Olympiad problems

2005 Taiwan TST Round 3, 2

Given a triangle $ABC$, we construct a circle $\Gamma$ through $B,C$ with center $O$. $\Gamma$ intersects $AC, AB$ at points $D$, $E$, respectively($D$, $E$ are distinct from $B$ and $C$). Let the intersection of $BD$ and $CE$ be $F$. Extend $OF$ so that it intersects the circumcircle of $\triangle ABC$ at $P$. Show that the incenters of triangles $PBD$ and $PCE$ coincide.

2013 Stanford Mathematics Tournament, 3

Tags: calculus , limit , trigonometry , analytic geometry

Suppose $a$ and $b$ are real numbers such that \[\lim_{x\to 0}\frac{\sin^2 x}{e^{ax}-bx-1}=\frac{1}{2}.\] Determine all possible ordered pairs $(a, b)$.

1997 All-Russian Olympiad Regional Round, 8.1

Tags: number theory , perfect square , combinatorics

Prove that the numbers from $1$ to $16$ can be written in a line, but cannot be written in a circle so that the sum of any two adjacent numbers is square of a natural number.

2004 India National Olympiad, 1

Tags: parallelogram , geometry

$ABCD$ is a convex quadrilateral. $K$, $L$, $M$, $N$ are the midpoints of the sides $AB$, $BC$, $CD$, $DA$. $BD$ bisects $KM$ at $Q$. $QA = QB = QC = QD$ , and$\frac{LK}{LM} = \frac{CD}{CB}$. Prove that $ABCD$ is a square

2017 ASDAN Math Tournament, 2

Tags: geometry test

An equilateral triangle $ABC$ shares a side with a square $BCDE$. If the resulting pentagon has a perimeter of $20$, what is the area of the pentagon? (The triangle and square do not overlap).

1950 Moscow Mathematical Olympiad, 183

Tags: geometry , perimeter , square , circles

A circle is inscribed in a triangle and a square is circumscribed around this circle so that no side of the square is parallel to any side of the triangle. Prove that less than half of the square’s perimeter lies outside the triangle.

1968 Miklós Schweitzer, 7

Tags: function , limit , advanced fields

For every natural number $ r$, the set of $ r$-tuples of natural numbers is partitioned into finitely many classes. Show that if $ f(r)$ is a function such that $ f(r)\geq 1$ and $ \lim _{r\rightarrow \infty} f(r)\equal{}\plus{}\infty$, then there exists an infinite set of natural numbers that, for all $ r$, contains $ r$-triples from at most $ f(r)$ classes. Show that if $ f(r) \not \rightarrow \plus{}\infty$, then there is a family of partitions such that no such infinite set exists. [i]P. Erdos, A. Hajnal[/i]

2001 Balkan MO, 4

Tags: 3d geometry , analytic geometry , combinatorics , geometry

A cube side 3 is divided into 27 unit cubes. The unit cubes are arbitrarily labeled 1 to 27 (each cube is given a different number). A move consists of swapping the cube labeled 27 with one of its 6 neighbours. Is it possible to find a finite sequence of moves at the end of which cube 27 is in its original position, but cube $n$ has moved to the position originally occupied by $27-n$ (for each $n = 1, 2, \ldots , 26$)?

2007 IMAR Test, 2

Tags: geometry , 3d geometry , sphere , combinatorics

Denote by $ \mathcal{C}$ the family of all configurations $ C$ of $ N > 1$ distinct points on the sphere $ S^2,$ and by $ \mathcal{H}$ the family of all closed hemispheres $ H$ of $ S^2.$ Compute: $ \displaystyle\max_{H\in\mathcal{H}}\displaystyle\min_{C\in\mathcal{C}}|H\cap C|$, $ \displaystyle\min_{H\in\mathcal{H}}\displaystyle\max_{C\in\mathcal{C}}|H\cap C|$ $ \displaystyle\max_{C\in\mathcal{C}}\displaystyle\min_{H\in\mathcal{H}}|H\cap C|$ and $ \displaystyle\min_{C\in\mathcal{C}}\displaystyle\max_{H\in\mathcal{H}}|H\cap C|.$

2023 Tuymaada Olympiad, 6

Tags: number theory , euclidean algorithm

An $\textit{Euclidean step}$ transforms a pair $(a, b)$ of positive integers, $a > b$, to the pair $(b, r)$, where $r$ is the remainder when a is divided by $b$. Let us call the $\textit{complexity}$ of a pair $(a, b)$ the number of Euclidean steps needed to transform it to a pair of the form $(s, 0)$. Prove that if $ad - bc = 1$, then the complexities of $(a, b)$ and $(c, d)$ differ at most by $2$.

2000 Tournament Of Towns, 6

Tags: tournament , combinatorics

In the spring round of the Tournament of Towns this year, $6$ problems were posed in the Senior A-Level paper. In a certain country, each problem was solved by exactly $1000$ participants, but no two participants solved all $6$ problems between them. What is the smallest possible number of participants from this country in the spring round Senior A-Level paper? (R Zhenodarov)

2005 Estonia Team Selection Test, 5

Tags: coloring , combinatorics

On a horizontal line, $2005$ points are marked, each of which is either white or black. For every point, one finds the sum of the number of white points on the right of it and the number of black points on the left of it. Among the $2005$ sums, exactly one number occurs an odd number of times. Find all possible values of this number.

2020 Canadian Mathematical Olympiad Qualification, 8

Tags: radical , rational , equation , algebra

Find all pairs $(a, b)$ of positive rational numbers such that $\sqrt[b]{a}= ab$

2013 Today's Calculation Of Integral, 897

Tags: calculus , integration , geometry , geometric transformation , rotation , calculus computations , logarithm

Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.

2019 LIMIT Category C, Problem 2

Tags: inequalities , absolute value

Let $x,y\in[0,\infty)$. Which of the following is true? $\textbf{(A)}~\left|\log\left(1+x^2\right)-\log\left(1+y^2\right)\right|\le|x-y|$ $\textbf{(B)}~\left|\sin^2x-\sin^2y\right|\le|x-y|$ $\textbf{(C)}~\left|\tan^{-1}x-\tan^{-1}y\right|\le|x-y|$ $\textbf{(D)}~\text{None of the above}$

2004 AIME Problems, 2

Tags: absolute value , p2 catalyst , wilkinson catalyst , birch reduction

Set $A$ consists of $m$ consecutive integers whose sum is $2m$, and set $B$ consists of $2m$ consecutive integers whose sum is $m$. The absolute value of the difference between the greatest element of $A$ and the greatest element of $B$ is $99$. Find $m$.

2000 Polish MO Finals, 2

Tags: geometry , trigonometry , circumcircle , analytic geometry , geometric transformation , reflection , cyclic quadrilateral

Let a triangle $ABC$ satisfy $AC = BC$; in other words, let $ABC$ be an isosceles triangle with base $AB$. Let $P$ be a point inside the triangle $ABC$ such that $\angle PAB = \angle PBC$. Denote by $M$ the midpoint of the segment $AB$. Show that $\angle APM + \angle BPC = 180^{\circ}$.

2006 Iran Team Selection Test, 5

Tags: circumcircle , trigonometry , parallelogram , angle bisector , geometry

Let $ABC$ be a triangle such that it's circumcircle radius is equal to the radius of outer inscribed circle with respect to $A$. Suppose that the outer inscribed circle with respect to $A$ touches $BC,AC,AB$ at $M,N,L$. Prove that $O$ (Center of circumcircle) is the orthocenter of $MNL$.

2022 Purple Comet Problems, 15

Tags:

Find the number of rearrangements of the nine letters $\text{AAABBBCCC}$ where no three consecutive letters are the same. For example, count $\text{AABBCCABC}$ and $\text{ACABBCCAB}$ but not $\text{ABABCCCBA}.$

2025 NEPALTST, 1

Tags: geometry

Consider a triangle $\triangle ABC$ and some point $X$ on $BC$. The perpendicular from $X$ to $AB$ intersects the circumcircle of $\triangle AXC$ at $P$ and the perpendicular from $X$ to $AC$ intersects the circumcircle of $\triangle AXB$ at $Q$. Show that the line $PQ$ does not depend on the choice of $X$. [i](Shining Sun, USA)[/i]

2012 National Olympiad First Round, 8

Tags: algebra , binomial theorem

In how many different ways can one select two distinct subsets of the set $\{1,2,3,4,5,6,7\}$, so that one includes the other? $ \textbf{(A)}\ 2059 \qquad \textbf{(B)}\ 2124 \qquad \textbf{(C)}\ 2187 \qquad \textbf{(D)}\ 2315 \qquad \textbf{(E)}\ 2316$

2009 Abels Math Contest (Norwegian MO) Final, 4a

Tags: algebra , inequalities

Show that $\left(\frac{2010}{2009}\right)^{2009}> 2$.

2009 Belarus Team Selection Test, 1

Tags: algebra , binary operation , operation

On R a binary algebraic operation ''*'' is defined which satisfies the following two conditions: i) for all $a,b \in R$, there exists a unique $x \in R$ such that $x *a=b$ (write $x=b/a$) ii) $(a*b)*c= (a*c)* (b*c)$ for all $a,b,c \in R$ a) Is this operation necesarily commutative (i.e. $a*b=b*a$ for all $a,b \in R$) ? b) Prove that $(a/b)/c = (a/c) / (b/c)$ and $(a/b)*c = (a*c) / (b*c)$ for all $a,b,c \in R$. A. Mirotin

2019 USEMO, 3

Tags: chessboard , combinatorics

Consider an infinite grid $\mathcal G$ of unit square cells. A [i]chessboard polygon[/i] is a simple polygon (i.e. not self-intersecting) whose sides lie along the gridlines of $\mathcal G$. Nikolai chooses a chessboard polygon $F$ and challenges you to paint some cells of $\mathcal G$ green, such that any chessboard polygon congruent to $F$ has at least $1$ green cell but at most $2020$ green cells. Can Nikolai choose $F$ to make your job impossible? [i]Nikolai Beluhov[/i]

1985 Tournament Of Towns, (092) T3

Tags: algebra , inequalities

Three real numbers $a, b$ and $c$ are given . It is known that $a + b + c >0 , bc+ ca + ab > 0$ and $abc > 0$ . Prove that $a > 0 , b > 0$ and $c > 0$ .