Olympiad problems

1985 IMO Longlists, 93

The sphere inscribed in tetrahedron $ABCD$ touches the sides $ABD$ and $DBC$ at points $K$ and $M$, respectively. Prove that $\angle AKB = \angle DMC$.

2021 Durer Math Competition Finals, 10

Tags: combinatorics

Billy owns some really energetic horses. They are hopping around on points of the plane having two integer coordinates. Each horse can make the following types of jumps. They can hop to a point obtained from their current position via a translation by vector $(15, 9)$, $(-9, 15)$, $(-15,-9)$ or $(9,-15)$. The horses are now standing on lattice points such that no two can meet by making jumps as above. What is the maximal possible number of horses Billy can own?

Russian TST 2016, P1

Tags: number theory , divisibility

Let $a{}$ and $b{}$ be natural numbers greater than one. Let $n{}$ be a natural number for which $a\mid 2^n-1$ and $b\mid 2^n+1$. Prove that there is no natural $k{}$ such that $a\mid 2^k+1$ and $b\mid 2^k-1$.

2012 Turkmenistan National Math Olympiad, 7

Tags: algebra

If $a,b,c$ are positive real numbers and satisfy: $\frac{a_1}{b_1}=\frac{a_2}{b_2}=...=\frac{a_n}{b_n}$ then prove that :$ \sum_{i=1}^{n} a^{2}_i \cdot \sum_{i=1}^{n} b^{2}_i =(\sum_{i=1}^{n} a_{i}b_{i})^2$

Ukraine Correspondence MO - geometry, 2012.7

Tags: geometry , bisects sements

Let $O$ and $H$ be the center of the circumcircle and the point of intersection of the altitudes of the acute triangle $ABC$ respectively, $D$ be the foot of the altitude drawn to $BC$, and $E$ be the midpoint of $AO$. Prove that the circumcircle of the triangle $ADE$ passes through the midpoint of the segment $OH$.

2016 District Olympiad, 1

Tags: geometry , 3d geometry , pyramid

Let be a pyramid having a square as its base and the projection of the top vertex to the base is the center of the square. Prove that two opposite faces are perpendicular if and only if the angle between two adjacent faces is $ 120^{\circ } . $

2018 Saudi Arabia GMO TST, 3

Tags: tangent , geometry , perpendicular , arc midpoint

Let $C$ be a point lies outside the circle $(O)$ and $CS, CT$ are tangent lines of $(O)$. Take two points $A, B$ on $(O)$ with $M$ is the midpoint of the minor arc $AB$ such that $A, B, M$ differ from $S, T$. Suppose that $MS, MT$ cut line $AB$ at $E, F$. Take $X \in OS$ and $Y \in OT$ such that $EX, FY$ are perpendicular to $AB$. Prove that $X Y$ and $C M$ are perpendicular.

2015 Junior Regional Olympiad - FBH, 1

Tags: counting years

Father is $42$ years old, and son has $14$ years. In how many years father will be twice as old as his son?

2000 Bosnia and Herzegovina Team Selection Test, 4

Tags: inequality , inequalities , algebra

Prove that for all positive real $a$, $b$ and $c$ holds: $$ \frac{bc}{a^2+2bc}+\frac{ac}{b^2+2ac}+\frac{ab}{c^2+2ab} \leq 1 \leq \frac{a^2}{a^2+2bc}+\frac{b^2}{b^2+2ac}+\frac{c^2}{c^2+2ab}$$

2013 Dutch IMO TST, 4

Tags: modular arithmetic , number theory

Determine all positive integers $n\ge 2$ satisfying $i+j\equiv\binom ni +\binom nj \pmod{2}$ for all $i$ and $j$ with $0\le i\le j\le n$.

2021 Azerbaijan IMO TST, 3

Tags: algebra , polynomial , algorithm , lagrange s interpolation

A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?

2013 Brazil Team Selection Test, 1

Tags: quadratic , number theory

Call admissible a set $A$ of integers that has the following property: If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$. Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. [i]Proposed by Warut Suksompong, Thailand[/i]

1985 IMO Longlists, 88

Tags: polynomial , calculus , binomial theorem , algebra

Determine the range of $w(w + x)(w + y)(w + z)$, where $x, y, z$, and $w$ are real numbers such that \[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\]

2013 Romania National Olympiad, 1

Tags: geometry , angle bisector , median , parallel

In the triangle $ABC$, the angle - bisector $AD$ ($D \in BC$) and the median $BE$ ($E \in AC$) intersect at point $P$. Lines $AB$ and $CP$ intesect at point $F$. The parallel through $B$ to $CF$ intersects $DF$ at point $M$. Prove that $DM = BF$

2021 Korea National Olympiad, P5

Tags: sequence , inequalities , algebra , polynomial

A real number sequence $a_1, \cdots ,a_{2021}$ satisfies the below conditions. $$a_1=1, a_2=2, a_{n+2}=\frac{2a_{n+1}^2}{a_n+a_{n+1}} (1\leq n \leq 2019)$$ Let the minimum of $a_1, \cdots ,a_{2021}$ be $m$, and the maximum of $a_1, \cdots ,a_{2021}$ be $M$. Let a 2021 degree polynomial $$P(x):=(x-a_1)(x-a_2) \cdots (x-a_{2021})$$ $|P(x)|$ is maximum in $[m, M]$ when $x=\alpha$. Show that $1<\alpha <2$.

2011 Greece Junior Math Olympiad, 2

Tags: number theory

We consider the set of four-digit positive integers $x =\overline{abcd}$ with digits different than zero and pairwise different. We also consider the integers $y = \overline{dcba}$ and we suppose that $x > y$. Find the greatest and the lowest value of the difference $x-y$, as well as the corresponding four-digit integers $x,y$ for which these values are obtained.

1998 All-Russian Olympiad, 1

Tags: analytic geometry , vieta , algebra

Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right. Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$.

2011 Swedish Mathematical Competition, 3

Tags: system of equations , system , algebra

Find all positive real numbers $x, y, z$, such that $$x - \frac{1}{y^2} = y - \frac{1}{z^2}= z - \frac{1}{x^2}$$

2010 Contests, 1

Tags: number theory

Find all primes $p,q$ such that $p^3-q^7=p-q$.

Geometry Mathley 2011-12, 11.2

Tags: equal segments , geometry

Let $ABC$ be a triangle inscribed in the circle $(O)$. Tangents at $B,C$ of the circles $(O)$ meet at $T$ . Let $M,N$ be the points on the rays $BT,CT$ respectively such that $BM = BC = CN$. The line through $M$ and $N$ intersects $CA,AB$ at $E, F$ respectively; $BE$ meets $CT$ at $P, CF$ intersects $BT$ at $Q$. Prove that $AP = AQ$. Trần Quang Hùng

2014 Brazil Team Selection Test, 3

Tags: circumcircle , isosceles triangle , angle chasing , inversion , geometry

Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA = \angle QBA = \angle ACB $ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $BQ$ for which $PD=PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \neq A$. Prove that $QB = QR$.

2020 LMT Spring, 26

Tags:

A magic $3 \times 5$ board can toggle its cells between black and white. Define a \textit{pattern} to be an assignment of black or white to each of the board's $15$ cells (so there are $2^{15}$ patterns total). Every day after Day 1, at the beginning of the day, the board gets bored with its black-white pattern and makes a new one. However, the board always wants to be unique and will die if any two of its patterns are less than $3$ cells different from each other. Furthermore, the board dies if it becomes all white. If the board begins with all cells black on Day $1$, compute the maximum number of days it can stay alive.