Olympiad problems

2014 Czech-Polish-Slovak Match, 5

Let all positive integers $n$ satisfy the following condition: for each non-negative integers $k, m$ with $k + m \le n$, the numbers $\binom{n-k}{m}$ and $\binom{n-m}{k}$ leave the same remainder when divided by $2$. (Poland) PS. The translation was done using Google translate and in case it is not right, there is the original text in Slovak

2009 USAMO, 6

Tags: number theory , greatest common divisor , Hi

Let $s_1, s_2, s_3, \dots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \dots.$ Suppose that $t_1, t_2, t_3, \dots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ is an integer for all $i$ and $j$. Prove that there exists a rational number $r$ such that $(s_i - s_j)r$ and $(t_i - t_j)/r$ are integers for all $i$ and $j$.

1982 Austrian-Polish Competition, 8

Tags: geometry , 3D geometry , solid geometry , geometric inequality , tetrahedron

Let $P$ be a point inside a regular tetrahedron ABCD with edge length $1$. Show that $$d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD) \ge \frac{3}{2} \sqrt2$$ , with equality only when $P$ is the centroid of $ABCD$. Here $d(P,XY)$ denotes the distance from point $P$ to line $XY$.

2005 Bulgaria Team Selection Test, 3

Tags: function , algebra proposed , algebra

Let $\mathbb{R}^{*}$ be the set of non-zero real numbers. Find all functions $f : \mathbb{R}^{*} \to \mathbb{R}^{*}$ such that $f(x^{2}+y) = (f(x))^{2} + \frac{f(xy)}{f(x)}$, for all $x,y \in \mathbb{R}^{*}$ and $-x^{2} \not= y$.

1999 Yugoslav Team Selection Test, Problem 3

Tags: algebra , Sequence

Consider the set $A_n=\{x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n\}$ of $2n$ variables. How many permutations of set $A_n$ are there for which it is possible to assign real values from the interval $(0,1)$ to the $2n$ variables so that: (i) $x_i+y_i=1$ for each $i$; (ii) $x_1<x_2<\ldots<x_n$; (iii) the $2n$ terms of the permutation form a strictly increasing sequence?

2016 ELMO Problems, 1

Tags: algebra , number theory

Cookie Monster says a positive integer $n$ is $crunchy$ if there exist $2n$ real numbers $x_1,x_2,\ldots,x_{2n}$, not all equal, such that the sum of any $n$ of the $x_i$'s is equal to the product of the other $n$ of the $x_i$'s. Help Cookie Monster determine all crunchy integers. [i]Yannick Yao[/i]

Denmark (Mohr) - geometry, 2002.4

Tags: geometry , isosceles , right triangle

In triangle $ABC$ we have $\angle C = 90^o$ and $AC = BC$. Furthermore $M$ is an interior pont in the triangle so that $MC = 1 , MA = 2$ and $MB =\sqrt2$. Determine $AB$

2011 Kazakhstan National Olympiad, 4

Tags: number theory unsolved , number theory

Prove that there are infinitely many natural numbers, the arithmetic mean and geometric mean of the divisors which are both integers.

1994 IMO Shortlist, 3

Tags: modular arithmetic , number theory , prime numbers , Combinatorial Number Theory , IMO , IMO 1994 , combinatorics

Show that there exists a set $ A$ of positive integers with the following property: for any infinite set $ S$ of primes, there exist [i]two[/i] positive integers $ m$ in $ A$ and $ n$ not in $ A$, each of which is a product of $ k$ distinct elements of $ S$ for some $ k \geq 2$.

1965 IMO, 4

Tags: system of equations , algebra , IMO , IMO 1965

Find all sets of four real numbers $x_1, x_2, x_3, x_4$ such that the sum of any one and the product of the other three is equal to 2.

2001 Switzerland Team Selection Test, 10

Tags: Subsets , number theory , Sum , power of 2

Prove that every $1000$-element subset $M$ of the set $\{0,1,...,2001\}$ contains either a power of two or two distinct numbers whose sum is a power of two.

2020 Vietnam National Olympiad, 4

Tags: geometry , geometric transformation , reflection , parallelogram

Let a non-isosceles acute triangle ABC with the circumscribed cycle (O) and the orthocenter H. D, E, F are the reflection of O in the lines BC, CA and AB. a) $H_a$ is the reflection of H in BC, A' is the reflection of A at O and $O_a$ is the center of (BOC). Prove that $H_aD$ and OA' intersect on (O). b) Let X is a point satisfy AXDA' is a parallelogram. Prove that (AHX), (ABF), (ACE) have a comom point different than A

2006 AIME Problems, 11

Tags: LaTeX , AMC , USA(J)MO , USAMO , AIME , geometry , AIME I

A sequence is defined as follows $a_1=a_2=a_3=1$, and, for all positive integers $n$, $a_{n+3}=a_{n+2}+a_{n+1}+a_n$. Given that $a_{28}=6090307$, $a_{29}=11201821$, and $a_{30}=20603361$, find the remainder when $\displaystyle \sum^{28}_{k=1} a_k$ is divided by 1000.

2011 All-Russian Olympiad, 4

Tags: geometry , Russia , All Russian Math Olympiad

Perimeter of triangle $ABC$ is $4$. Point $X$ is marked at ray $AB$ and point $Y$ is marked at ray $AC$ such that $AX=AY=1$. Line segments $BC$ and $XY$ intersectat point $M$. Prove that perimeter of one of triangles $ABM$ or $ACM$ is $2$. (V. Shmarov).

2017 Taiwan TST Round 1, 4

Tags: geometry

Two line $BC$ and $EF$ are parallel. Let $D$ be a point on segment $BC$ different from $B$,$C$. Let $I$ be the intersection of $BF$ ans $CE$. Denote the circumcircle of $\triangle CDE$ and $\triangle BDF$ as $K$,$L$. Circle $K$,$L$ are tangent with $EF$ at $E$,$F$,respectively. Let $A$ be the other intersection of circle $K$ and $L$. Let $DF$ and circle $K$ intersect again at $Q$, and $DE$ and circle $L$ intersect again at $R$. Let $EQ$ and $FR$ intersect at $M$.\\ Prove that $I$, $A$, $M$ are collinear.

2012 Thailand Mathematical Olympiad, 9

Tags: algebra , polynomial , Real Roots , inequalities

Let $n$ be a positive integer and let $P(x) = x^n + a_{n-1}x^{n-1} +... + a_1x + 1$ be a polynomial with positive real coefficients. Under the assumption that the roots of $P$ are all real, show that $P(x) \ge (x + 1)^n$ for all $x > 0$.

1994 India Regional Mathematical Olympiad, 3

Tags:

Find all 6-digit numbers $a_1a_2a_3a_4a_5a_6$ formed by using the digits $1,2,3,4,5,6$ once each such that the number $a_1a_2a_2\ldots a_k$ is divisible by $k$ for $1 \leq k \leq 6$.

1960 Miklós Schweitzer, 3

Tags: college contests

[b]3.[/b] Let $f(z)$ with $f(0)=1$ be regular in the unit disk and let $\left [\frac{\partial^2 \mid f(z)\mid}{\partial x\partial y} \right ] _{z=0} =1$. Show thatthe area of the image of the unit disk by $w= f(z)$ (taken with multiplicity) is not less than $\frac {1} {2}$ .[b](f. 6)[/b]

2010 Brazil National Olympiad, 2

Tags: algebra , polynomial , ceiling function , number theory , prime numbers , algebra unsolved

Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.

2006 Switzerland - Final Round, 7

Tags: geometry , equal segments , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral with $\angle ABC = 60^o$ and $| BC | = | CD |$. Prove that $|CD| + |DA| = |AB|$

2011 Mathcenter Contest + Longlist, 1 sl1

Tags: inequalities , algebra

Let $a,b,c \in \mathbb{R}$. Prove that $$\sum_{cyc} (a^3-b^3)^2+3\sum_{cyc}(a^2-b^2)^2+6(a-b)(b-c)(c-a)(ab+ bc+ca) \ge 0.$$ [i](LightLucifer)[/i]

2011 LMT, 9