Olympiad problems

1992 Bundeswettbewerb Mathematik, 3

Tags: trigonometry , geometry

Provided a convex equilateral pentagon. On every side of the pentagon We construct equilateral triangles which run through the interior of the pentagon. Prove that at least one of the triangles does not protrude the pentagon's boundary.

2010 Balkan MO Shortlist, N3

Tags: function , greatest common divisor , relatively prime , totient function , number theory

For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$. Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.

1981 Yugoslav Team Selection Test, Problem 2

Tags: geometry

Suppose that there is a point $S$ inside a quadrilateral $ABCD$ such that segments $SA,SB,SC,SD$ divide the quadrilateral into four triangles of equal areas. Prove that one of the diagonals of the quadrilateral bisects the other one.

2021 JBMO Shortlist, G1

Tags: geometry

Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to the side $BC$. The lines $BC$ and $AO$ intersect at $E$. Let $s$ be the line through $E$ perpendicular to $AO$. The line $s$ intersects $AB$ and $AC$ at $K$ and $L$, respectively. Denote by $\omega$ the circumcircle of triangle $AKL$. Line $AD$ intersects $\omega$ again at $X$. Prove that $\omega$ and the circumcircles of triangles $ABC$ and $DEX$ have a common point.

2016 CHMMC (Fall), 9

Tags: geometry

In quadrilateral $ABCD$, $AB = DB$ and $AD = BC$. If $\angle ABD = 36^{\circ}$ and $\angle BCD = 54^{\circ}$, find $\angle ADC$ in degrees.

2000 Austrian-Polish Competition, 10

Tags: combinatorial geometry , combinatorics

The plan of the castle in Baranow Sandomierski can be presented as the graph with $16$ vertices on the picture. A night guard plans a closed round along the edges of this graph. (a) How many rounds passing through each vertex exactly once are there? The directions are irrelevant. (b) How many non-selfintersecting rounds (taking directions into account) containing each edge of the graph exactly once are there? [img]https://cdn.artofproblemsolving.com/attachments/1/f/27ca05fc689fd8d873130db9d8cc52acf49bb4.png[/img]

2013 Romania National Olympiad, 2

Tags: superior algebra , polynomial , algebra

Given a ring $\left( A,+,\cdot \right)$ that meets both of the following conditions: (1) $A$ is not a field, and (2) For every non-invertible element $x$ of $ A$, there is an integer $m>1$ (depending on $x$) such that $x=x^2+x^3+\ldots+x^{2^m}$. Show that (a) $x+x=0$ for every $x \in A$, and (b) $x^2=x$ for every non-invertible $x\in A$.

1990 Federal Competition For Advanced Students, P2, 2

Tags: algebra , inequalities

Show that for all integers $ n \ge 2$, $ \sqrt { 2\sqrt[3]{3 \sqrt[4]{4...\sqrt[n]{n}}}}<2$

2017 CIIM, Problem 2

Tags: college math , real analysis

Let $f :\mathbb{R} \to \mathbb{R}$ a derivable function such that $f(0) = 0$ and $|f'(x)| \leq |f(x)\cdot log |f(x)||$ for every $x \in \mathbb{R}$ such that $0 < |f(x)| < 1/2.$ Prove that $f(x) = 0$ for every $x \in \mathbb{R}$.

2023 Brazil EGMO TST -wrong source, 3

Tags: combinatorics

There are $n$ cards. Max and Lewis play, alternately, the following game Max starts the game, he removes exactly $1$ card, in each round the current player can remove any quantity of cards, from $1$ card to $t+1$ cards, which $t$ is the number of removed cards by the previous player, and the winner is the player who remove the last card. Determine all the possible values of $n$ such that Max has the winning strategy.

2022 Stanford Mathematics Tournament, 7

Tags:

Let \[A_j=\left\{(x,y):0\le x\sin\left(\frac{j\pi}{3}\right)+y\cos\left(\frac{j\pi}{3}\right)\le6-\left(x\cos\left(\frac{j\pi}{3}\right)-y\sin\left(\frac{j\pi}{3}\right)\right)^2\right\}\] The area of $\cup_{j=0}^5A_j$ can be expressed as $m\sqrt{n}$. What is the area?

2000 Turkey Team Selection Test, 3

Tags: function , limit , inequalities , algebra

Suppose $f:\mathbb{R} \to \mathbb{R}$ is a function such that \[|f(x+y)-f(x)-f(y)|\le 1\ \ \ \text{for all} \ \ x, y \in\mathbb R.\] Prove that there is a function $g:\mathbb{R}\to\mathbb{R}$ such that $|f(x)-g(x)|\le 1$ and $g(x+y)=g(x)+g(y)$ for all $x,y \in\mathbb R.$

2003 China Second Round Olympiad, 2

Tags: geometry , perimeter , floor function , calculus , integration , modular arithmetic , algebra

Let the three sides of a triangle be $\ell, m, n$, respectively, satisfying $\ell>m>n$ and $\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$, where $\{x\}=x-\lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$. Find the minimum perimeter of such a triangle.

2018 Thailand TST, 3

Tags: geometry , incenter

A convex quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_a, I_b, I_c$ and $I_d$ be the incenters of the triangles $DAB, ABC, BCD$ and $CDA$, respectively. Suppose that the common external tangents of the circles $AI_bI_d$ and $CI_bI_d$ meet at $X$, and the common external tangents of the circles $BI_aI_c$ and $DI_aI_c$ meet at $Y$. Prove that $\angle{XIY}=90^{\circ}$.

2006 National Olympiad First Round, 31

Tags:

Let $P(x)=x^3+ax^2+bx+c$ where $a,b,c$ are positive real numbers. If $P(1)\geq 2$ and $P(3)\leq 31$, how many of integers can $P(4)$ take? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ \text{None of above} $

2017 Purple Comet Problems, 15

Tags: polynomial , algebra

For real numbers $a, b$, and $c$ the polynomial $p(x) = 3x^7 - 291x^6 + ax^5 + bx^4 + cx^2 + 134x - 2$ has $7$ real roots whose sum is $97$. Find the sum of the reciprocals of those $7$ roots.

2017 Kosovo National Mathematical Olympiad, 3

Tags: combinatorics

$n$ teams participated in a basketball tournament. Each team has played with each team exactly one game. There was no tie. If in the end of the tournament the $i$-th team has $x_{i}$ wins and $y_{i}$ loses $(1\leq i \leq n)$ prove that: $\sum_{i=1}^{n} {x_{i}}^2=\sum_{i=1}^{n} {y_{i}}^2$

2021 All-Russian Olympiad, 2

Tags: number theory

Let $n$ be a natural number. An integer $a>2$ is called $n$-decomposable, if $a^n-2^n$ is divisible by all the numbers of the form $a^d+2^d$, where $d\neq n$ is a natural divisor of $n$. Find all composite $n\in \mathbb{N}$, for which there's an $n$-decomposable number.

2022 Romania National Olympiad, P3

Tags: complex number , algebra

Let $Z\subset \mathbb{C}$ be a set of $n$ complex numbers, $n\geqslant 2.$ Prove that for any positive integer $m$ satisfying $m\leqslant n/2$ there exists a subset $U$ of $Z$ with $m$ elements such that\[\Bigg|\sum_{z\in U}z\Bigg|\leqslant\Bigg|\sum_{z\in Z\setminus U}z\Bigg|.\][i]Vasile Pop[/i]

2021 Thailand TSTST, 2

Tags: combinatorics , combinatorial geometry , geometry

Let $n$ be a positive integer and let $0\leq k\leq n$ be an integer. Show that there exist $n$ points in the plane with no three on a line such that the points can be divided into two groups satisfying the following properties. $\text{(i)}$ The first group has $k$ points and the distance between any two distinct points in this group is irrational. $\text{(ii)}$ The second group has $n-k$ points and the distance between any two distinct points in this group is an integer. $\text{(iii)}$ The distance between a point in the first group and a point in the second group is irrational.

2008 Postal Coaching, 1

Tags: floor function , number theory , natural , algebra , irrational , fraction

For each positive $ x \in \mathbb{R}$, define $ E(x)=\{[nx]: n\in \mathbb{N}\}$ Find all irrational $ \alpha >1$ with the following property: If a positive real $ \beta$ satisfies $ E(\beta) \subset E(\alpha)$. then $ \frac{\beta}{\alpha}$ is a natural number.

2021 Kazakhstan National Olympiad, 3

Tags: algebra

Let $(a_n)$ and $(b_n)$ be sequences of real numbers, such that $a_1 = b_1 = 1$, $a_{n+1} = a_n + \sqrt{a_n}$, $b_{n+1} = b_n + \sqrt[3]{b_n}$ for all positive integers $n$. Prove that there is a positive integer $n$ for which the inequality $a_n \leq b_k < a_{n+1}$ holds for exactly 2021 values of $k$.

2024 Sharygin Geometry Olympiad, 9

Tags: geometry , trapezoid

Let $ABCD$ ($AD \parallel BC$) be a trapezoid circumscribed around a circle $\omega$, which touches the sides $AB, BC, CD, $ and $AD$ at points $P, Q, R, S$ respectively. The line passing through $P$ and parallel to the bases of the trapezoid meets $QR$ at point $X$. Prove that $AB, QS$ and $DX$ concur.

2006 MOP Homework, 1

Tags: geometry

Triangle $ABC$ is inscribed in circle $w$. Line $l_{1}$ bisects $\angle BAC$ and meets segments $BC$ and $w$ in $D$ and $M$,respectively. Let $y$ denote the circle centered at $M$ with radius $BM$. Line $l_{2}$ passes through $D$ and meets circle $y$ at $X$ and $Y$. Prove that line $l_{1}$ also bisects $\angle XAY$

2021 ISI Entrance Examination, 8

Tags: differential equation , calculus

A pond has been dug at the Indian Statistical Institute as an inverted truncated pyramid with a square base (see figure below). The depth of the pond is 6m. The square at the bottom has side length 2m and the top square has side length 8m. Water is filled in at a rate of $\tfrac{19}{3}$ cubic meters per hour. At what rate is the water level rising exactly $1$ hour after the water started to fill the pond? [img]https://cdn.artofproblemsolving.com/attachments/0/9/ff8cac4bb4596ec6c030813da7e827e9a09dfd.png[/img]