Olympiad problems

2018 Yasinsky Geometry Olympiad, 1

In the triangle $ABC$, $AD$ is altitude, $M$ is the midpoint of $BC$. It is known that $\angle BAD = \angle DAM = \angle MAC$. Find the values of the angles of the triangle $ABC$

2022 USAMO, 6

Tags: AMC , USA(J)MO , USAMO , combinatorics , graph theory , Hi

There are $2022$ users on a social network called Mathbook, and some of them are Mathbook-friends. (On Mathbook, friendship is always mutual and permanent.) Starting now, Mathbook will only allow a new friendship to be formed between two users if they have [i]at least two[/i] friends in common. What is the minimum number of friendships that must already exist so that every user could eventually become friends with every other user?

1994 All-Russian Olympiad, 6

Tags: combinatorics , board

Cards numbered with numbers $1$ to $1000$ are to be placed on the cells of a $1\times 1994$ rectangular board one by one, according to the following rule: If the cell next to the cell containing the card $n$ is free, then the card $n+1$ must be put on it. Prove that the number of possible arrangements is not more than half a mllion.

1939 Eotvos Mathematical Competition, 2

Tags: number theory , factorial , power of 2 , divides

Determine the highest power of $2$ that divides $2^n!$.

1989 Tournament Of Towns, (235) 3

Tags: number theory , Perfect Square , combinatorics

Do there exist $1000 000$ distinct positive integers such that the sum of any collection of these numbers is never an exact square?

2001 Moldova National Olympiad, Problem 8

Tags: inequalities

Suppose that $a,b,c$ are real numbers such that $\left|ax^2+bx+c\right|\le1$ for $-1\le x\le1$. Prove that $\left|cx^2+bx+a\right|\le2$ for $-1\le x\le1$.

2011 Pre-Preparation Course Examination, 2

Tags: function , geometric series , advanced fields , advanced fields unsolved

by using the formula $\pi cot(\pi z)=\frac{1}{z}+\sum_{n=1}^{\infty}\frac{2z}{z^2-n^2}$ calculate values of $\zeta(2k)$ on terms of bernoli numbers and powers of $\pi$.

2014 Peru Iberoamerican Team Selection Test, P3

Tags: number theory

A positive integer $n$ is called $special$ if there exist integers $a > 1$ and $b > 1$ such that $n=a^b + b$. Is there a set of $2014$ consecutive positive integers that contains exactly $2012$ $special$ numbers?

JOM 2015 Shortlist, N5

Tags: number theory

Let $ a,b,c $ be pairwise coprime positive integers. Find all positive integer values of $$ \frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b} $$

2016 PUMaC Individual Finals A, 1

Tags: combinatorics

There are $12$ candies on the table, four of which are rare candies. Chad has a friend who can tell rare candies apart from regular candies, but Chad can’t. Chad’s friend is allowed to take four candies from the table, but may not take any rare candies. Can his friend always take four candies in such a way that Chad will then be able to identify the four rare candies? If so, describe a strategy. If not, prove that it cannot be done. Note that Chad does not know anything about how the candies were selected (e.g. the order in which they were selected). However, Chad and his friend may communicate beforehand.

2021/2022 Tournament of Towns, P6

Tags: combinatorics , Tournament of Towns , Kvant

There are 20 buns with jam and 20 buns with treacle arranged in a row in random order. Alice and Bob take in turn a bun from any end of the row. Alice starts, and wants to finally obtain 10 buns of each type; Bob tries to prevent this. Is it true for any order of the buns that Alice can win no matter what are the actions of Bob? [i]Alexandr Gribalko[/i]

2021 Ukraine National Mathematical Olympiad, 5

Tags: number theory , Sum

Find all sets of $n\ge 2$ consecutive integers $\{a+1,a+2,...,a+n\}$ where $a\in Z$, in which one of the numbers is equal to the sum of all the others. (Bogdan Rublev)

2016 HMIC, 3

Tags: HMMT , algebra , Functional inequality , functional equation , HMIC

Denote by $\mathbb{N}$ the positive integers. Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a function such that, for any $w,x,y,z \in \mathbb{N}$, \[ f(f(f(z)))f(wxf(yf(z)))=z^{2}f(xf(y))f(w). \] Show that $f(n!) \ge n!$ for every positive integer $n$. [i]Pakawut Jiradilok[/i]

1985 Traian Lălescu, 1.3

Tags: geometry , Pure geometry

Let $ H $ be the orthocenter of $ ABC $ and $ A',B',C', $ the symmetric points of $ A,B,C $ with respect to $ H. $ The intersection of the segments $ BC,CA, AB $ with the circles of diameter $ A'H,B'H, $ respectively, $ C'H, $ consists of $ 6 $ points. Prove that these are concyclic.

2015 AMC 12/AHSME, 4

Tags: AMC

David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$-person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6$th place. Who finished in $8$th place? $\textbf{(A) } \text{David} \qquad\textbf{(B) } \text{Hikmet} \qquad\textbf{(C) } \text{Jack} \qquad\textbf{(D) } \text{Rand} \qquad\textbf{(E) } \text{Todd} $

2019 Kosovo National Mathematical Olympiad, 1

Tags: algebra

Calculate $1^2-2^2+3^2-4^2+...-2018^2+2019^2$.

2015 Mathematical Talent Reward Programme, MCQ: P 12

Tags: trigonometry , algebra , Trigonometric Equations

Maximum value of $\sin^4\theta +\cos^6\theta $ will be ? [list=1] [*] $\frac{1}{2\sqrt{2}}$ [*] $\frac{1}{2}$ [*] $\frac{1}{\sqrt{2}}$ [*] 1 [/list]

2018 Balkan MO Shortlist, A1

Tags: algebra , Inequality , TST , Balkan MO Shortlist , AZE JBMO TST

Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

2017 China Team Selection Test, 6

Tags: combinatorics

Every cell of a $2017\times 2017$ grid is colored either black or white, such that every cell has at least one side in common with another cell of the same color. Let $V_1$ be the set of all black cells, $V_2$ be the set of all white cells. For set $V_i (i=1,2)$, if two cells share a common side, draw an edge with the centers of the two cells as endpoints, obtaining graphs $G_i$. If both $G_1$ and $G_2$ are connected paths (no cycles, no splits), prove that the center of the grid is one of the endpoints of $G_1$ or $G_2$.

2017 Hong Kong TST, 3

Tags: combinatorics

At a mathematical competition $n$ students work on 6 problems each one with three possible answers. After the competition, the Jury found that for every two students the number of the problems, for which these students have the same answers, is 0 or 2. Find the maximum possible value of $n$.

2009 Turkey Junior National Olympiad, 2

Tags:

In the beginnig, each square of a strip formed by $n$ adjacent squares contains $0$ or $1$. At each step, we are writing $1$ to the squares containing $0$ and to the squares having exactly one neighbour containing $1$, and we are writing $0$s into the other squares. Determine all possible values of $n$ such that whatever the initial arrangement of $0$ and $1$ is, after finite number of steps, all squares can turn into $0$.

2017 Balkan MO Shortlist, A5

Tags: polynomial , algebra , Integer Polynomial , Perfect Powers , Perfect power

Consider integers $m\ge 2$ and $n\ge 1$. Show that there is a polynomial $P(x)$ of degree equal to $n$ with integer coefficients such that $P(0),P(1),...,P(n)$ are all perfect powers of $m$ .

2011 Sharygin Geometry Olympiad, 23

Tags: geometry , Nine Point Circle , projections , midpoints , Circumcenter , collinear

Given are triangle $ABC$ and line $\ell$ intersecting $BC, CA$ and $AB$ at points $A_1, B_1$ and $C_1$ respectively. Point $A'$ is the midpoint of the segment between the projections of $A_1$ to $AB$ and $AC$. Points $B'$ and $C'$ are defined similarly. (a) Prove that $A', B'$ and $C'$ lie on some line $\ell'$. (b) Suppose $\ell$ passes through the circumcenter of $\triangle ABC$. Prove that in this case $\ell'$ passes through the center of its nine-points circle. [i]M. Marinov and N. Beluhov[/i]

PEN L Problems, 2

Tags: number theory , greatest common divisor , Linear Recurrences

The Fibonacci sequence $\{F_{n}\}$ is defined by \[F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.\] Show that $\gcd (F_{m}, F_{n})=F_{\gcd (m, n)}$ for all $m, n \in \mathbb{N}$.

1998 Tournament Of Towns, 2

Tags: geometry , parallelogram , equal segments , equal angles

$ABCD$ is a parallelogram. A point $M$ is found on the side $AB$ or its extension such that $\angle MAD = \angle AMO$ where $O$ is the intersection point of the diagonals of the parallelogram. Prove that $MD = MG$. (M Smurov)