Olympiad problems

2011 Indonesia TST, 1

For all positive integer $n$, define $f_n(x)$ such that $f_n(x) = \sum_{k=1}^n{|x - k|}$. Determine all solution from the inequality $f_n(x) < 41$ for all positive $2$-digit integers $n$ (in decimal notation).

1978 IMO Shortlist, 11

Tags: function , algebra , concave , mean

A function $f : I \to \mathbb R$, defined on an interval $I$, is called concave if $f(\theta x + (1 - \theta)y) \geq \theta f(x) + (1 - \theta)f(y)$ for all $x, y \in I$ and $0 \leq \theta \leq 1$. Assume that the functions $f_1, \ldots , f_n$, having all nonnegative values, are concave. Prove that the function $(f_1f_2 \cdots f_n)^{1/n}$ is concave.

1989 Bulgaria National Olympiad, Problem 1

Tags: triangle , geometry

In triangle $ABC$, point $O$ is the center of the excircle touching the side $BC$, while the other two excircles touch the sides $AB$ and $AC$ at points $M$ and $N$ respectively. A line through $O$ perpendicular to $MN$ intersects the line $BC$ at $P$. Determine the ratio $AB/AC$, given that the ratio of the area of $\triangle ABC$ to the area of $\triangle MNP$ is $2R/r$, where $R$ is the circumradius and $r$ the inradius of $\triangle ABC$.

2008 Irish Math Olympiad, 3

Tags: number theory

Determine, with proof, all integers $ x$ for which $ x(x\plus{}1)(x\plus{}7)(x\plus{}8)$ is a perfect square.

2011 Romania National Olympiad, 1

Tags: inequalities , algebra , factorization , sum of cubes , formula

Let be a natural number $ n $ and $ n $ real numbers $ a_1,a_2,\ldots ,a_n $ such that $$ a_m+a_{m+1} +\cdots +a_n\ge \frac{(m+n)(n-m+1)}{2} ,\quad\forall m\in\{ 1,2,\ldots ,n \} . $$ Prove that $ a_1^2+a_2^2+\cdots +a_n^2\ge\frac{n(n+1)(2n+1)}{6} . $

2024 New Zealand MO, 7

Tags: combinatorics , lattice points

Some of the $80960$ lattice points in a $40\times2024$ lattice are coloured red. It is known that no four red lattice points are vertices of a rectangle with sides parallel to the axes of the lattice. What is the maximum possible number of red points in the lattice?

2024 AMC 8 -, 16

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Minh enters the numbers from 1 to 81 in a $9\times9$ grid in some order. She calculates the product of the numbers in each row and column. What is the least number of rows and columns that could have a product divisible by 3? $\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }12$

2025 Romania EGMO TST, P3

Tags: ratio , geometry

$BE$ and $CF$ are the altitudes of the acute scalene $\triangle ABC$, $O$ is its circumcenter and $M$ is the midpoint of the side $BC$. If point, which is symmetric to $M$ with respect to $O$, lies on the line $EF$, find all possible values of the ratio $\dfrac{AM}{AO}$. [i]Proposed by Fedir Yudin[/i]

1936 Moscow Mathematical Olympiad, 028

Tags: perimeter , geometric construction , line construction , geometry

Given an angle less than $180^o$, and a point $M$ outside the angle. Draw a line through $M$ so that the triangle, whose vertices are the vertex of the angle and the intersection points of its legs with the line drawn, has a given perimeter.

2012 Stars of Mathematics, 2

Tags: number theory

Prove the value of the expression $$\displaystyle \dfrac {\sqrt{n + \sqrt{0}} + \sqrt{n + \sqrt{1}} + \sqrt{n + \sqrt{2}} + \cdots + \sqrt{n + \sqrt{n^2-1}} + \sqrt{n + \sqrt{n^2}}} {\sqrt{n - \sqrt{0}} + \sqrt{n - \sqrt{1}} + \sqrt{n - \sqrt{2}} + \cdots + \sqrt{n - \sqrt{n^2-1}} + \sqrt{n - \sqrt{n^2}}}$$ is constant over all positive integers $n$. ([i]Folklore (also Philippines Olympiad)[/i])

1968 Putnam, B3

Tags: geometry , construction

Given that a $60^{\circ}$ angle cannot be trisected with ruler and compass, prove that a $\frac{120^{\circ}}{n}$ angle cannot be trisected with ruler and compass for $n=1,2,\ldots$

2025 Bulgarian Spring Mathematical Competition, 12.4

Tags: geometry , mixtilinear , angle chasing , tangent circle

Let $ABC$ be an acute-angled triangle $ ABC $ with $ AC > BC $ and incenter $ I $. Let $ \omega $ be the mixtilinear circle at vertex $ C $, i.e. the circle internally tangent to the circumcircle of $ \triangle ABC $ and also tangent to lines $ AC $ and $ BC $. A circle $ \Gamma $ passes through points $ A $ and $ B $ and is tangent to $ \omega $ at point $ T $, with $ C \notin \Gamma $ and $ I $ being inside $ \triangle ATB $. Prove that: $$\angle CTB + \angle ATI = 180^\circ + \angle BAI - \angle ABI.$$

2013 IMC, 5

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Consider a circular necklace with $\displaystyle{2013}$ beads. Each bead can be paintes either green or white. A painting of the necklace is called [i]good[/i] if among any $\displaystyle{21}$ successive beads there is at least one green bead. Prove that the number of good paintings of the necklace is odd. [b]Note.[/b] Two paintings that differ on some beads, but can be obtained from each other by rotating or flipping the necklace, are counted as different paintings. [i]Proposed by Vsevolod Bykov and Oleksandr Rybak, Kiev.[/i]

1949-56 Chisinau City MO, 10

Tags: root , algebra , rational

Get rid of irrationality in the denominator of a fraction $$\frac{1}{\sqrt[3]{4}+\sqrt[3]{2}+2}$$.

1993 Baltic Way, 20

Tags: geometry , 3d geometry , tetrahedron , geometric transformation , dilation

Let $ \mathcal Q$ be a unit cube. We say that a tetrahedron is [b]good[/b] if all its edges are equal and all of its vertices lie on the boundary of $ \mathcal Q$. Find all possible volumes of good tetrahedra.

2024 Chile TST IMO, 3

Tags: geometry

Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.

2015 ASDAN Math Tournament, 10

Tags: geometry test

Triangle $ABC$ has $\angle BAC=90^\circ$. A semicircle with diameter $XY$ is inscribed inside $\triangle ABC$ such that it is tangent to a point $D$ on side $BC$, with $X$ on $AB$ and $Y$ on $AC$. Let $O$ be the midpoint of $XY$. Given that $AB=3$, $AC=4$, and $AX=\tfrac{9}{4}$, compute the length of $AO$.

2015 CCA Math Bonanza, L5.2

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If a train carrying $27$ passengers leaves Grand Central Station at $8:00$ AM and travels $900$ miles due west to Chicago, arriving at $5:00$ PM, what is the average speed of the train in miles per hour? [i]2015 CCA Math Bonanza Lightning Round #5.2[/i]

2014 AMC 12/AHSME, 6

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Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is $50\%$ more than the regular. After both consume $\tfrac{3}{4}$ of their drinks, Ann gives Ed a third of what she has left, and $2$ additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together? ${ \textbf{(A)}\ 30\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}}\ 40\qquad\textbf{(E)}\ 50 $

1970 All Soviet Union Mathematical Olympiad, 129

Tags: geometry , geometric construction

Given a circle, its diameter $[AB]$ and a point $C$ on it. Construct (with the help of compasses and ruler) two points $X$ and $Y$, that are symmetric with respect to $(AB)$ line, such that $(YC)$ is orthogonal to $(XA)$.

1996 Tournament Of Towns, (497) 4

Tags: geometry , 3d geometry , combinatorics , combinatorial geometry

Is it possible to tile space using a combination of regular tetrahedra and regular octahedra? (A Belov)

2010 Kyiv Mathematical Festival, 2

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Denote by $S(n)$ the sum of digits of integer $n.$ Find 1) $S(3)+S(6)+S(9)+\ldots+S(300);$ 2) $S(3)+S(6)+S(9)+\ldots+S(3000).$

PEN O Problems, 36

Tags: induction , inequalities

Let a and b be non-negative integers such that $ab \ge c^{2}$ where $c$ is an integer. Prove that there is a positive integer n and integers $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$, $y_{1}$, $y_{2}$, $\cdots$, $y_{n}$ such that \[{x_{1}}^{2}+\cdots+{x_{n}}^{2}=a,\;{y_{1}}^{2}+\cdots+{y_{n}}^{2}=b,\; x_{1}y_{1}+\cdots+x_{n}y_{n}=c\]

2013 Stars Of Mathematics, 4

Tags: induction , algebra , functional equation , combinatorics

Given a (fixed) positive integer $N$, solve the functional equation \[f \colon \mathbb{Z} \to \mathbb{R}, \ f(2k) = 2f(k) \textrm{ and } f(N-k) = f(k), \ \textrm{for all } k \in \mathbb{Z}.\] [i](Dan Schwarz)[/i]

2017 AMC 10, 1

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Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11.$ Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number? $\textbf{(A)} \text{ 11} \qquad \textbf{(B)} \text{ 12} \qquad \textbf{(C)} \text{ 13} \qquad \textbf{(D)} \text{ 14} \qquad \textbf{(E)} \text{ 15}$