Olympiad problems

2014 JBMO TST - Macedonia, 2

Tags: geometry

Point $M$ is an arbitrary point in the plane and let points $G$ and $H$ be the intersection points of the tangents from point M and the circle $k$. Let $O$ be the center of the circle $k$ and let $K$ be the orthocenter of the triangle $MGH$. Prove that ${\angle}GMH={\angle}OGK$.

2006 Denmark MO - Mohr Contest, 2

Tags: algebra , system of equations , system

Determine all sets of real numbers $(x,y,z)$ which fulfills $$\begin{cases} x + y =2 \\ xy -z^2= 1\end{cases}$$

2022 Bosnia and Herzegovina IMO TST, 4

Tags: combinatorics , counting

In each square of a $4 \times 4$ table a number $0$ or $1$ is written, such that the product of every two neighboring squares is $0$ (neighboring by side). $a)$ In how many ways is this possible to do if the middle $2\times 2$ is filled with $4$ zeros? $b)$ In general, in how many ways is this possible to do (regardless of the middle $2 \times 2$)?

2014 ELMO Shortlist, 6

Tags: inequalities

Let $a,b,c$ be positive reals such that $a+b+c=ab+bc+ca$. Prove that \[ (a+b)^{ab-bc}(b+c)^{bc-ca}(c+a)^{ca-ab} \ge a^{ca}b^{ab}c^{bc}. \][i]Proposed by Sammy Luo[/i]

2019 Caucasus Mathematical Olympiad, 8

Tags: number theory

Determine if there exist positive integers $a_1,a_2,...,a_{10}$, $b_1,b_2,...,b_{10}$ satisfying the following property: for each non-empty subset $S$ of $\{1,2,\ldots,10\}$ the sum $\sum\limits_{i\in S}a_i$ divides $\left( 12+\sum\limits_{i\in S}b_i \right)$.

2001 Hong kong National Olympiad, 1

Tags: circumcircle , geometry

A triangle $ABC$ is given. A circle $\Gamma$, passing through $A$, is tangent to side $BC$ at point $P$ and intersects sides $AB$ and $AC$ at $M$ and $N$ respectively. Prove that the smaller arcs $MP$ and $NP$ of $\Gamma$ are equal iff $\Gamma$ is tangent to the circumcircle of $\Delta ABC$ at $A$.

2024 Ukraine National Mathematical Olympiad, Problem 1

Tags: number theory , least common multiple , greatest common divisor

Find all pairs $a, b$ of positive integers, for which $$(a, b) + 3[a, b] = a^3 - b^3$$ Here $(a, b)$ denotes the greatest common divisor of $a, b$, and $[a, b]$ denotes the least common multiple of $a, b$. [i]Proposed by Oleksiy Masalitin[/i]

2019 IMC, 7

Tags: number theory , convergence

Let $C=\{4,6,8,9,10,\ldots\}$ be the set of composite positive integers. For each $n\in C$ let $a_n$ be the smallest positive integer $k$ such that $k!$ is divisible by $n$. Determine whether the following series converges: $$\sum_{n\in C}\left(\frac{a_n}{n}\right)^n.$$ [i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan[/i]

2011 Hanoi Open Mathematics Competitions, 1

Tags: geometry , combinatorial geometry

Three lines are drawn in a plane. Which of the following could NOT be the total number of points of intersections? (A) $0$ (B) $1$ (C) $2$ (D) $3$ (E) They all could.

1991 Spain Mathematical Olympiad, 4

Tags: incircle , geometry , angle

The incircle of $ABC$ touches the sides $BC,CA,AB$ at $A' ,B' ,C'$ respectively. The line $A' C'$ meets the angle bisector of $\angle A$ at $D$. Find $\angle ADC$.

2016 Junior Balkan Team Selection Tests - Romania, 1

Tags: geometry

The altitudes $AA_1$,$BB_1$,$CC_1$ of $\triangle{ABC}$ intersect at $H$.$O$ is the circumcenter of $\triangle{ABC}$.Let $A_2$ be the reflection of $A$ wrt $B_1C_1$.Prove that: a)$O$,$A_2$,$B_1$,$C$ are all on a circle b)$O$,$H$,$A_1$,$A_2$ are all on a circle

2014 District Olympiad, 4

Tags: function , induction , algebra

Find all functions $f:\mathbb{Q}\to \mathbb{Q}$ such that \[ f(x+3f(y))=f(x)+f(y)+2y \quad \forall x,y\in \mathbb{Q}\]

2016 239 Open Mathematical Olympiad, 7

Tags: number theory

A set is called $six\ square$ if it has six pair-wise coprime numbers and for any partition of it into two set with three elements, the sum of the numbers in one of them is perfect square. Prove that there exist infinitely many $six\ square$.

2011 Today's Calculation Of Integral, 675

Tags: calculus , trapezoid , geometry , analytic geometry , integration , conic , parabola

In the coordinate plane with the origin $O$, consider points $P(t+2,\ 0),\ Q(0, -2t^2-2t+4)\ (t\geq 0).$ If the $y$-coordinate of $Q$ is nonnegative, then find the area of the region swept out by the line segment $PQ$. [i]2011 Ritsumeikan University entrance exam/Pharmacy[/i]

1936 Moscow Mathematical Olympiad, 025

Tags: geometry , geometric construction , diameter

Consider a circle and a point $P$ outside the circle. The angle of given measure with vertex at $P$ subtends a diameter of the circle. Construct the circle’s diameter with ruler and compass.

1999 Tournament Of Towns, 2

Tags: tangent , circumcircle , geometry , parallelogram

Let $O$ be the intersection point of the diagonals of a parallelogram $ABCD$ . Prove that if the line $BC$ is tangent to the circle passing through the points $A, B$, and $O$, then the line $CD$ is tangent to the circle passing through the points $B, C$ and $O$. (A Zaslavskiy)

2021 Harvard-MIT Mathematics Tournament., 1

Tags: combi

Leo the fox has a $5$ by $5$ checkerboard grid with alternating red and black squares. He fills in the grid with the numbers $1, 2, 3, \dots, 25$ such that any two consecutive numbers are in adjacent squares (sharing a side) and each number is used exactly once. He then computes the sum of the numbers in the $13$ squares that are the same color as the center square. Compute the maximum possible sum Leo can obtain.

1992 Tournament Of Towns, (351) 3

Tags: algebra , combinatorics

We are given a finite number of functions of the form $y = c2^{-|x-d|}$. In each case $c$ and $d$ are parameters with $c > 0$. The function $f(x)$ is defined on the interval $[a, b]$ as follows: For each $x$ in $[a, b]$, $f(x)$ is the maximum value taken by any of the given functions $y$ (defined above) at that point $x$. It is known that $f(a) = f(b)$. Prove that the total length of the intervals in which the function $f$ is increasing is equal to the total length of the intervals in which it is decreasing (that is, both are equal to $(b- a)/2$ ). (NB Vasiliev)

2001 India National Olympiad, 5

Tags: trig identities , circumcircle , trigonometry , law of sines , geometry

$ABC$ is a triangle. $M$ is the midpoint of $BC$. $\angle MAB = \angle C$, and $\angle MAC = 15^{\circ}$. Show that $\angle AMC$ is obtuse. If $O$ is the circumcenter of $ADC$, show that $AOD$ is equilateral.

2007 Gheorghe Vranceanu, 1

Tags: number theory , base 10 number theory , base 10

Given an arbitrary natural number $ n, $ is there a multiple of $ n $ whose base $ 10 $ representation can be written only with the digits $ 0,2,7? $ Explain.

2023 CMIMC Algebra/NT, 5

Tags: algebra , number theory

Let $\mathcal{P}$ be a parabola that passes through the points $(0, 0)$ and $(12, 5)$. Suppose that the directrix of $\mathcal{P}$ takes the form $y = b$. (Recall that a parabola is the set of points equidistant from a point called the focus and line called the directrix) Find the minimum possible value of $|b|$. [i]Proposed by Kevin You[/i]

1997 AMC 8, 25

Tags:

All of the even numbers from 2 to 98 inclusive, excluding those ending in 0, are multiplied together. What is the rightmost digit (the units digit) of the product? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$

1949-56 Chisinau City MO, 36

Tags: algebra , sum

Calculate the sum: $1+ 2q + 3q^2 +...+nq^{n-1}$

2018 Irish Math Olympiad, 10

Tags: combinatorics , game strategy

The game of Greed starts with an initial configuration of one or more piles of stones. Player $1$ and Player $2$ take turns to remove stones, beginning with Player $1$. At each turn, a player has two choices: • take one stone from any one of the piles (a simple move); • take one stone from each of the remaining piles (a greedy move). The player who takes the last stone wins. Consider the following two initial configurations: (a) There are $2018$ piles, with either $20$ or $18$ stones in each pile. (b) There are four piles, with $17, 18, 19$, and $20$ stones, respectively. In each case, find an appropriate strategy that guarantees victory to one of the players.

2013 NIMO Problems, 1

Tags:

Find the sum of all primes that can be written both as a sum of two primes and as a difference of two primes. [i]Anonymous Proposal[/i]