Olympiad problems

1991 IMO Shortlist, 17

Tags: modular arithmetic , algebra , number theory , equation , IMO Shortlist

Find all positive integer solutions $ x, y, z$ of the equation $ 3^x \plus{} 4^y \equal{} 5^z.$

2012 Pre - Vietnam Mathematical Olympiad, 2

Tags: limit , topology , algebra proposed , algebra

Compute $\mathop {\lim }\limits_{n \to \infty } \left\{ {{{\left( {2 + \sqrt 3 } \right)}^n}} \right\}$

1992 Tournament Of Towns, (352) 1

Tags: number theory , consecutive , Perfect Square

Prove that there exists a sequence of $100$ different integers such that the sum of the squares of any two consecutive terms is a perfect square. (S Tokarev)

2021 Taiwan TST Round 1, 3

Tags: Diophantine equation , number theory , Taiwan

Find all triples $(x, y, z)$ of positive integers such that \[x^2 + 4^y = 5^z. \] [i]Proposed by Li4 and ltf0501[/i]

2019 BMT Spring, 10

Tags: number theory

Compute the remainder when the product of all positive integers less than and relatively prime to $2019$ is divided by $2019$.

2024 Dutch IMO TST, 1

Tags: number theory , number theory proposed , Divisors , coprime

For a positive integer $n$, let $\alpha(n)$ be the arithmetic mean of the divisors of $n$, and let $\beta(n)$ be the arithmetic mean of the numbers $k \le n$ with $\text{gcd}(k,n)=1$. Determine all positive integers $n$ with $\alpha(n)=\beta(n)$.

2008 Germany Team Selection Test, 1

Tags: inequalities , algebra unsolved , algebra

Determine $ Q \in \mathbb{R}$ which is so big that a sequence with non-negative reals elements $ a_1 ,a_2, \ldots$ which satisfies the following two conditions: [b](i)[/b] $ \forall m,n \geq 1$ we have $ a_{m \plus{} n} \leq 2 \left(a_m \plus{} a_n \right)$ [b](ii)[/b] $ \forall k \geq 0$ we have $ a_{2^k} \leq \frac {1}{(k \plus{} 1)^{2008}}$ such that for each sequence element we have the inequality $ a_n \leq Q.$

2021 Bulgaria EGMO TST, 4

Tags: induction , geometry , combinatorics proposed , combinatorics

In a convex $n$-gon, several diagonals are drawn. Among these diagonals, a diagonal is called [i]good[/i] if it intersects exactly one other diagonal drawn (in the interior of the $n$-gon). Find the maximum number of good diagonals.

2017 AMC 8, 19

Tags: AMC 8

For any positive integer $M$, the notation $M!$ denotes the product of the integers $1$ through $M$. What is the largest integer $n$ for which $5^n$ is a factor of the sum $98!+99!+100!$ ? $\textbf{(A) }23\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$

2018 Poland - Second Round, 6

Tags: algebra , analysis , Poland , Sequence

Let $k$ be a positive integer and $a_1, a_2, ...$ be a sequence of terms from set $\{ 0, 1, ..., k \}$. Let $b_n = \sqrt[n] {a_1^n + a_2^n + ... + a_n^n}$ for all positive integers $n$. Prove, that if in sequence $b_1, b_2, b_3, ...$ are infinitely many integers, then all terms of this series are integers.

2008 District Olympiad, 1

Tags: geometry , rhombus , Plane , 3D geometry , tetrahedron , square

A regular tetrahedron is sectioned with a plane after a rhombus. Prove that the rhombus is square.

2013 QEDMO 13th or 12th, 9

Tags: number theory , primes

Are there infinitely many different natural numbers $a_1,a_2, a_3,...$ so that for every integer $k$ only finitely many of the numbers $a_1 + k$,$a_2 + k$,$a_3 + k$,$...$ are numbers prime?

2021 Azerbaijan EGMO TST, 4

Tags: algebra

Let $P(x), Q(x)$ be distinct polynomials of degree $2020$ with non-zero coefficients. Suppose that they have $r$ common real roots counting multiplicity and $s$ common coefficients. Determine the maximum possible value of $r + s$. [i]Demetres Christofides, Cyprus[/i]

2009 Postal Coaching, 5

Tags: number theory , least common multiple , LCM , inequalities

For positive integers $n, k$ with $1 \le k \le n$, define $$L(n, k) = Lcm \,(n, n - 1, n -2, ..., n - k + 1)$$ Let $f(n)$ be the largest value of $k$ such that $L(n, 1) < L(n, 2) < ... < L(n, k)$. Prove that $f(n) < 3\sqrt{n}$ and $f(n) > k$ if $n > k! + k$.

1995 Cono Sur Olympiad, 2

Tags:

There are ten points marked on a circumference, numbered from $1$ to $10$ and join all points with segments. I color the segments, with red someones and others with blue. Without changing the colors of the segments, renumber all the points from the $1$ to the $10$. Will be possible to color the segments and to renumber the points so that those numbers that were jointed with red are jointed now with blue and the numbers that were jointed with blue they are jointed now with red?

2008 Balkan MO, 4

Tags: inequalities , algebra , polynomial , modular arithmetic , induction , number theory , relatively prime

Let $ c$ be a positive integer. The sequence $ a_1,a_2,\ldots$ is defined as follows $ a_1\equal{}c$, $ a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3$ for all positive integers $ n$. Find all $ c$ so that there are integers $ k\ge1$ and $ m\ge2$ so that $ a_k^2\plus{}c^3$ is the $ m$th power of some integer.

2018 Iran Team Selection Test, 4

Tags:

Let $ABC$ be a triangle ($\angle A\neq 90^\circ$). $BE,CF$ are the altitudes of the triangle. The bisector of $\angle A$ intersects $EF,BC$ at $M,N$. Let $P$ be a point such that $MP\perp EF$ and $NP\perp BC$. Prove that $AP$ passes through the midpoint of $BC$. [i]Proposed by Iman Maghsoudi, Hooman Fattahi[/i]

2022 Philippine MO, 2

Tags: number theory , combinatorics , permutations , PMO

The PMO Magician has a special party game. There are $n$ chairs, labelled $1$ to $n$. There are $n$ sheets of paper, labelled $1$ to $n$. [list] [*] On each chair, she attaches exactly one sheet whose number does not match the number on the chair. [*] She then asks $n$ party guests to sit on the chairs so that each chair has exactly one occupant. [*] Whenever she claps her hands, each guest looks at the number on the sheet attached to their current chair, and moves to the chair labelled with that number. [/list] Show that if $1 < m \leq n$, where $m$ is not a prime power, it is always possible for the PMO Magician to choose which sheet to attach to each chair so that everyone returns to their original seats after exactly $m$ claps.

2021 CMIMC Integration Bee, 10

Tags: calculus , integration , CMIMC , integration bee

$$\int_{-\infty}^\infty\frac{x\arctan(x)}{x^4+1}\,dx$$ [i]Proposed by Connor Gordon[/i]

OMMC POTM, 2023 1

Tags: Coloring , combinatorics , ommc

Define a $100 \times 100$ square grid $G$. Initially color all cells of $G$ white. A move consists of selecting a $1 \times 7$ or $7 \times 1$ subgrid of $G$ and flipping the colors of all cells in this subgrid from white to black or vice versa. Is it possible that after a series of moves, all cells are colored black? [i]Proposed by Evan Chang (squareman), USA[/i]

2016 Puerto Rico Team Selection Test, 1

Tags: algebra

The integers $1, 2, 3,. . . , 2016$ are written in a board. You can choose any pair of numbers in the board and replace them with their average. For example, you can replace $1$ and $2$ with $1.5$, or you can replace $1$ and $3$ with a second copy of $2$. After such replacements, the board will have only one number. (a) Prove that there is a sequence of substitutions that will make let the final number be $2$. (b) Prove that there is a sequence of substitutions that will make let the final number be $1000$.

2021 AIME Problems, 11

Tags: AIME , AIME II

A teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher announced to the class that the numbers in $S$ were four consecutive two-digit positive integers, that some number in $S$ was divisible by $6$, and a different number in $S$ was divisible by $7$. The teacher then asked if any of the students could deduce what $S$ is, but in unison, all of the students replied no. However, upon hearing that all four students replied no, each student was able to determine the elements of $S$. Find the sum of all possible values of the greatest element of $S$.

2019 LIMIT Category B, Problem 9

Tags: trigonometry , algebra

The number of solutions of the equation $\tan x+\sec x=2\cos x$, where $0\le x\le\pi$, is $\textbf{(A)}~0$ $\textbf{(B)}~1$ $\textbf{(C)}~2$ $\textbf{(D)}~3$

2021 Science ON all problems, 2

Tags: Rings , abstract algebra , superior algebra

Consider an odd prime $p$. A comutative ring $(A,+, \cdot)$ has the property that $ab=0$ implies $a^p=0$ or $b^p=0$. Moreover, $\underbrace{1+1+\cdots +1}_{p \textnormal{ times}} =0$. Take $x,y\in A$ such that there exist $m,n\geq 1$, $m\neq n$ with $x+y=x^my=x^ny$, and also $y$ is not invertible. \\ \\ $\textbf{(a)}$ Prove that $(a+b)^p=a^p+b^p$ and $(a+b)^{p^2}=a^{p^2}+b^{p^2}$ for all $a,b\in A$.\\ $\textbf{(b)}$ Prove that $x$ and $y$ are nilpotent.\\ $\textbf{(c)}$ If $y$ is invertible, does the conclusion that $x$ is nilpotent stand true? \\ \\ [i] (Bogdan Blaga)[/i]

1982 National High School Mathematics League, 9

Tags: geometry , 3D geometry , tetrahedron

In tetrahedron $SABC$, $\angle ASB=\frac{\pi}{2}, \angle ASC=\alpha(0<\alpha<\frac{\pi}{2}), \angle BSC=\beta(0<\beta<\frac{\pi}{2})$. Let $\theta=A-SC-B$, prove that $\theta=-\arccos(\cot\alpha\cdot\cot\beta)$.