Olympiad problems

2008 SEEMOUS, Problem 1

Tags: function

Let $f:[1,\infty)\to(0,\infty)$ be a continuous function. Assume that for every $a>0$, the equation $f(x)=ax$ has at least one solution in the interval $[1,\infty)$. (a) Prove that for every $a>0$, the equation $f(x)=ax$ has infinitely many solutions. (b) Give an example of a strictly increasing continuous function $f$ with these properties.

Kvant 2022, M2708 b)

Tags: combinatorics , combinatorial geometry

Do there exist 100 points on the plane such that the pairwise distances between them are pairwise distinct consecutive integer numbers larger than 2022?

2008 Pre-Preparation Course Examination, 5

Tags: probability , factorial , expected value , probability and stats

A permutation $ \pi$ is selected randomly through all $ n$-permutations. a) if \[ C_a(\pi)\equal{}\mbox{the number of cycles of length }a\mbox{ in }\pi\] then prove that $ E(C_a(\pi))\equal{}\frac1a$ b) Prove that if $ \{a_1,a_2,\dots,a_k\}\subset\{1,2,\dots,n\}$ the probability that $ \pi$ does not have any cycle with lengths $ a_1,\dots,a_k$ is at most $ \frac1{\sum_{i\equal{}1}^ka_i}$

2017 Macedonia National Olympiad, Problem 3

Tags: inequalities

Let $x,y,z \in \mathbb{R}$ such that $xyz = 1$. Prove that: $$\left(x^4 + \frac{z^2}{y^2}\right)\left(y^4 + \frac{x^2}{z^2}\right)\left(z^4 + \frac{y^2}{x^2}\right) \ge \left(\frac{x^2}{y} + 1 \right)\left(\frac{y^2}{z} + 1 \right)\left(\frac{z^2}{x} + 1 \right).$$

2015 Hanoi Open Mathematics Competitions, 8

Tags: algebra , equation

Solve the equation $(2015x -2014)^3 = 8(x-1)^3 + (2013x -2012)^3$

2018 AMC 12/AHSME, 16

Tags: geometry

The solutions to the equation $(z+6)^8=81$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled $A,B,$ and $C$. What is the least possible area of $\triangle ABC?$ $\textbf{(A) } \frac{1}{6}\sqrt{6} \qquad \textbf{(B) } \frac{3}{2}\sqrt{2}-\frac{3}{2} \qquad \textbf{(C) } 2\sqrt3-3\sqrt2 \qquad \textbf{(D) } \frac{1}{2}\sqrt{2} \qquad \textbf{(E) } \sqrt 3-1$

2018 Irish Math Olympiad, 4

Tags: combinatorics , combinatorial geometry , rectangle

We say that a rectangle with side lengths $a$ and $b$ [i]fits inside[/i] a rectangle with side lengths $c$ and $d$ if either ($a \le c$ and $b \le d$) or ($a \le d$ and $b \le c$). For instance, a rectangle with side lengths $1$ and $5$ [i]fits inside[/i] another rectangle with side lengths $1$ and $5$, and also [i]fits inside[/i] a rectangle with side lengths $6$ and $2$. Suppose $S$ is a set of $2019$ rectangles, all with integer side lengths between $1$ and $2018$ inclusive. Show that there are three rectangles $A$, $B$, and $C$ in $S$ such that $A$ fits inside $B$, and $B$ [i]fits inside [/i]$C$.

2011 Argentina National Olympiad, 5

Tags: number theory , divide , divisible

Find all integers $n$ such that $1<n<10^6$ and $n^3-1$ is divisible by $10^6 n-1$.

2017 IMAR Test, 2

Tags: number theory , abstract algebra

For every $k\leq n$ define $r_k$ the residue of $2^n$ modulo $k$. Prove that $\sum r_i> \frac{n*log_2(\frac{n}{3})}{2}-n$, for any $n\geq 2$

2021 Math Prize for Girls Problems, 7

Tags:

Compute the value of the infinite series \[ \sum_{k=0}^{\infty} \frac{\cos(k \pi / 4)}{2^k} \, . \]

2023 Indonesia TST, G

Tags: geometry , circumcircle , concurrency

Given an acute triangle $ABC$ with altitudes $AD$ and $BE$ intersecting at $H$, $M$ is the midpoint of $AB$. A nine-point circle of $ABC$ intersects with a circumcircle of $ABH$ on $P$ and $Q$ where $P$ lays on the same side of $A$ (with respect to $CH$). Prove that $ED, PH, MQ$ are concurrent on circumcircle $ABC$

1996 IMC, 3

Tags: vector , linear algebra

The linear operator $A$ on a finite-dimensional vector space $V$ is called an involution if $A^{2}=I$, where $I$ is the identity operator. Let $\dim V=n$. i) Prove that for every involution $A$ on $V$, there exists a basis of $V$ consisting of eigenvectors of $A$. ii) Find the maximal number of distinct pairwise commuting involutions on $V$.

2019 PUMaC Combinatorics B, 2

Tags: number theory , probability , combinatorics , relatively prime

Suppose Alan, Michael, Kevin, Igor, and Big Rahul are in a running race. It is given that exactly one pair of people tie (for example, two people both get second place), so that no other pair of people end in the same position. Each competitor has equal skill; this means that each outcome of the race, given that exactly two people tie, is equally likely. The probability that Big Rahul gets first place (either by himself or he ties for first) can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

2010 Poland - Second Round, 3

Tags: combinatorics , ceiling function , graph theory

The $n$-element set of real numbers is given, where $n \geq 6$. Prove that there exist at least $n-1$ two-element subsets of this set, in which the arithmetic mean of elements is not less than the arithmetic mean of elements in the whole set.

1995 All-Russian Olympiad Regional Round, 10.2

Tags: number theory , lcm , gcd

Natural numbers $m$ and $n$ satisfy $$gcd(m,n)+lcm(m,n) = m+n. $$Prove that one of numbers $m,n$ divides the other.

PEN A Problems, 23

Tags: number theory , quadratic , gauss , modular arithmetic , relatively prime , divisibility theory

(Wolstenholme's Theorem) Prove that if \[1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}\] is expressed as a fraction, where $p \ge 5$ is a prime, then $p^{2}$ divides the numerator.

2020 USEMO, 1

Tags:

Which positive integers can be written in the form \[\frac{\operatorname{lcm}(x, y) + \operatorname{lcm}(y, z)}{\operatorname{lcm}(x, z)}\] for positive integers $x$, $y$, $z$?

2023 Israel National Olympiad, P3

Tags: geometry , reflection , geometric transformation , congruent triangles

A triangle $ABC$ is given together with an arbitrary circle $\omega$. Let $\alpha$ be the reflection of $\omega$ with respect to $A$, $\beta$ the reflection of $\omega$ with respect to $B$, and $\gamma$ the reflection of $\omega$ with respect to $C$. It is known that the circles $\alpha, \beta, \gamma$ don't intersect each other. Let $P$ be the meeting point of the two internal common tangents to $\beta, \gamma$ (see picture). Similarly, $Q$ is the meeting point of the internal common tangents of $\alpha, \gamma$, and $R$ is the meeting point of the internal common tangents of $\alpha, \beta$. Prove that the triangles $PQR, ABC$ are congruent.

2021 USMCA, 28

Tags:

How many functions $f : \mathbb{Z} \rightarrow \{0, 1, 2, \cdots, 2020 \}$ are there such that $f(n) = f(n+2021)$ and $2021 \mid f(2n) - f(n) - f(n-1)$ for all integers $n$?

1996 IMO Shortlist, 3

Tags: combinatorics , additive combinatorics , additive number theory , extremal combinatorics , subset

Let $ k,m,n$ be integers such that $ 1 < n \leq m \minus{} 1 \leq k.$ Determine the maximum size of a subset $ S$ of the set $ \{1,2,3, \ldots, k\minus{}1,k\}$ such that no $ n$ distinct elements of $ S$ add up to $ m.$

2002 Baltic Way, 4

Tags: inequalities

Let $n$ be a positive integer. Prove that \[\sum_{i=1}^nx_i(1-x_i)^2\le\left(1-\frac{1}{n}\right)^2 \] for all nonnegative real numbers $x_1,x_2,\ldots ,x_n$ such that $x_1+x_2+\ldots x_n=1$.

2020 Olympic Revenge, 1

Tags: algebra

Let $n$ be a positive integer and $a_1, a_2, \dots, a_n$ non-zero real numbers. What is the least number of non-zero coefficients that the polynomial $P(x) = (x - a_1)(x - a_2)\cdots(x - a_n)$ can have?

2013 Saudi Arabia BMO TST, 5

Tags: sum of squares , digit , perfect square , number theory

We call a positive integer [i]good[/i ] if it doesn’t have a zero digit and the sum of the squares of its digits is a perfect square. For example, $122$ and $34$ are good and $304$ and $12$ are not not good. Prove that there exists a $n$-digit good number for every positive integer $n$.

1979 IMO Longlists, 65

Tags: function , inequalities , algebra

Given a function $f$ such that $f(x)\le x\forall x\in\mathbb{R}$ and $f(x+y)\le f(x)+f(y)\forall \{x,y\}\in\mathbb{R}$, prove that $f(x)=x\forall x\in\mathbb{R}$.

1989 Greece Junior Math Olympiad, 1

Tags: number theory

Let $A$ be the sum of three consecutive integers and $B$ be the sum of the exact three consecutive integers. Is it possible to have $AB=33333$ ?