Olympiad problems

1982 All Soviet Union Mathematical Olympiad, 338

Cucumber river in the Flower city has parallel banks with the distance between them $1$ metre. It has some islands with the total perimeter $8$ metres. Mr. Know-All claims that it is possible to cross the river in a boat from the arbitrary point, and the trajectory will not exceed $3$ metres. Is he right?

2014 USA TSTST, 4

Tags: modular arithmetic , linear algebra , floor function , vector , pigeonhole principle , polynomial , algebra

Let $P(x)$ and $Q(x)$ be arbitrary polynomials with real coefficients, and let $d$ be the degree of $P(x)$. Assume that $P(x)$ is not the zero polynomial. Prove that there exist polynomials $A(x)$ and $B(x)$ such that: (i) both $A$ and $B$ have degree at most $d/2$ (ii) at most one of $A$ and $B$ is the zero polynomial. (iii) $\frac{A(x)+Q(x)B(x)}{P(x)}$ is a polynomial with real coefficients. That is, there is some polynomial $C(x)$ with real coefficients such that $A(x)+Q(x)B(x)=P(x)C(x)$.

2018 AMC 10, 19

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A number $m$ is randomly selected from the set $\{11,13,15,17,19\}$, and a number $n$ is randomly selected from $\{1999,2000,2001,\ldots,2018\}$. What is the probability that $m^n$ has a units digit of $1$? $\textbf{(A) } \frac{1}{5} \qquad \textbf{(B) } \frac{1}{4} \qquad \textbf{(C) } \frac{3}{10} \qquad \textbf{(D) } \frac{7}{20} \qquad \textbf{(E) } \frac{2}{5} $

2015 Iran Team Selection Test, 4

Tags: combinatorics , number theory

$n$ is a fixed natural number. Find the least $k$ such that for every set $A$ of $k$ natural numbers, there exists a subset of $A$ with an even number of elements which the sum of it's members is divisible by $n$.

1975 Canada National Olympiad, 3

Tags: function

For each real number $ r$, $ [r]$ denotes the largest integer less than or equal to $ r$, e.g. $ [6] \equal{} 6, [\pi] \equal{} 3, [\minus{}1.5] \equal{} \minus{}2$. Indicate on the $ (x,y)$-plane the set of all points $ (x,y)$ for which $ [x]^2 \plus{} [y]^2 \equal{} 4$.

Estonia Open Junior - geometry, 2013.2.3

Tags: right triangle , geometry , equal angles , isosceles

In an isosceles right triangle $ABC$ the right angle is at vertex $C$. On the side $AC$ points $K, L$ and on the side $BC$ points $M, N$ are chosen so that they divide the corresponding side into three equal segments. Prove that there is exactly one point $P$ inside the triangle $ABC$ such that $\angle KPL = \angle MPN = 45^o$.

2020 Romania EGMO TST, P2

Tags: product , number theory

Let $n$ be a positive integer. Prove that $n^2 + n + 1$ cannot be written as the product of two positive integers of which the difference is smaller than $2\sqrt{n}$.

2023 Junior Balkan Team Selection Tests - Romania, P3

Tags: algebra

Initially the numbers $i^3-i$ for $i=2,3 \ldots 2n+1$ are written on a blackboard, where $n\geq 2$ is a positive integer. On one move we can delete three numbers $a, b, c$ and write the number $\frac{abc} {ab+bc+ca}$. Prove that when two numbers remain on the blackboard, their sum will be greater than $16$.

2025 Euler Olympiad, Round 1, 7

Tags: algebra

Let $s(n)$ be the final value obtained after repeatedly summing the digits of $n$ until a single-digit number is reached. (For example: $s(187) = 7$, because the digit sum of $187$ is $16$ and the digit sum of $16$ is $7$). Evaluate the sum: $$ s(1^2) + s(2^2) + s(3^2) + \ldots + s(2025^2)$$ [i]Proposed by Lia Chitishvili, Georgia [/i]

2018 IOM, 1

Tags: algebra

Solve the system of equations in real numbers: \[ \begin{cases*} (x - 1)(y - 1)(z - 1) = xyz - 1,\\ (x - 2)(y - 2)(z - 2) = xyz - 2.\\ \end{cases*} \] [i]Vladimir Bragin[/i]

2025 China National Olympiad, 5

Tags: prime number , number theory

Let $p$ be a prime number and $f$ be a bijection from $\left\{0,1,\ldots,p-1\right\}$ to itself. Suppose that for integers $a,b \in \left\{0,1,\ldots,p-1\right\}$, $|f(a) - f(b)|\leqslant 2024$ if $p \mid a^2 - b$. Prove that there exists infinite many $p$ such that there exists such an $f$ and there also exists infinite many $p$ such that there doesn't exist such an $f$.

2004 Estonia Team Selection Test, 1

Tags: algebra , functional equation , functional , polynomial

Let $k > 1$ be a fixed natural number. Find all polynomials $P(x)$ satisfying the condition $P(x^k) = (P(x))^k$ for all real numbers $x$.

1993 AMC 12/AHSME, 28

Tags: geometry , analytic geometry

How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1 \le x \le 4$ and $1 \le y \le 4$? $ \textbf{(A)}\ 496 \qquad\textbf{(B)}\ 500 \qquad\textbf{(C)}\ 512 \qquad\textbf{(D)}\ 516 \qquad\textbf{(E)}\ 560 $

2016 Taiwan TST Round 3, 2

Tags: algebra , functional equation , function

Determine all functions $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ satisfying $f(x+y+f(y))=4030x-f(x)+f(2016y), \forall x,y \in \mathbb{R}^+$.

2010 Postal Coaching, 2

Tags: arithmetic sequence , number theory

Call a triple $(a, b, c)$ of positive integers a [b]nice[/b] triple if $a, b, c$ forms a non-decreasing arithmetic progression, $gcd(b, a) = gcd(b, c) = 1$ and the product $abc$ is a perfect square. Prove that given a nice triple, there exists some other nice triple having at least one element common with the given triple.

2017 ASDAN Math Tournament, 22

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Let $x=2\sin8^\circ+2\sin16^\circ+\dots+2\sin176^\circ$. What is $\arctan(x)$?

2009 Ukraine Team Selection Test, 4

Tags: algebra , functional equation , functional

Let $n$ be some positive integer. Find all functions $f:{{R}^{+}}\to R$ (i.e., functions defined by the set of all positive real numbers with real values) for which equality holds $f\left( {{x}^{n+1}}+ {{y}^{n+1}} \right)={{x}^{n}}f\left( x \right)+{{y}^{n}}f\left( y \right)$ for any positive real numbers $x, y$

2011 AIME Problems, 4

Tags: graphing lines , slope , ratio , geometry , analytic geometry

In triangle $ABC$, $AB=\frac{20}{11} AC$. The angle bisector of $\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of the intersection of $AC$ and $BM$. The ratio of $CP$ to $PA$ can be expressed in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

1973 IMO Shortlist, 10

Tags: sequence , inequality , geometric sequence , algebra , construction

Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that: [i]a.)[/i] $ a_{k} < b_{k}$ for all $k = 1, \ldots, n,$ [i]b.)[/i] $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$ [i]c.)[/i] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$

2004 AMC 10, 24

Tags: easiest p24

Let $a_1, a_2, \cdots$, be a sequence with the following properties. I. $a_1 = 1$, and II. $a_{2n}=n\cdot a_n$ for any positive integer $n$. What is the value of $a_{2^{100}}$? $ \textbf{(A)}\; 1\qquad \textbf{(B)}\; 2^{99}\qquad \textbf{(C)}\; 2^{100}\qquad \textbf{(D)}\; 2^{4950}\qquad \textbf{(E)}\; 2^{9999} $

2016 LMT, 8

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Consider the function $f:[0,1)\rightarrow[0,1)$ defined by $f(x)=2x-\lfloor 2x\rfloor$, where $\lfloor 2x\rfloor$ is the greatest integer less than or equal to $2x$. Find the sum of all values of $x$ such that $f^{17}(x)=x.$ [i]Proposed by Matthew Weiss

1968 Swedish Mathematical Competition, 5

Tags: min , inequalities , trigonometry , algebra

Let $a, b$ be non-zero integers. Let $m(a, b)$ be the smallest value of $\cos ax + \cos bx$ (for real $x$). Show that for some $r$, $m(a, b) \le r < 0$ for all $a, b$.

2015 IFYM, Sozopol, 3

Tags: function , algebra

Find all functions $f:\mathbb R^{+} \longrightarrow \mathbb R^{+}$ so that $f(xy + f(x^y)) = x^y + xf(y)$ for all positive reals $x,y$.

1985 IMO Longlists, 25

Tags: number theory

Find eight positive integers $n_1, n_2, \dots , n_8$ with the following property: For every integer $k$, $-1985 \leq k \leq 1985$, there are eight integers $a_1, a_2, \dots, a_8$, each belonging to the set $\{-1, 0, 1\}$, such that $k=\sum_{i=1}^{8} a_i n_i .$

2013 India IMO Training Camp, 2

Tags: number theory , equation , modular arithmetic

An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers. [b]a)[/b] Prove that there are at least $500$ friendly integers in the set $\{ 1,2,\ldots ,2012\}$. [b]b)[/b] Decide whether $a=2$ is friendly.