Olympiad problems

2009 Czech-Polish-Slovak Match, 4

Given a circle, let $AB$ be a chord that is not a diameter, and let $C$ be a point on the longer arc $AB$. Let $K$ and $L$ denote the reflections of $A$ and $B$, respectively, about lines $BC$ and $AC$, respectively. Prove that the distance between the midpoint of $AB$ and the midpoint of $KL$ is independent of the choice of $C$.

2009 Iran MO (2nd Round), 1

Tags: geometry , rectangle , induction , combinatorics

We have a $ (n+2)\times n $ rectangle and we’ve divided it into $ n(n+2) \ \ 1\times1 $ squares. $ n(n+2) $ soldiers are standing on the intersection points ($ n+2 $ rows and $ n $ columns). The commander shouts and each soldier stands on its own location or gaits one step to north, west, east or south so that he stands on an adjacent intersection point. After the shout, we see that the soldiers are standing on the intersection points of a $ n\times(n+2) $ rectangle ($ n $ rows and $ n+2 $ columns) such that the first and last row are deleted and 2 columns are added to the right and left (To the left $1$ and $1$ to the right). Prove that $ n $ is even.

1989 Tournament Of Towns, (221) 5

Tags: combinatorics , combinatorial geometry , line

We are given $N$ lines ($N > 1$ ) in a plane, no two of which are parallel and no three of which have a point in common. Prove that it is possible to assign, to each region of the plane determined by these lines, a non-zero integer of absolute value not exceeding $N$ , such that the sum of the integers o n either side of any of the given lines is equal to $0$ . (S . Fomin, Leningrad)

2008 Korea - Final Round, 4

Tags: modular arithmetic , number theory

For any positive integer $m\ge2$ define $A_m=\{m+1, 3m+2, 5m+3, 7m+4, \ldots, (2k-1)m + k, \ldots\}$. (1) For every $m\ge2$, prove that there exists a positive integer $a$ that satisfies $1\le a<m$ and $2^a\in A_m$ or $2^a+1\in A_m$. (2) For a certain $m\ge2$, let $a, b$ be positive integers that satisfy $2^a\in A_m$, $2^b+1\in A_m$. Let $a_0, b_0$ be the least such pair $a, b$. Find the relation between $a_0$ and $b_0$.

2015 Costa Rica - Final Round, G3

Tags: geometry , parallel , trisector , trisect

Let $\vartriangle A_1B_1C_1$ and $l_1, m_1, n_1$ be the trisectors closest to $A_1B_1$, $B_1C_1$, $C_1A_1$ of the angles $A_1, B_1, C_1$ respectively. Let $A_2 = l_1 \cap n_1$, $B_2 = m_1 \cap l_1$, $C_2 = n_1 \cap m_1$. So on we create triangles $\vartriangle A_nB_nC_n$ . If $\vartriangle A_1B_1C_1$ is equilateral prove that exists $n \in N$, such that all the sides of $\vartriangle A_nB_nC_n$ are parallel to the sides of $\vartriangle A_1B_1C_1$.

1993 All-Russian Olympiad Regional Round, 11.3

Tags: 3d geometry , sphere , pyramid , geometry

Point $O$ is the foot of the altitude of a quadrilateral pyramid. A sphere with center $O$ is tangent to all lateral faces of the pyramid. Points $A,B,C,D$ are taken on successive lateral edges so that segments $AB$, $BC$, and $CD$ pass through the three corresponding tangency points of the sphere with the faces. Prove that the segment $AD$ passes through the fourth tangency point

2014 Gulf Math Olympiad, 4

Tags: combinatorics , sum , chessboard

The numbers from $1$ to $64$ must be written on the small squares of a chessboard, with a different number in each small square. Consider the $112$ numbers you can make by adding the numbers in two small squares which have a common edge. Is it possible to write the numbers in the squares so that these $112$ sums are all different?

2018-IMOC, C1

Tags: combinatorics

IMOC is a small country without any lake. One day, the king decides to divide IMOC into many regions so that each region borders the sea. Prove that the map is $3$-colorable.

1996 Iran MO (2nd round), 3

Tags: trigonometry , geometry

Let $N$ be the midpoint of side $BC$ of triangle $ABC$. Right isosceles triangles $ABM$ and $ACP$ are constructed outside the triangle, with bases $AB$ and $AC$. Prove that $\triangle MNP$ is also a right isosceles triangle.

2009 Stars Of Mathematics, 2

Tags: geometry , circumcircle , circumcenter

Let $\omega$ be a circle in the plane and $A,B$ two points lying on it. We denote by $M$ the midpoint of $AB$ and let $P \ne M$ be a new point on $AB$. Build circles $\gamma$ and $\delta$ tangent to $AB$ at $P$ and to $\omega$ at $C$, respectively $D$. Consider $E$ to be the point diametrically opposed to $D$ in $\omega$. Prove that the circumcenter of $\triangle BMC$ lies on the line $BE$.

2016 Singapore Senior Math Olympiad, 1

Tags: geometry , angle bisector

Let $\triangle ABC$ be a triangle with $AB < AC$. Let the angle bisector of $\angle BAC$ meet $BC$ at $D$ , and let $M$ be the midpoint of $BC$ . Let $P$ be the foot of the perpendicular from $B$ to $AD$ . $Q$ the intersection of $BP$ and $AM$ . Show that : $(DQ) // (AB) $ .

1941 Moscow Mathematical Olympiad, 077

Tags: integer root , integer polynomial , polynomial , algebra

A polynomial $P(x)$ with integer coefficients takes odd values at $x = 0$ and $x = 1$. Prove that $P(x)$ has no integer roots.

2020 Brazil Team Selection Test, 4

Tags: combinatorics

Let $n$ be an odd positive integer. Some of the unit squares of an $n\times n$ unit-square board are colored green. It turns out that a chess king can travel from any green unit square to any other green unit squares by a finite series of moves that visit only green unit squares along the way. Prove that it can always do so in at most $\tfrac{1}{2}(n^2-1)$ moves. (In one move, a chess king can travel from one unit square to another if and only if the two unit squares share either a corner or a side.) [i]Proposed by Nikolai Beluhov[/i]

2020 Durer Math Competition Finals, 1

Tags: combinatorics , tiling , tiles

How many ways are there to tile a $3 \times 3$ square with $4$ dominoes of size $1 \times 2$ and $1$ domino of size $1 \times 1$? Tilings that can be obtained from each other by rotating the square are considered different. Dominoes of the same size are completely identical

2010 Mexico National Olympiad, 2

Tags: analytic geometry , invariant , induction , combinatorics

In each cell of an $n\times n$ board is a lightbulb. Initially, all of the lights are off. Each move consists of changing the state of all of the lights in a row or of all of the lights in a column (off lights are turned on and on lights are turned off). Show that if after a certain number of moves, at least one light is on, then at this moment at least $n$ lights are on.

2010 Today's Calculation Of Integral, 580

Tags: calculus , integration , analytic geometry , geometry , calculus computations

Let $ k$ be a positive constant number. Denote $ \alpha ,\ \beta \ (0<\beta <\alpha)$ the $ x$ coordinates of the curve $ C: y\equal{}kx^2\ (x\geq 0)$ and two lines $ l: y\equal{}kx\plus{}\frac{1}{k},\ m: y\equal{}\minus{}kx\plus{}\frac{1}{k}$.　Find the minimum area of the part bounded by the curve $ C$ and two lines $ l,\ m$.

2013 AMC 12/AHSME, 24

Tags: trigonometry , geometry , angle bisector , similar triangles , trig identities , law of cosines

Let $ABC$ be a triangle where $M$ is the midpoint of $\overline{AC}$, and $\overline{CN}$ is the angle bisector of $\angle ACB$ with $N$ on $\overline{AB}$. Let $X$ be the intersection of the median $\overline{BM}$ and the bisector $\overline{CN}$. In addition $\bigtriangleup BXN$ is equilateral and $AC=2$. What is $BN^2$? $\textbf{(A)}\ \frac{10-6\sqrt{2}}{7} \qquad\textbf{(B)}\ \frac{2}{9} \qquad\textbf{(C)}\ \frac{5\sqrt{2} - 3\sqrt{3}}{8} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{6} \qquad\textbf{(E)}\ \frac{3\sqrt{3} - 4}{5}$.

2013 Argentina Cono Sur TST, 3

Tags: analytic geometry , geometry , rectangle , inequalities , pythagorean theorem , combinatorics

$1390$ ants are placed near a line, such that the distance between their heads and the line is less than $1\text{cm}$ and the distance between the heads of two ants is always larger than $2\text{cm}$. Show that there is at least one pair of ants such that the distance between their heads is at least $10$ meters (consider the head of an ant as point).

2020-2021 OMMC, 6

Tags:

Find the minimum possible value of $$\left(\sqrt{x^2+4} + \sqrt{x^2+7\sqrt{3}x + 49}\right)^2$$ over all real numbers.

2016 Poland - Second Round, 6

Tags: graph theory , combinatorics

$n$ ($n \ge 4$) green points are in a data space and no $4$ green points lie on one plane. Some segments which connect green points have been colored red. Number of red segments is even. Each two green points are connected with polyline which is build from red segments. Show that red segments can be split on pairs, such that segments from one pair have common end.

2015 CCA Math Bonanza, T2

Tags:

How many ways are there to rearrange the letters of the word RAVEN such that no two vowels are consecutive? [i]2015 CCA Math Bonanza Team Round #2[/i]

1998 Korea Junior Math Olympiad, 7

Tags: geometry , circumcircle , angle bisector

$O$ is a circumcircle of non-isosceles triangle $ABC$ and the angle bisector of $A$ meets $BC$ at $D$. If the line perpendicular to $BC$ passing through $D$ meets $AO$ at $E$, show that $ADE$ is an isosceles triangle.

2019 Miklós Schweitzer, 10

Tags: linear algebra

Let $A$ and $B$ be positive self-adjoint operators on a complex Hilbert space $H$. Prove that \[\limsup_{n \to \infty} \|A^n x\|^{1/n} \le \limsup_{n \to \infty} \|B^n x\|^{1/n}\] holds for every $x \in H$ if and only if $A^n \le B^n$ for each positive integer $n$.

1987 All Soviet Union Mathematical Olympiad, 462

Tags: algebra , inequalities

Prove that for every natural $n$ the following inequality is held: $$(2n + 1)^n \ge (2n)^n + (2n - 1)^n$$

2016 IFYM, Sozopol, 3

Tags: functional equation , function , algebra

Let $f: \mathbb{R}^2\rightarrow \mathbb{R}$ be a function for which for arbitrary $x,y,z\in \mathbb{R}$ we have that $f(x,y)+f(y,z)+f(z,x)=0$. Prove that there exist function $g:\mathbb{R}\rightarrow \mathbb{R}$ for which: $f(x,y)=g(x)-g(y),\, \forall x,y\in \mathbb{R}$.