Olympiad problems

2021 Saint Petersburg Mathematical Olympiad, 6

A line $\ell$ passes through vertex $C$ of the rhombus $ABCD$ and meets the extensions of $AB, AD$ at points $X,Y$. Lines $DX, BY$ meet $(AXY)$ for the second time at $P,Q$. Prove that the circumcircle of $\triangle PCQ$ is tangent to $\ell$ [i]A. Kuznetsov[/i]

2004 Argentina National Olympiad, 6

Tags: arithmetic progression , algebra , arithmetic sequence

Decide if it is possible to generate an infinite sequence of positive integers $a_n$ such that in the sequence there are no three terms that are in arithmetic progression and that for all $n$ $\left |a_n-n^2\right | <\frac{n}{2}$. Clarification: Three numbers $a$, $b$, $c$ are in arithmetic progression if and only if $2b=a+c$.

2008 Sharygin Geometry Olympiad, 15

Tags: symmetry , geometry

(M.Volchkevich, 9--11) Given two circles and point $ P$ not lying on them. Draw a line through $ P$ which cuts chords of equal length from these circles.

2002 Tuymaada Olympiad, 8

Tags: tangent , geometry , area of a triangle , area

The circle with the center of $ O $ touches the sides of the angle $ A $ at the points of $ K $ and $ M $. The tangent to the circle intersects the segments $ AK $ and $ AM $ at points $ B $ and $ C $ respectively, and the line $ KM $ intersects the segments $ OB $ and $ OC $ at the points $ D $ and $ E $. Prove that the area of the triangle $ ODE $ is equal to a quarter of the area of a triangle $ BOC $ if and only if the angle $ A $ is $ 60^\circ $.

2010 Germany Team Selection Test, 2

Tags: geometry , cyclic quadrilateral , projective geometry , tangent , power of a point

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

1999 Abels Math Contest (Norwegian MO), 3

Tags: isosceles , equilateral , triangle area , area

An isosceles triangle $ABC$ with $AB = AC$ and $\angle A = 30^o$ is inscribed in a circle with center $O$. Point $D$ lies on the shorter arc $AC$ so that $\angle DOC = 30^o$, and point $G$ lies on the shorter arc $AB$ so that $DG = AC$ and $AG < BG$. The line $BG$ intersects $AC$ and $AB$ at $E$ and $F$, respectively. (a) Prove that triangle $AFG$ is equilateral. (b) Find the ratio between the areas of triangles $AFE$ and $ABC$.

2011 Iran MO (2nd Round), 1

Tags: geometry , analytic geometry , combinatorics

We have a line and $1390$ points around it such that the distance of each point to the line is less than $1$ centimeters and the distance between any two points is more than $2$ centimeters. prove that there are two points such that their distance is at least $10$ meters ($1000$ centimeters).

MOAA Team Rounds, 2021.2

Tags: team

Four students Alice, Bob, Charlie, and Diana want to arrange themselves in a line such that Alice is at either end of the line, i.e., she is not in between two students. In how many ways can the students do this? [i]Proposed by Nathan Xiong[/i]

2014 Iran MO (3rd Round), 2

Tags: function , induction , algebra

Find all continuous function $f:\mathbb{R}^{\geq 0}\rightarrow \mathbb{R}^{\geq 0}$ such that : \[f(xf(y))+f(f(y)) = f(x)f(y)+2 \: \: \forall x,y\in \mathbb{R}^{\geq 0}\] [i]Proposed by Mohammad Ahmadi[/i]

MOAA Accuracy Rounds, 2023.6

Tags:

Let $b$ be a positive integer such that 2032 has 3 digits when expressed in base $b$. Define the function $S_k(n)$ as the sum of the digits of the base $k$ representation of $n$. Given that $S_b(2032)+S_{b^2}(2032) = 14$, find $b$. [i]Proposed by Anthony Yang[/i]

2001 Saint Petersburg Mathematical Olympiad, 10.1

Tags: inequalities , quadratic , quadratic trinomial , algebra

Quadratic trinomials $f$ and $g$ with integer coefficients obtain only positive values and the inequality $\dfrac{f(x)}{g(x)}\geq \sqrt{2}$ is true $\forall x\in\mathbb{R}$. Prove that $\dfrac{f(x)}{g(x)}>\sqrt{2}$ is true $\forall x\in\mathbb{R}$ [I]Proposed by A. Khrabrov[/i]

2014 Contests, 1

Tags: geometry , geometric transformation , parallelogram , reflection , asymptote

Let $ABC$ be an acute triangle, and let $X$ be a variable interior point on the minor arc $BC$ of its circumcircle. Let $P$ and $Q$ be the feet of the perpendiculars from $X$ to lines $CA$ and $CB$, respectively. Let $R$ be the intersection of line $PQ$ and the perpendicular from $B$ to $AC$. Let $\ell$ be the line through $P$ parallel to $XR$. Prove that as $X$ varies along minor arc $BC$, the line $\ell$ always passes through a fixed point. (Specifically: prove that there is a point $F$, determined by triangle $ABC$, such that no matter where $X$ is on arc $BC$, line $\ell$ passes through $F$.) [i]Robert Simson et al.[/i]

2023 Romania EGMO TST, P4

Tags: inequalities , absolute value , algebra

Let $n\geqslant 3$ be an integer and $a_1,\ldots,a_n$ be nonzero real numbers, with sum $S{}$. Prove that \[\sum_{i=1}^n\left|\frac{S-a_i}{a_i}\right|\geqslant\frac{n-1}{n-2}.\]

2004 AMC 12/AHSME, 2

Tags:

In the expression $ c\cdot a^b\minus{}d$, the values of $ a$, $ b$, $ c$, and $ d$ are $ 0$, $ 1$, $ 2$, and $ 3$, although not necessarily in that order. What is the maximum possible value of the result? $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ 10$

VI Soros Olympiad 1999 - 2000 (Russia), 11.2

Tags: algebra , number theory

Find the sum of all possible products of natural numbers of the form $k_1k_2...k_{999}$, where in each product $ k_1 < k_2 < ... < k_{999} <1999$, and there are no $k_i$ and $k_j$ such that $k_i + k_j =1999$.

2011 District Olympiad, 2

Tags: complex number , absolute value , geometry , rectangle , trigonometry

[b]a)[/b] Show that if four distinct complex numbers have the same absolute value and their sum vanishes, then they represent a rectangle. [b]b)[/b] Let $ x,y,z,t $ be four real numbers, and $ k $ be an integer. Prove the following implication: $$ \sum_{j\in\{ x,y,z,t\}} \sin j = 0 = \sum_{j\in\{ x,y,z,t\}} \cos j\implies \sum_{j\in\{ x,y,z,t\}} \sin (1+2n)j. $$

2020 Miklós Schweitzer, 4

Tags: geometry , combinatorics

Consider horizontal and vertical segments in the plane that may intersect each other. Let $n$ denote their total number. Suppose that we have $m$ curves starting from the origin that are pairwise disjoint except for their endpoints. Assume that each curve intersects exactly two of the segments, a different pair for each curve. Prove that $m=O(n)$.

2022 Bulgarian Autumn Math Competition, Problem 8.1

Tags: algebra

Solve the equation: \[4x^2+|9-6x|=|10x-15|+6(2x+1)\]

2020 South Africa National Olympiad, 4

Tags: number theory , diophantine equation

A positive integer $k$ is said to be [i]visionary[/i] if there are integers $a > 0$ and $b \geq 0$ such that $a \cdot k + b \cdot (k + 1) = 2020.$ How many visionary integers are there?

1975 USAMO, 3

Tags: algebra , polynomial

If $ P(x)$ denotes a polynomial of degree $ n$ such that $ P(k)\equal{}\frac{k}{k\plus{}1}$ for $ k\equal{}0,1,2,\ldots,n$, determine $ P(n\plus{}1)$.

2020 AMC 12/AHSME, 15

Tags: complex plane

In the complex plane, let $A$ be the set of solutions to $z^3 - 8 = 0$ and let $B$ be the set of solutions to $z^3 - 8z^2 - 8z + 64 = 0$. What is the greatest distance between a point of $A$ and a point of $B?$ $\textbf{(A) } 2\sqrt{3} \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 2\sqrt{21} \qquad \textbf{(E) } 9 + \sqrt{3}$

2024 Dutch IMO TST, 2

Tags: ratio , incenter , sides of a triangle , geometry

Let $ABC$ be a triangle. A point $P$ lies on the segment $BC$ such that the circle with diameter $BP$ passes through the incenter of $ABC$. Show that $\frac{BP}{PC}=\frac{c}{s-c}$ where $c$ is the length of segment $AB$ and $2s$ is the perimeter of $ABC$.

2001 USA Team Selection Test, 4

Tags: modular arithmetic , induction , pigeonhole principle , combinatorics

There are 51 senators in a senate. The senate needs to be divided into $n$ committees so that each senator is on one committee. Each senator hates exactly three other senators. (If senator A hates senator B, then senator B does [i]not[/i] necessarily hate senator A.) Find the smallest $n$ such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee.

2013 AMC 10, 9

Tags:

Three positive integers are each greater than $1$, have a product of $27000$, and are pairwise relatively prime. What is their sum? $\textbf{(A) }100\qquad \textbf{(B) } 137\qquad\textbf{(C) } 156\qquad\textbf{(D) }160\qquad\textbf{(E) }165$

Kyiv City MO Seniors 2003+ geometry, 2016.10.4

Tags: geometry , locus , collinear , circles

On the circle with diameter $AB$, the point $M$ was selected and fixed. Then the point ${{Q} _ {i}}$ is selected, for which the chord $M {{Q} _ {i}}$ intersects $AB$ at the point ${{K} _ {i}}$ and thus $ \angle M {{K} _ {i}} B <90 {} ^ \circ$. A chord that is perpendicular to $AB$ and passes through the point ${{K} _ {i}}$ intersects the line $B {{Q} _ {i}}$ at the point ${{P } _ {i}}$. Prove that the points ${{P} _ {i}}$ in all possible choices of the point ${{Q} _ {i}}$ lie on the same line. (Igor Nagel)