Explain how this may be done.

2006 Baltic Way, 2

Tags: function , inequalities

Suppose that the real numbers $a_i\in [-2,17],\ i=1,2,\ldots,59,$ satisfy $a_1+a_2+\ldots+a_{59}=0.$ Prove that \[a_1^2+a_2^2+\ldots+a_{59}^2\le 2006\]

2019 Durer Math Competition Finals, 13

Tags: combinatorics

There are $12$ chairs arranged in a circle, numbered from $ 1$ to $ 12$. How many ways are there to select some of the chairs in such a way that our selection includes $3$ consecutive chairs somewhere?

2006 MOP Homework, 6

Tags: number theory

Find all integers $n$ for which there exists an equiangular $n$-gon whose side lengths are distinct rational numbers.

1993 Irish Math Olympiad, 2

Tags: prime number , number theory

A positive integer $ n$ is called $ good$ if it can be uniquely written simultaneously as $ a_1\plus{}a_2\plus{}...\plus{}a_k$ and as $ a_1 a_2...a_k$, where $ a_i$ are positive integers and $ k \ge 2$. (For example, $ 10$ is good because $ 10\equal{}5\plus{}2\plus{}1\plus{}1\plus{}1\equal{}5 \cdot 2 \cdot 1 \cdot 1 \cdot 1$ is a unique expression of this form). Find, in terms of prime numbers, all good natural numbers.

1993 AMC 12/AHSME, 13

Tags: perimeter , geometry

A square of perimeter $20$ is inscribed in a square of perimeter $28$. What is the greatest distance between a vertex of the inner square and a vertex of the outer square? $ \textbf{(A)}\ \sqrt{58} \qquad\textbf{(B)}\ \frac{7\sqrt{5}}{2} \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ \sqrt{65} \qquad\textbf{(E)}\ 5\sqrt{3} $

2018 CMIMC Number Theory, 10

Tags: algebra , polynomial

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 $S$.