Olympiad problems

1956 Czech and Slovak Olympiad III A, 2

In a given plane $\varrho$ consider a convex quadrilateral $ABCD$ and denote $E=AC\cap BD.$ Moreover, consider a point $V\notin\varrho$. On rays $VA,VB,VC,VD$ find points $A',B',C',D'$ respectively such that $E,A',B',C',D'$ are coplanar and $A'B'C'D'$ is a parallelogram. Discuss conditions of solvability.

2017 Tournament Of Towns, 7

Tags: combinatorics , geometry , chessboard

$1\times 2$ dominoes are placed on an $8 \times 8$ chessboard without overlapping. They may partially stick out from the chessboard but the center of each domino must be strictly inside the chessboard (not on its border). Place on the chessboard in such a way: a) at least $40$ dominoes, (3 points) b) at least $41$ dominoes, (3 points) c) more than $41$ dominoes. (6 points) [i](Mikhail Evdokimov)[/i]

2015 All-Russian Olympiad, 6

Tags: inequalities

Let a,b,c,d be real numbers satisfying $|a|,|b|,|c|,|d|>1$ and $abc+abd+acd+bcd+a+b+c+d=0$. Prove that $\frac {1} {a-1}+\frac {1} {b-1}+ \frac {1} {c-1}+ \frac {1} {d-1} >0$

2023 Ukraine National Mathematical Olympiad, 9.1

Tags: combinatorics , equality

$n \ge 4$ real numbers are arranged in a circle. It turned out that for any four consecutive numbers $a, b, c, d$, that lie on the circle in this order, holds $a+d = b+c$. For which $n$ does it follow that all numbers on the circle are equal? [i]Proposed by Oleksiy Masalitin[/i]

2008 Gheorghe Vranceanu, 2

Tags: inequalities , kronecker , number theory , fractional part

Show that there is a natural number $ n $ that satisfies the following inequalities: $$ \sqrt{3} -\frac{1}{10}<\{ n\sqrt 3\} +\{ (n+1)\sqrt 3 \} <\sqrt 3. $$

1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 6

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How many 11-digit bank account numbers are there consisting of 1's and 2's only, and such that there are no two consecutive 1's? A. 64 B. 233 C. 1024 D. 1279 E. 1365

2004 India IMO Training Camp, 2

Tags: polynomial , algebra

Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ and $Q(x) = x^2 + px + q$be two real polynomials. Suppose that there exista an interval $(r,s)$ of length greater than $2$ SUCH THAT BOTH $P(x)$ AND $Q(x)$ ARE nEGATIVE FOR $X \in (r,s)$ and both are positive for $x > s$ and $x<r$. Show that there is a real $x_0$ such that $P(x_0) < Q(x_0)$

LMT Speed Rounds, 2010.7

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Let $ABCD$ be a square with $AB=6.$ A point $P$ in the interior is $2$ units away from side $BC$ and $3$ units away from side $CD.$ What is the distance from $P$ to $A?$

1987 AIME Problems, 2

Tags: geometry , 3d geometry , sphere , inequalities , triangle inequality

What is the largest possible distance between two points, one on the sphere of radius 19 with center $(-2, -10, 5)$ and the other on the sphere of radius 87 with center $(12, 8, -16)$?

2006 Germany Team Selection Test, 2

Tags: inequalities , combinatorics

In a room, there are $2005$ boxes, each of them containing one or several sorts of fruits, and of course an integer amount of each fruit. [b]a)[/b] Show that we can find $669$ boxes, which altogether contain at least a third of all apples and at least a third of all bananas. [b]b)[/b] Can we always find $669$ boxes, which altogether contain at least a third of all apples, at least a third of all bananas and at least a third of all pears?

2018 Sharygin Geometry Olympiad, 6

Tags: geometry , circumcircle

Let $\omega$ be the circumcircle of $ABC$, and $KL$ be the diameter of $\omega$ passing through $M$ midpoint of $AB$ ($K,C$ lies on different sides of $AB$). A circle passing through $L$ and $M$ meets $CK$ at points $P$ and $Q$ ($Q$ lies on $KP$). Let $LQ$ meet the circumcircle of $KMQ$ again at $R$. Prove that $APBR$ is cyclic.

2020-21 IOQM India, 23

Tags: geometry , inradius , tangent circle

The incircle $\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D, CA$ at $E$ and $AB$ at $F$. Let $r_A$ be the radius of the circle inside $ABC$ which is tangent to $\Gamma$ and the sides $AB$ and $AC$. Define $r_B$ and $r_C$ similarly. If $r_A = 16, r_B = 25$ and $r_C = 36$, determine the radius of $\Gamma$.

2013 IFYM, Sozopol, 1

Tags: geometry

Point D is from AC of triangle ABC so that 2AD=DC. Let DE be perpendicular to BC and AE intersects BD at F. It is known that triangle BEF is equilateral. Find <ADB?

2018 Saint Petersburg Mathematical Olympiad, 2

Tags: combinatorics

Vasya has $100$ cards of $3$ colors, and there are not more than $50$ cards of same color. Prove that he can create $10\times 10$ square, such that every cards of same color have not common side.

2012 May Olympiad, 5

Tags: floor function , combinatorics

There are 12 people such that for every person A and person B there exists a person C that is a friend to both of them. Determine the minimum number of pairs of friends and construct a graph where the edges represent friendships.

2006 MOP Homework, 6

Tags: geometry

Suppose there are $18$ light houses on the Mexican gulf. Each of the lighthouses lightens an angle with size $20$ degrees. Prove that we can choose the directions of the lighthouses such that the whole gulf is lightened.

VMEO IV 2015, 12.2

Tags: geometry , parallel , angle bisector

Given a triangle $ABC$ inscribed in circle $(O)$ and let $P$ be a point on the interior angle bisector of $BAC$. $PB$, $PC$ cut $CA$, $AB$ at $E,F$ respectively. Let $EF$ meet $(O)$ at $M,N$. The line that is perpendicular to $PM$, $PN$ at $M,N$ respectively intersect $(O)$ at $S, T$ different from $M,N$. Prove that $ST \parallel BC$.

Kyiv City MO Seniors 2003+ geometry, 2004.10.5

Tags: geometry , centroid , orthocenter

Let the points $M$ and $N$ in the triangle $ABC$ be the midpoints of the sides $BC$ and $AC$, respectively. It is known that the point of intersection of the altitudes of the triangle $ABC$ coincides with the point of intersection of the medians of the triangle $AMN$. Find the value of the angle $ABC$.

2013 QEDMO 13th or 12th, 5

Tags: combinatorial geometry , combinatorics , chessboard

$16$ pieces of the shape $1\times 3$ are placed on a $7\times 7$ chessboard, each of which is exactly three fields. One field remains free. Find all possible positions of this field.

2014 JHMMC 7 Contest, 25

Tags: three altitude triangle , bashing , perimeter

If a triangle has three altitudes of lengths $6, 6, \text{and} 6,$ what is its perimeter?

2009 South africa National Olympiad, 6

Tags: function , linear equation , absolute value , algebra

Let $A$ denote the set of real numbers $x$ such that $0\le x<1$. A function $f:A\to \mathbb{R}$ has the properties: (i) $f(x)=2f(\frac{x}{2})$ for all $x\in A$; (ii) $f(x)=1-f(x-\frac{1}{2})$ if $\frac{1}{2}\le x<1$. Prove that (a) $f(x)+f(1-x)\ge \frac{2}{3}$ if $x$ is rational and $0<x<1$. (b) There are infinitely many odd positive integers $q$ such that equality holds in (a) when $x=\frac{1}{q}$.

2017 Vietnam Team Selection Test, 2

Tags: binomial coefficient , integer sequence , periodic sequence

For each positive integer $n$, set $x_n=\binom{2n}{n}$. a. Prove that if $\frac{2017^k}{2}<n<2017^k$ for some positive integer $k$ then $2017$ divides $x_n$. b. Find all positive integer $h>1$ such that there exists positive integers $N,T$ such that $(x_n)_{n>N}$ is periodic mod $h$ with period $T$.

1998 AMC 8, 19

Tags: probability

Tamika selects two different numbers at random from the set $ \{ 8,9,10\} $ and adds them. Carlos takes two different numbers at random from the set $ \{ 3,5,6\} $ and multiplies them. What is the probability that Tamika's result is greater than Carlos' result? $ \text{(A)}\ \frac{4}{9}\qquad\text{(B)}\ \frac{5}{9}\qquad\text{(C)}\ \frac{1}{2}\qquad\text{(D)}\ \frac{1}{3}\qquad\text{(E)}\ \frac{2}{3} $

2013 Baltic Way, 17

Tags: modular arithmetic , number theory

Let $c$ and $n > c$ be positive integers. Mary's teacher writes $n$ positive integers on a blackboard. Is it true that for all $n$ and $c$ Mary can always label the numbers written by the teacher by $a_1,\ldots, a_n$ in such an order that the cyclic product $(a_1-a_2)\cdot(a_2-a_3)\cdots(a_{n-1}-a_n)\cdot(a_n-a_1)$ would be congruent to either $0$ or $c$ modulo $n$?

2002 Junior Balkan MO, 4

Tags: function , rearrangement inequality , 3-variable inequality , cyclic inequality , inequalities

Prove that for all positive real numbers $a,b,c$ the following inequality takes place \[ \frac{1}{b(a+b)}+ \frac{1}{c(b+c)}+ \frac{1}{a(c+a)} \geq \frac{27}{2(a+b+c)^2} . \] [i]Laurentiu Panaitopol, Romania[/i]