Olympiad problems

1986 Miklós Schweitzer, 10

Let $X_1, X_2$ be independent, identically distributed random variables such that $X_i\geq 0$ for all $i$. Let $\mathrm EX_i=m$, $\mathrm{Var} (X_i)=\sigma ^2<\infty$. Show that, for all $0<\alpha\leq 1$ $$\lim_{n\to\infty} n\,\mathrm{Var} \left( \left[ \frac{X_1+\ldots +X_n}{n}\right] ^\alpha\right)=\frac{\alpha ^ 2 \sigma ^ 2}{m^{2(1-\alpha)}}$$ [Gy. Michaletzki]

1989 India National Olympiad, 3

Tags: combinatorics unsolved , combinatorics

Let $ A$ denote a subset of the set $ \{ 1,11,21,31, \dots ,541,551 \}$ having the property that no two elements of $ A$ add up to $ 552$. Prove that $ A$ can't have more than $ 28$ elements.

1968 IMO Shortlist, 19

Tags: combinatorics , circle , algebra , approximation , IMO Shortlist

We are given a fixed point on the circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0, 1, 2, \ldots $ from it we obtain points with abscisas $n = 0, 1, 2, .\ldots$ respectively. How many points among them should we take to ensure that some two of them are less than the distance $\frac 15$ apart ?

1997 Turkey Team Selection Test, 2

Tags: modular arithmetic , abstract algebra , number theory proposed , number theory

Show that for each prime $p \geq 7$, there exist a positive integer $n$ and integers $x_{i}$, $y_{i}$ $(i = 1, . . . , n)$, not divisible by $p$, such that $x_{i}^{2}+ y_{i}^{2}\equiv x_{i+1}^{2}\pmod{p}$ where $x_{n+1} = x_{1}$

2009 Today's Calculation Of Integral, 501

Tags: calculus , integration , geometry , 3D geometry , sphere , dodecahedron , calculus computations

Find the volume of the uion $ A\cup B\cup C$ of the three subsets $ A,\ B,\ C$ in $ xyz$ space such that: \[ A\equal{}\{(x,\ y,\ z)\ |\ |x|\leq 1,\ y^2\plus{}z^2\leq 1\}\] \[ B\equal{}\{(x,\ y,\ z)\ |\ |y|\leq 1,\ z^2\plus{}x^2\leq 1\}\] \[ C\equal{}\{(x,\ y,\ z)\ |\ |z|\leq 1,\ x^2\plus{}y^2\leq 1\}\]

1985 IMO Longlists, 96

Tags: function , limit , algebra unsolved , algebra

Determine all functions $f : \mathbb R \to \mathbb R$ satisfying the following two conditions: (a) $f(x + y) + f(x - y) = 2f(x)f(y)$ for all $x, y \in \mathbb R$, and (b) $\lim_{x\to \infty} f(x) = 0$.

2008 Dutch IMO TST, 4

Tags: Perfect Square , number theory , radical

Let $n$ be positive integer such that $\sqrt{1 + 12n^2}$ is an integer. Prove that $2 + 2\sqrt{1 + 12n^2}$ is the square of an integer.

2014 France Team Selection Test, 6

Tags: inequalities proposed , inequalities

Let $n$ be a positive integer and $x_1,x_2,\ldots,x_n$ be positive reals. Show that there are numbers $a_1,a_2,\ldots, a_n \in \{-1,1\}$ such that the following holds: \[a_1x_1^2+a_2x_2^2+\cdots+a_nx_n^2 \ge (a_1x_1+a_2x_2 +\cdots+a_nx_n)^2\]

2019 Online Math Open Problems, 16

Tags: Online Math Open

Let $ABC$ be a scalene triangle with inradius $1$ and exradii $r_A$, $r_B$, and $r_C$ such that \[20\left(r_B^2r_C^2+r_C^2r_A^2+r_A^2r_B^2\right)=19\left(r_Ar_Br_C\right)^2.\] If \[\tan\frac{A}{2}+\tan\frac{B}{2}+\tan\frac{C}{2}=2.019,\] then the area of $\triangle{ABC}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Tristan Shin[/i]

2009 South East Mathematical Olympiad, 4

Tags: geometry , combinatorics proposed , combinatorics

Given 12 red points on a circle , find the mininum value of $n$ such that there exists $n$ triangles whose vertex are the red points . Satisfies: every chord whose points are the red points is the edge of one of the $n$ triangles .

1991 China National Olympiad, 4

Tags: number theory unsolved , number theory

Find all positive integer solutions $(x,y,z,n)$ of equation $x^{2n+1}-y^{2n+1}=xyz+2^{2n+1}$, where $n\ge 2$ and $z \le 5\times 2^{2n}$.

1974 USAMO, 4

Tags: probability

A father, a mother and son hold a family tournament, playing a two person board game with no ties. The tournament rules are: (i) The weakest player chooses the first two contestants. (ii) The winner of any game plays the next game against the person left out. (iii) The first person to win two games wins the tournament. The father is the weakest player, the son the strongest, and it is assumed that any player's probability of winning an individual game from another player does not change during the tournament. Prove that the father's optimal strategy for winning the tournament is to play the first game with his wife.

2020 Jozsef Wildt International Math Competition, W39

Tags: inequalities

Prove that: i) $$\sum_{k=1}^{n-1}(1+\ln k)\le n^2-n+1$$ ii) $$\sum_{k=1}^{n-1}\sqrt{\ln k}\le\frac{n^2-n+1}2$$ [i]Proposed by Laurențiu Modan[/i]

2022 Korea Junior Math Olympiad, 6

Tags: geometry , circumcircle , Concyclic , tangent

Let $ABC$ be a isosceles triangle with $\overline{AB}=\overline{AC}$. Let $D(\neq A, C)$ be a point on the side $AC$, and circle $\Omega$ is tangent to $BD$ at point $E$, and $AC$ at point $C$. Denote by $F(\neq E)$ the intersection of the line $AE$ and the circle $\Omega$, and $G(\neq a)$ the intersection of the line $AC$ and the circumcircle of the triangle $ABF$. Prove that points $D, E, F,$ and $G$ are concyclic.

2017 Kosovo National Mathematical Olympiad, 5

Tags: geometry

A sphere with ray $R$ is cut by two parallel planes. such that the center of the sphere is outside the region determined by these planes. Let $S_{1}$ and $S_{2}$ be the areas of the intersections, and $d$ the distance between these planes. Find the area of the intersection of the sphere with the plane parallel with these two planes, with equal distance from them.

2019 Latvia Baltic Way TST, 1

Tags: inequalities

Prove that for all positive real numbers $a, b, c$ with $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} =1$ the following inequality holds: $$3(ab+bc+ca)+\frac{9}{a+b+c} \le \frac{9abc}{a+b+c} + 2(a^2+b^2+c^2)+1$$

2011 Mathcenter Contest + Longlist, 9 sl13

Tags: algebra , inequalities

Let $a,b,c\in\mathbb{R^+}$ If $3=a+b+c\le 3abc$ , prove that $$\frac{1}{\sqrt{2a+1}}+ \frac{1}{\sqrt{2b+1}}+\frac{1}{\sqrt{2c+1}}\le \left( \frac32\right)^{3/2}$$ [i](Real Matrik)[/i]

1994 IMC, 4

Tags: IMC , linear algebra , matrix

Let $A$ be a $n\times n$ diagonal matrix with characteristic polynomial $$(x-c_1)^{d_1}(x-c_2)^{d_2}\ldots (x-c_k)^{d_k}$$ where $c_1, c_2, \ldots, c_k$ are distinct (which means that $c_1$ appears $d_1$ times on the diagonal, $c_2$ appears $d_2$ times on the diagonal, etc. and $d_1+d_2+\ldots + d_k=n$). Let $V$ be the space of all $n\times n$ matrices $B$ such that $AB=BA$. Prove that the dimension of $V$ is $$d_1^2+d_2^2+\cdots + d_k^2$$

2024 Serbia National Math Olympiad, 1

Tags: geometric sequence , number theory

Find all positive integers $n$, such that if their divisors are $1=d_1<d_2<\ldots<d_k=n$ for $k \geq 4$, then the numbers $d_2-d_1, d_3-d_2, \ldots, d_k-d_{k-1}$ form a geometric progression in some order.

2003 Junior Tuymaada Olympiad, 2

Tags: number theory , prime , Perfect Squares , Perfect Square

Find all natural $ x $ for which $ 3x+1 $ and $ 6x-2 $ are perfect squares, and the number $ 6x^2-1 $ is prime.

2024 Nigerian MO Round 3, Problem 1

Tags: algebra , mod , number theory , modular arithmetic

Find the value of $$(2^{40}+12^{41}+23^{42}+67^{43}+87^{44})^{45!+46}\mod11$$ (variation but same answer) [hide=Answer]3[/hide]

1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 8

Tags: geometry

Place three discs with radius $ r$ in a square with sides of length 1 so that the discs do not intersect: as on the figure. What is the greatest possible value of $ r$? [img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1997Number8.jpg[/img] A. $ \frac {1}{3}$ B. $ \frac {1}{4}$ C. $ \frac {\sqrt {2}}{6}$ D. $ 2 \sqrt {2} \minus{} \sqrt {6}$ E. $ \frac {\sqrt {2}}{1 \plus{} 2 \sqrt {2} \plus{} \sqrt {3}}$

1986 Tournament Of Towns, (110) 4

Tags: geometry , circumcircle , square , diagonal

We are given the square $ABCD$. On sides $AB$ and $CD$ we are given points $ K$ and $L$ respectively, and on segment $KL$ we are given point $M$ . Prove that the second intersection point (i.e. the one other than $M$) of the intersection points of circles circumscribed around triangles $AKM$ and $MLC$ lies on the diagonal $AC$. (V . N . Dubrovskiy)

2015 Balkan MO Shortlist, A5

Tags: inequalities , algebra , Exponential inequality , exponential , Exponential equation

Let $m, n$ be positive integers and $a, b$ positive real numbers different from $1$ such thath $m > n$ and $$\frac{a^{m+1}-1}{a^m-1} = \frac{b^{n+1}-1}{b^n-1} = c$$. Prove that $a^m c^n > b^n c^{m}$ (Turkey)

2001 Tuymaada Olympiad, 4

Tags: function , combinatorics proposed , combinatorics

Is it possible to colour all positive real numbers by 10 colours so that every two numbers with decimal representations differing in one place only are of different colours? (We suppose that there is no place in a decimal representations such that all digits starting from that place are 9's.) [i]Proposed by A. Golovanov[/i]