Olympiad problems

2005 China National Olympiad, 1

Tags: trigonometry , inequalities

Suppose $\theta_{i}\in(-\frac{\pi}{2},\frac{\pi}{2}), i = 1,2,3,4$. Prove that, there exist $x\in \mathbb{R}$, satisfying two inequalities \begin{eqnarray*} \cos^2\theta_1\cos^2\theta_2-(\sin\theta\sin\theta_2-x)^2 &\geq& 0, \\ \cos^2\theta_3\cos^2\theta_4-(\sin\theta_3\sin\theta_4-x)^2 & \geq & 0 \end{eqnarray*} if and only if \[ \sum^4_{i=1}\sin^2\theta_i\leq2(1+\prod^4_{i=1}\sin\theta_i + \prod^4_{i=1}\cos\theta_i). \]

1996 Israel National Olympiad, 8

Tags: function , algebra , max

Consider the function $f : N \to N$ given by (i) $f(1) = 1$, (ii) $f(2n) = f(n)$ for any $n \in N$, (iii) $f(2n+1) = f(2n)+1$ for any $n \in N$. (a) Find the maximum value of $f(n)$ for $1 \le n \le 1995$; (b) Find all values of $f$ on this interval.

1998 Belarus Team Selection Test, 1

Tags: number theory , composition , combinatorics

Let $n\ge 2$ be positive integer. Find the least possible number of elements of tile set $A =\{1,2,...,2n-1,2n\}$ that should be deleted in order to the sum of any two different elements remained be a composite number.

1977 IMO Longlists, 18

Tags: geometry

Given an isosceles triangle $ABC$ with a right angle at $C,$ construct the center $M$ and radius $r$ of a circle cutting on segments $AB, BC, CA$ the segments $DE, FG,$ and $HK,$ respectively, such that $\angle DME + \angle FMG + \angle HMK = 180^\circ$ and $DE : FG : HK = AB : BC : CA.$

Kyiv City MO Juniors 2003+ geometry, 2014.7.41

Tags: triangle inequality , geometry , geometric inequality

The sides of triangles $ABC$ and $ACD$ satisfy the following conditions: $AB = AD = 3$ cm, $BC = 7$ cm, $DC = 11$ cm. What values can the side length $AC$ take if it is an integer number of centimeters, is the average in $\Delta ACD$ and the largest in $\Delta ABC$?

2012 IFYM, Sozopol, 6

Tags: geometry , concentric , equilateral triangle

Let $A_1 B_1 C_1$ and $A_2 B_2 C_2$ be two oppositely oriented concentric equilateral triangles. Prove that the lines $A_1 A_2$ , $B_1 B_2$ , and $C_1 C_2$ intersect in one point.

2016 India IMO Training Camp, 2

Tags: number theory , divisibility

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

VI Soros Olympiad 1999 - 2000 (Russia), 9.1

Tags: algebra

A car and a motorcyclist left point $A$ in the direction of point $B$ at $10$ o'clock, and half an hour later a cyclist left point $B$ (in the direction of point A) and a pedestrian left (in the direction of point $A$) The car met the cyclist at $11$ o'clock hour and half an hour later overtook the pedestrian, and the motorcyclist overtook the pedestrian at $12:30$ p.m. At what time did the motorcyclist and the cyclist meet? (Speeds and directions of movement of ALL participants)

2023-IMOC, G6

Tags: geometry

Triangle $ABC$ has circumcenter $O$. $D$ is the foot from $A$ to $BC$, and $P$ is apoint on $AD$. The feet from $P$ to $CA, AB$ are $E, F$, respectively, and the foot from $D$ to $EF$ is $T$. $AO$ meets $(ABC)$ again at $A'$. $A'D$ meets $(ABC)$ again at $R$. If $Q$ is a point on $AO$ satisfying $\angle ABP = \angle QBC$, prove that $D, P, T, R$ lie on acircle and $DQ$ is tangent to it.

2009 IMAC Arhimede, 4

Tags: algebra , functional equation , functional

Let $m,n \in Z, m\ne n, m \ne 0, n \ne 0$ . Find all $f: Z \to R$ such that $f(mx+ny)=mf(x)+nf(y)$ for all $x,y \in Z$ .

1998 Romania Team Selection Test, 2

Tags: number theory , modular arithmetic , perfect square , perfect cube , arithmetic sequence

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

2002 Paraguay Mathematical Olympiad, 3

Tags: number theory

With three different digits, six-digit numbers are written, multiples of $3$. One of the the digits are in the unit's place, another in the hundred's place, and the third in the remaining places. If we take out two units from the hundred's digit and add these to the unit's digit, the number is left with all the same digits. Find the numbers.

2019 Singapore MO Open, 2

Tags: functional equation , function , algebra

find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $f(-f(x)-f(y)) = 1-x-y$ $\quad \forall x,y \in \mathbb{Z}$

1997 Singapore Team Selection Test, 1

Tags: number theory , prime number , game , invariant

Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers $ a,b,c,d$ are replaced by $ a\minus{}b,b\minus{}c,c\minus{}d,d\minus{}a.$ Is it possible after 1996 such to have numbers $ a,b,c,d$ such the numbers $ |bc\minus{}ad|, |ac \minus{} bd|, |ab \minus{} cd|$ are primes?

2006 Tournament of Towns, 7

Tags: combinatorics

A Magician has a deck of $52$ cards. Spectators want to know the order of cards in the deck(without specifying face-up or face-down). They are allowed to ask the questions “How many cards are there between such-and-such card and such-and-such card?” One of the spectators knows the card order. Find the minimal number of questions he needs to ask to be sure that the other spectators can learn the card order. (9)

2022 Bundeswettbewerb Mathematik, 2

Tags: combinatorics , combinatorial geometry , triangle

Eva draws an equilateral triangle and its altitudes. In a first step she draws the center triangle of the equilateral triangle, in a second step the center triangle of this center triangle and so on. After each step Eva counts all triangles whose sides lie completely on drawn lines. What is the minimum number of center triangles she must have drawn so that the figure contains more than 2022 such triangles?

2017 JBMO Shortlist, G2

Tags: geometry , circumcenter , equal angles , circumcircle

Let $ABC$ be an acute triangle such that $AB$ is the shortest side of the triangle. Let $D$ be the midpoint of the side $AB$ and $P$ be an interior point of the triangle such that $\angle CAP = \angle CBP = \angle ACB$. Denote by M and $N$ the feet of the perpendiculars from $P$ to $BC$ and $AC$, respectively. Let $p$ be the line through $ M$ parallel to $AC$ and $q$ be the line through $N$ parallel to $BC$. If $p$ and $q$ intersect at $K$ prove that $D$ is the circumcenter of triangle $MNK$.

2010 Saudi Arabia Pre-TST, 2.1

Tags: diophantine , system of equations , system , number theory

Find all triples $(x,y,z)$ of positive integers such that $$\begin{cases} x + y +z = 2010 \\x^2 + y^2 + z^2 - xy - yz - zx =3 \end{cases}$$

LMT Team Rounds 2021+, A13

Tags:

In a round-robin tournament, where any two players play each other exactly once, the fact holds that among every three students $A$, $B$, and $C$, one of the students beats the other two. Given that there are six players in the tournament and Aidan beats Zach but loses to Andrew, find how many ways there are for the tournament to play out. Note: The order in which the matches take place does not matter. [i]Proposed by Kevin Zhao[/i]

2006 Princeton University Math Competition, 9

Tags: combinatorics

A stick of length $10$ is marked with $9$ evenly spaced marks (so each is one unit apart). An ant is placed at every mark and at the endpoints, randomly facing either right or left. Suddenly, all the ants start walking simultaneously at a rate of $ 1$ unit per second. If two ants collide head-on, they immediately reverse direction (assume that turning takes no time). Ants fall off the stick as soon as they walk past the endpoints (so the two on the end don’t fall off immediately unless they are facing outwards). On average, how long (in seconds) will it take until all of the ants fall off of the stick?

1988 AMC 12/AHSME, 1

Tags:

$\sqrt{8}+\sqrt{18}=$ $\textbf{(A)}\ \sqrt{20} \qquad \textbf{(B)}\ 2(\sqrt{2}+\sqrt{3}) \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 5\sqrt{2} \qquad \textbf{(E)}\ 2\sqrt{13}$

2009 China Team Selection Test, 2

Tags: circumcircle , geometric transformation , reflection , ratio , trigonometry , geometry

In acute triangle $ ABC,$ points $ P,Q$ lie on its sidelines $ AB,AC,$ respectively. The circumcircle of triangle $ ABC$ intersects of triangle $ APQ$ at $ X$ (different from $ A$). Let $ Y$ be the reflection of $ X$ in line $ PQ.$ Given $ PX>PB.$ Prove that $ S_{\bigtriangleup XPQ}>S_{\bigtriangleup YBC}.$ Where $ S_{\bigtriangleup XYZ}$ denotes the area of triangle $ XYZ.$

2015 CCA Math Bonanza, L1.1

Tags:

What is the value of $(2^{-1})^{-2}$? [i]2015 CCA Math Bonanza Lightning Round #1.1[/i]

2008 China Team Selection Test, 5

Tags: function , inequalities , cauchy inequality , algebra

For two given positive integers $ m,n > 1$, let $ a_{ij} (i = 1,2,\cdots,n, \; j = 1,2,\cdots,m)$ be nonnegative real numbers, not all zero, find the maximum and the minimum values of $ f$, where \[ f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i= 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\sum_{j = 1}^{m}a_{ij})^2 + mn\sum_{i = 1}^{n}\sum_{j=1}^{m}a_{ij}^2}. \]

Gheorghe Țițeica 2024, P1

Tags: algebra

Let $n\geq 3$ and $A=\{1,2,\dots ,n\}$. For any function $f:A\rightarrow A$ we define $$A_f=\{|f(1)-f(2)|,|f(2)-f(3)|,\dots ,|f(n-1)-f(n)|,|f(n)-f(1)|\}.$$ Determine the smallest and greatest value of the cardinal of $A_f$ as $f$ goes through all bijective functions from $A$ to $A$. [i]Silviu Cristea[/i]