Olympiad problems

2013 CHMMC (Fall), 9

Tags: combinatorics

A $ 7 \times 7$ grid of unit-length squares is given. Twenty-four $1 \times 2$ dominoes are placed in the grid, each covering two whole squares and in total leaving one empty space. It is allowed to take a domino adjacent to the empty square and slide it lengthwise to fill the whole square, leaving a new one empty and resulting in a different configuration of dominoes. Given an initial configuration of dominoes for which the maximum possible number of distinct configurations can be reached through any number of slides, compute the maximum number of distinct configurations.

2013 Hanoi Open Mathematics Competitions, 10

Tags: perimeter , rectangle , geometry

Consider the set of all rectangles with a given area $S$. Find the largest value o $ M = \frac{S}{2S+p + 2}$ where $p$ is the perimeter of the rectangle.

1997 Bundeswettbewerb Mathematik, 4

Tags: grid , combinatorics

There are $10000$ trees in a park, arranged in a square grid with $100$ rows and $100$ columns. Find the largest number of trees that can be cut down, so that sitting on any of the tree stumps one cannot see any other tree stump.

2008 Tournament Of Towns, 3

Tags: symmetry , combinatorics

There are $N$ piles each consisting of a single nut. Two players in turns play the following game. At each move, a player combines two piles that contain coprime numbers of nuts into a new pile. A player who can not make a move, loses. For every $N > 2$ determine which of the players, the first or the second, has a winning strategy.

2005 South africa National Olympiad, 5

Tags: induction , n-variable inequality , inequalities

Let $x_1,x_2,\dots,x_n$ be positive numbers with product equal to 1. Prove that there exists a $k\in\{1,2,\dots,n\}$ such that \[\frac{x_k}{k+x_1+x_2+\cdots+x_k}\ge 1-\frac{1}{\sqrt[n]{2}}.\]

2013 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: number theory , rational , square root

Find all integers $a$ such that $\sqrt{\frac{9a+4}{a-6}}$ is rational number

2012 India Regional Mathematical Olympiad, 1

Tags: geometry

Let ABCD be a unit square. Draw a quadrant of a circle with A as centre and B;D as end points of the arc. Similarly, draw a quadrant of a circle with B as centre and A;C as end points of the arc. Inscribe a circle ? touching the arc AC internally, the arc BD internally and also touching the side AB. Find the radius of the circle ?.

2011 Indonesia TST, 1

Tags: inequalities , sum , algebra

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

Tags:

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

Tags:

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

Tags:

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]