Olympiad problems

1979 Kurschak Competition, 3

Tags: combinatorics

An $n \times n$ array of letters is such that no two rows are the same. Show that it must be possible to omit a column, so that the remaining table has no two rows the same.

2010 Tournament Of Towns, 6

Tags: perpendicular bisector , incenter , geometric transformation , geometry , reflection

In acute triangle $ABC$, an arbitrary point $P$ is chosen on altitude $AH$. Points $E$ and $F$ are the midpoints of sides $CA$ and $AB$ respectively. The perpendiculars from $E$ to $CP$ and from $F$ to $BP$ meet at point $K$. Prove that $KB = KC$.

2016 Azerbaijan National Mathematical Olympiad, 4

Tags: algebra , function

Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation $$\sum_{i=1}^{2015} f(x_i + x_{i+1}) + f\left( \sum_{i=1}^{2016} x_i \right) \le \sum_{i=1}^{2016} f(2x_i)$$ for all real numbers $x_1, x_2, ... , x_{2016}.$

2018 239 Open Mathematical Olympiad, 10-11.5

Tags: geometry

Given a trapezoid $ABCD$, with $AB\parallel CD$. Lines $AC$ and $BD$ intersect at point $E$, and lines $AD$ and $BC$ intersect at point $F$. It turns out that the circle with diameter $EF$ is tangent to the midline of the trapezoid. Prove that there exists a square such that there is a mutual correspondence between all six lines containing pairs of its vertices, and points $A$, $B$, $C$, $D$, $E$, and $F$: each line corresponds to a point lying on it. [i]Proposed by V. Mokin[/i]

2006 Miklós Schweitzer, 10

Tags: real analysis , convex , compactness , euclidean space

Let $K_1,...,K_d$ be convex, compact sets in $R^d$ with non-empty interior. Suppose they are strongly separated, which means for any choice of $x_1 \in K_1, x_2 \in K_2, ...$, their affine hull is a hyperplane in $R^d$. Also let $0< \alpha_i <1$. A half-space H is called an $\alpha$-cut if $vol(K_i \cap H) = \alpha_i\cdot vol(K_i)$ for all i. How many $\alpha$-cuts are there?

2024 India IMOTC, 5

Tags: geometry

Let $ABC$ be an acute angled triangle with $AC>AB$ and incircle $\omega$. Let $\omega$ touch the sides $BC, CA,$ and $AB$ at $D, E,$ and $F$ respectively. Let $X$ and $Y$ be points outside $\triangle ABC$ satisfying \[\angle BDX = \angle XEA = \angle YDC = \angle AFY = 45^{\circ}.\] Prove that the circumcircles of $\triangle AXY, \triangle AEF$ and $\triangle ABC$ meet at a point $Z\ne A$. [i]Proposed by Atul Shatavart Nadig and Shantanu Nene[/i]

1970 All Soviet Union Mathematical Olympiad, 137

Tags: combinatorics , number theory , divisible , sum

Prove that from every set of $200$ integers you can choose a subset of $100$ with the total sum divisible by $100$.

2021 New Zealand MO, 2

Tags: inequalities , algebra

Prove that $$x^2 +\frac{8}{xy}+ y^2 \ge 8$$ for all positive real numbers $x$ and $y$.

2015 BMT Spring, P2

Tags: geometry

Suppose that fixed circle $C_1$ with radius $a > 0$ is tangent to the fixed line $\ell$ at $A$. Variable circle $C_2$, with center $X$, is externally tangent to $C_1$ at $B \ne A$ and $\ell$ at $C$. Prove that the set of all $X$ is a parabola minus a point

1983 Austrian-Polish Competition, 1

Tags: algebra , inequalities

Nonnegative real numbers $a, b,x,y$ satisfy $a^5 + b^5 \le $1 and $x^5 + y^5 \le 1$. Prove that $a^2x^3 + b^2y^3 \le 1$.

2023 Nordic, P1

Tags: combinatorics

Alice and Bianca have one hundred marbles. At the start of the game they split these hundred marbles into two piles. Thereafter, a move consists of choosing a pile, then choosing a positive integer not larger than half of the number of marbles in that pile, and finally removing that number of marbles from the chosen pile. The first player unable to remove any marbles loses. Alice makes the first move of the game. Determine all initial pile sizes for which Bianca has a winning strategy.

1967 Dutch Mathematical Olympiad, 3

Tags: geometry , parallel , pentagon

The convex pentagon $ABC DE$ is given, such that $AB,BC,CD$ and $DE$ are parallel to one of the diagonals. Prove that this also applies to $EA$.

2015 AIME Problems, 10

Tags: algebra , absolute value , polynomial

Let $f(x)$ be a third-degree polynomial with real coefficients satisfying \[|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.\] Find $|f(0)|$.

2013 AMC 12/AHSME, 4

Tags: algebra

What is the value of \[\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}?\] $ \textbf{(A)}\ -1\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ \frac{5}{3}\qquad\textbf{(D)}\ 2013\qquad\textbf{(E)}\ 2^{4024} $

2018 Korea National Olympiad, 1

Tags: geometry , incenter

Let there be an acute triangle $\triangle ABC$ with incenter $I$. $E$ is the foot of the perpendicular from $I$ to $AC$. The line which passes through $A$ and is perpendicular to $BI$ hits line $CI$ at $K$. The line which passes through $A$ and is perpendicular to $CI$ hits the line which passes through $C$ and is perpendicular to $BI$ at $L$. Prove that $E, K, L$ are colinear.

2021 China Team Selection Test, 3

Tags: algebra , number theory , diophantine equation

Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation $$ax+by+cz=n.$$ Prove that there exists constants $\alpha, \beta, \gamma \in \mathbb{R}$ such that for any non-negative integer $n$, $$|f(n)- \left( \alpha n^2+ \beta n + \gamma \right) | < \frac{1}{12} \left( a+b+c \right).$$

2010 Today's Calculation Of Integral, 545

Tags: function , calculus , calculus computations , integration

(1) Evaluate $ \int_0^1 xe^{x^2}dx$. (2) Let $ I_n\equal{}\int_0^1 x^{2n\minus{}1}e^{x^2}dx$. Express $ I_{n\plus{}1}$ in terms of $ I_n$.

2022 Balkan MO Shortlist, N4

Tags: number theory

A hare and a tortoise run in the same direction, at constant but different speeds, around the base of a tall square tower. They start together at the same vertex, and the run ends when both return to the initial vertex simultaneously for the first time. Suppose the hare runs with speed 1, and the tortoise with speed less than 1. For what rational numbers $q{}$ is it true that, if the tortoise runs with speed $q{}$, the fraction of the entire run for which the tortoise can see the hare is also $q{}$?

2018 BAMO, A

Tags: combinatorics , grid , square table

Twenty-five people of different heights stand in a $5\times 5$ grid of squares, with one person in each square. We know that each row has a shortest person, suppose Ana is the tallest of these five people. Similarly, we know that each column has a tallest person, suppose Bev is the shortest of these five people. Assuming Ana and Bev are not the same person, who is taller: Ana or Bev? Prove that your answer is always correct.

2014 Moldova Team Selection Test, 3

Tags: circumcircle , trigonometry , geometry

Let $\triangle ABC$ be a triangle with $\angle A$-acute. Let $P$ be a point inside $\triangle ABC$ such that $\angle BAP = \angle ACP$ and $\angle CAP =\angle ABP$. Let $M, N$ be the centers of the incircle of $\triangle ABP$ and $\triangle ACP$, and $R$ the radius of the circumscribed circle of $\triangle AMN$. Prove that $\displaystyle \frac{1}{R}=\frac{1}{AB}+\frac{1}{AC}+\frac{1}{AP}. $

2000 JBMO ShortLists, 19

Tags: geometry

Let $ABC$ be a triangle. Find all the triangles $XYZ$ with vertices inside triangle $ABC$ such that $XY,YZ,ZX$ and six non-intersecting segments from the following $AX, AY, AZ, BX, BY, BZ, CX, CY, CZ$ divide the triangle $ABC$ into seven regions with equal areas.

1984 Spain Mathematical Olympiad, 8

Tags: polynomial , algebra , determinant , remainder

Find the remainder upon division by $x^2-1$ of the determinant $$\begin{vmatrix} x^3+3x & 2 & 1 & 0 \\ x^2+5x & 3 & 0 & 2 \\x^4 +x^2+1 & 2 & 1 & 3 \\x^5 +1 & 1 & 2 & 3 \\ \end{vmatrix}$$

2004 Bulgaria National Olympiad, 4

Tags: combinatorics , geometry , induction , function , invariant , geometric transformation , group theory

In a word formed with the letters $a,b$ we can change some blocks: $aba$ in $b$ and back, $bba$ in $a$ and backwards. If the initial word is $aaa\ldots ab$ where $a$ appears 2003 times can we reach the word $baaa\ldots a$, where $a$ appears 2003 times.

1997 Canadian Open Math Challenge, 5

Tags:

Two cubes have their faces painted either red or blue. The 1st cube has five red faces and one blue face. When the two cubes are rolled simultaneously, the probability that the two top faces show the same color is $\frac{1}{2}$. How many red faces are there on the second cube?

2023 Turkey EGMO TST, 3

Tags: inequalities , minimum value

Let $x,y,z$ be positive real numbers that satisfy at least one of the inequalities, $2xy>1$, $yz>1$. Find the least possible value of $$xy^3z^2+\frac{4z}{x}-8yz-\frac{4}{yz}$$ .