Olympiad problems

2021 Iranian Geometry Olympiad, 1

Tags: geometry

Let $ABC$ be a triangle with $AB = AC$. Let $H$ be the orthocenter of $ABC$. Point $E$ is the midpoint of $AC$ and point $D$ lies on the side $BC$ such that $3CD = BC$. Prove that $BE \perp HD$. [i]Proposed by Tran Quang Hung - Vietnam[/i]

2024 Belarusian National Olympiad, 8.6

Tags: number theory

For each number $x$ we denote by $S(x)$ the sum of digits from its decimal representation. Find all positive integers $m$ for each of which there exists a positive integer $n$, such that $$S(n^2-2n+10)=m$$ [i]Chernov S.[/i]

2004 Italy TST, 2

Tags: combinatorics

Let $\mathcal{P}_0=A_0A_1\ldots A_{n-1}$ be a convex polygon such that $A_iA_{i+1}=2^{[i/2]}$ for $i=0, 1,\ldots ,n-1$ (where $A_n=A_0$). Define the sequence of polygons $\mathcal{P}_k=A_0^kA_1^k\ldots A_{n-1}^k$ as follows: $A_i^1$ is symmetric to $A_i$ with respect to $A_0$, $A_i^2$ is symmetric to $A_i^1$ with respect to $A_1^1$, $A_i^3$ is symmetric to $A_i^2$ with respect to $A_2^2$ and so on. Find the values of $n$ for which infinitely many polygons $\mathcal{P}_k$ coincide with $\mathcal{P}_0$.

2003 USAMO, 1

Tags: induction , number theory

Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.

1992 Tournament Of Towns, (356) 5

Tags: geometry

The bisector of the angle $A$ of triangle $ABC$ intersects its circumscribed circle at the point $D$. Suppose $P$ is the point symmetric to the incentre of the triangle with respect to the midpoint of the side $BC$, and $M$ is the second intersection point of the line $PD$ with the circumscribed circle. Prove that one of the distances $AM$, $BM$, $CM$ is equal to the sum of two other distances. (VO Gordon)

2009 Germany Team Selection Test, 1

Tags: geometry , rectangle , combinatorial geometry , extremal combinatorics , combinatorics

In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[/i]. Two boxes [i]intersect[/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

1997 Canada National Olympiad, 3

Tags: induction , limit , floor function , inequalities

Prove that $\frac{1}{1999}< \prod_{i=1}^{999}{\frac{2i-1}{2i}}<\frac{1}{44}$.

2011 Junior Balkan Team Selection Tests - Romania, 2

Tags: combinatorics , square

We consider an $n \times n$ ($n \in N, n \ge 2$) square divided into $n^2$ unit squares. Determine all the values of $k \in N$ for which we can write a real number in each of the unit squares such that the sum of the $n^2$ numbers is a positive number, while the sum of the numbers from the unit squares of any $k \times k$ square is a negative number.

2012 Thailand Mathematical Olympiad, 12

Tags: number theory , integer , perfect cube

Let $a, b, c$ be positive integers. Show that if $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer then $abc$ is a perfect cube.

1992 National High School Mathematics League, 2

Tags:

The equation of unit circle in Quadrant I, III, IV ($(-1,0),(1,0),(0,-1),(0,1)$ included) is $\text{(A)}(x+\sqrt{1-y^2})(y+\sqrt{1-x^2})=0$ $\text{(B)}(x-\sqrt{1-y^2})(y-\sqrt{1-x^2})=0$ $\text{(C)}(x+\sqrt{1-y^2})(y-\sqrt{1-x^2})=0$ $\text{(D)}(x-\sqrt{1-y^2})(y+\sqrt{1-x^2})=0$

1966 IMO Shortlist, 21

Tags: geometry , 3d geometry , area , volume , geometric inequality

Prove that the volume $V$ and the lateral area $S$ of a right circular cone satisfy the inequality \[\left( \frac{6V}{\pi}\right)^2 \leq \left( \frac{2S}{\pi \sqrt 3}\right)^3\] When does equality occur?

1984 Bundeswettbewerb Mathematik, 2

Tags: geometry , combinatorial geometry

Determine all bounded closed subsets $F$ of the plane with the following property: $F$ consists of at least two points and always contains two points $A$ and $B$ as well as at least one of the two semicircular arcs over the segment $AB$. Definitions: A subset of the $F$ of the plane is said to be closed if: For every point $P$ of the plane that is not an element of $F$ , there is a (non-degenerate) disc with center $P$ that has no elements of $F$.

2007 Grigore Moisil Intercounty, 1

Tags: linear algebra

Let be two distinct $ 2\times 2 $ real matrices having the property that there exists a natural number such that these matrices raised to this number are equal, and these matrices raised to the successor of this number are also equal. Prove that these matrices raised to any power greater than $ 2 $ are equal.

ICMC 3, 3

Tags:

Consider a grid of points where each point is coloured either white or black, such that no two rows have the same sequence of colours and no two columns have the same sequence of colours. Let a [i]table[/i] denote four points on the grid that form the vertices of a rectangle with sides parallel to those of the grid. A table is called [i]balanced[/i] if one diagonal pair of points are coloured white and the other diagonal pair black. Determine all possible values of $k \geq 2$ for which there exists a colouring of a $k\times 2019$ grid with no balanced tables. [i]proposed by the ICMC Problem Committee[/i]

1995 IMO Shortlist, 4

Tags: function , inequalities , optimization , system of equations , 133

Find all of the positive real numbers like $ x,y,z,$ such that : 1.) $ x \plus{} y \plus{} z \equal{} a \plus{} b \plus{} c$ 2.) $ 4xyz \equal{} a^2x \plus{} b^2y \plus{} c^2z \plus{} abc$ Proposed to Gazeta Matematica in the 80s by VASILE CÎRTOAJE and then by Titu Andreescu to IMO 1995.

2002 Stanford Mathematics Tournament, 2

Tags: geometry , rectangle , ratio

Upon cutting a certain rectangle in half, you obtain two rectangles that are scaled down versions of the original. What is the ratio of the longer side length to the shorter side length?

1990 IMO Longlists, 46

Tags: analytic geometry , inequalities , geometry

For each $P$ inside the triangle $ABC$, let $A(P), B(P)$, and $C(P)$ be the points of intersection of the lines $AP, BP$, and $CP$ with the sides opposite to $A, B$, and $C$, respectively. Determine $P$ in such a way that the area of the triangle $A(P)B(P)C(P)$ is as large as possible.

2023 Bangladesh Mathematical Olympiad, P5

Tags: algebra , integration , functional equation , function

Consider an integrable function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $af(a)+bf(b)=0$ when $ab=1$. Find the value of the following integration: $$ \int_{0}^{\infty} f(x) \,dx $$

2021 Thailand Online MO, P1

Tags: combinatorics

There is a fence that consists of $n$ planks arranged in a line. Each plank is painted with one of the available $100$ colors. Suppose that for any two distinct colors $i$ and $j$, there is a plank with color $i$ located to the left of a (not necessarily adjacent) plank with color $j$. Determine the minimum possible value of $n$.

2016 Harvard-MIT Mathematics Tournament, 17

Tags:

Compute the sum of all integers $1 \le a \le 10$ with the following property: there exist integers $p$ and $q$ such that $p$, $q$, $p^2+a$ and $q^2+a$ are all distinct prime numbers.

1990 IMO Longlists, 80

Tags: function , induction , algebra

Function $f(x, y): \mathbb N \times \mathbb N \to \mathbb Q$ satisfies the conditions: (i) $f(1, 1) =1$, (ii) $f(p + 1, q) + f(p, q + 1) = f(p, q)$ for all $p, q \in \mathbb N$, and (iii) $qf(p + 1, q) = pf(p, q + 1)$ for all $p, q \in \mathbb N$. Find $f(1990, 31).$

2008 IMO Shortlist, 3

Tags: combinatorics , extremal combinatorics , point set , bezout s identity , pigeonhole principle , geometry

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

2011 Postal Coaching, 5

Tags: circumcircle , geometry

Let $H$ be the orthocentre and $O$ be the circumcentre of an acute triangle $ABC$. Let $AD$ and $BE$ be the altitudes of the triangle with $D$ on $BC$ and $E$ on $CA$. Let $K =OD \cap BE, L = OE \cap AD$. Let $X$ be the second point of intersection of the circumcircles of triangles $HKD$ and $HLE$, and let $M$ be the midpoint of side $AB$. Prove that points $K, L, M$ are collinear if and only if $X$ is the circumcentre of triangle $EOD$.

2004 Junior Tuymaada Olympiad, 1

Tags: algebra , irrational number

A positive rational number is written on the blackboard. Every minute Vasya replaces the number $ r $ written on the board with $ \sqrt {r + 1} $. Prove that someday he will get an irrational number.

2009 Philippine MO, 3

Tags: combinatorics , pigeonhole principle

Each point of a circle is colored either red or blue. [b](a)[/b] Prove that there always exists an isosceles triangle inscribed in this circle such that all its vertices are colored the same. [b](b)[/b] Does there always exist an equilateral triangle inscribed in this circle such that all its vertices are colored the same?