Olympiad problems

2016 Tournament Of Towns, 4

A designer took a wooden cube $5 \times 5 \times 5$, divided each face into unit squares and painted each square black, white or red so that any two squares with a common side have different colours. What is the least possible number of black squares? (Squares with a common side may belong to the same face of the cube or to two different faces.) [i](8 points)[/i] [i]Mikhail Evdokimov[/i]

2009 Oral Moscow Geometry Olympiad, 4

Tags: concurrency , diameter , median , geometry

Three circles are constructed on the medians of a triangle as on diameters. It is known that they intersect in pairs. Let $C_1$ be the intersection point of the circles built on the medians $AM_1$ and $BM_2$, which is more distant from the vertex $C$. Points $A_1$ and $B_1$ are defined similarly. Prove that the lines $AA_1, BB_1$ and $CC_1$ intersect at one point. (D. Tereshin)

1982 National High School Mathematics League, 2

Tags: analytic geometry

In polar coordinates, the equation $\rho=\frac{1}{1-\cos\theta+\sin\theta}$ stands for a $\text{(A)}$circle $\text{(B)}$ellipse $\text{(C)}$hyperbola $\text{(D)}$parabola

2020 Czech and Slovak Olympiad III A, 1

Tags: game , ratio , combinatorics

Two positive integers $m$ and $n$ are written on the board. We replace one of two numbers in each step on the board by either their sum, or product, or ratio (if it is an integer). Depending on the numbers $m$ and $n$, specify all the pairs that can appear on the board in pairs. (Radovan Švarc)

2006 Putnam, A5

Tags: vieta , trigonometry , induction , polynomial , algebra , ratio

Let $n$ be a positive odd integer and let $\theta$ be a real number such that $\theta/\pi$ is irrational. Set $a_{k}=\tan(\theta+k\pi/n),\ k=1,2\dots,n.$ Prove that \[\frac{a_{1}+a_{2}+\cdots+a_{n}}{a_{1}a_{2}\cdots a_{n}}\] is an integer, and determine its value.

2013 District Olympiad, 4

Tags: inequalities , algebra , number theory

For a given a positive integer $n$, find all integers $x_1, x_2,... , x_n$ subject to $0 < x_1 < x_2 < ...< x_n < x_{n+1}$ and $$x_nx_{n+1} \le 2(x_1 + x_2 + ... + x_n).$$

1999 Mediterranean Mathematics Olympiad, 1

Tags: combinatorics

Do there exist a circle and an infinite set of points on it such that the distance between any two of the points is rational?

2018 Azerbaijan BMO TST, 2

Tags: inequalities , three variable inequality , algebra

Let $M = \{(a,b,c)\in R^3 :0 <a,b,c<\frac12$ with $a+b+c=1 \}$ and $f: M\to R$ given as $$f(a,b,c)=4\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)-\frac{1}{abc}$$ Find the best (real) bounds $\alpha$ and $\beta$ such that $f(M) = \{f(a,b,c): (a,b,c)\in M\}\subseteq [\alpha,\beta]$ and determine whether any of them is achievable.

2002 Taiwan National Olympiad, 4

Tags: inequalities

Let $0<x_{1},x_{2},x_{3},x_{4}\leq\frac{1}{2}$ are real numbers. Prove that $\frac{x_{1}x_{2}x_{3}x_{4}}{(1-x_{1})(1-x_{2})(1-x_{3})(1-x_{4})}\leq\frac{x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4}}{(1-x_{1})^{4}+(1-x_{2})^{4}+(1-x_{3})^{4}+(1-x_{4})^{4}}$.

2010 Indonesia TST, 3

Tags: geometry , homothety , geometric transformation

Two parallel lines $r,s$ and two points $P \in r$ and $Q \in s$ are given in a plane. Consider all pairs of circles $(C_P, C_Q)$ in that plane such that $C_P$ touches $r$ at $P$ and $C_Q$ touches $s$ at $Q$ and which touch each other externally at some point $T$. Find the locus of $T$.

1995 Bundeswettbewerb Mathematik, 1

Tags: lattice , combinatorics , lattice points

Starting at $(1,1)$, a stone is moved in the coordinate plane according to the following rules: (i) From any point $(a,b)$, the stone can move to $(2a,b)$ or $(a,2b)$. (ii) From any point $(a,b)$, the stone can move to $(a-b,b)$ if $a > b$, or to $(a,b-a)$ if $a < b$. For which positive integers $x,y$ can the stone be moved to $(x,y)$?

2017 AIME Problems, 3

Tags:

A triangle has vertices $A(0,0)$, $B(12,0)$, and $C(8,10)$. The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

2024 Argentina Iberoamerican TST, 4

Tags: combinatorics

Find all natural numbers $n \geqslant 2$ with the property that there are two permutations $(a_1, a_2,\ldots, a_n) $ and $(b_1, b_2,\ldots, b_n)$ of the numbers $1, 2,\ldots, n$ such that $(a_1 + b_1, a_2 +b_2,\ldots, a_n + b_n)$ are consecutive natural numbers.

2019 Israel National Olympiad, 7

Tags: circles , circumcircle , geometry

In the plane points $A,B,C$ are marked in blue and points $P,Q$ are marked in red (no 3 marked points lie on a line, and no 4 marked points lie on a circle). A circle is called [b]separating[/b] if all points of one color are inside it, and all points of the other color are outside of it. Denote by $O$ the circumcenter of $ABC$ and by $R$ the circumradius of $ABC$. Prove that [b]exactly one[/b] of the following holds: [list] [*] There exists a separating circle; [*] There exists a point $X$ on the segment $PQ$ which also lies inside the triangle $ABC$, for which $PX\cdot XQ = R^2-OX^2$.

2024 Australian Mathematical Olympiad, P2

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral. Point $P$ is on line $CB$ such that $CP=CA$and $B$ lies between $C$ and $P$. Point $Q$ is on line $CD$ such that $CQ=CA$ and $D$ lies between $C$ and $Q$. Prove that the incentre of triangle $ABD$ lies on line $PQ.$

2024 Sharygin Geometry Olympiad, 9.2

Tags: geometry , geo

Points $A, B, C, D$ on the plane do not form a rectangle. Let the sidelengths of triangle $T$ equal $AB+CD$, $AC+BD$, $AD+BC$. Prove that the triangle $T$ is acute-angled.

2021 Harvard-MIT Mathematics Tournament., 1

Tags: analytic geometry

A circle contains the points $(0, 11)$ and $(0, -11)$ on its circumference and contains all points $(x, y)$ with $x^2+y^2<1$ in its interior. Compute the largest possible radius of the circle.

2001 VJIMC, Problem 3

Tags: inequalities

Let $n\ge2$ be a natural number. Prove that $$\prod_{k=2}^n\ln k<\frac{\sqrt{n!}}n.$$

Russian TST 2018, P1

Tags: number theory

Let $k>1$ be the given natural number and $p\in \mathbb{P}$ such that $n=kp+1$ is composite number. Given that $n\mid 2^{n-1}-1.$ Prove that $n<2^k.$

2005 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: number theory

Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$. (Note: $n$ is written in the usual base ten notation.)

1990 Chile National Olympiad, 1

Tags: geometry , isosceles triangle , isosceles

Show that any triangle can be subdivided into isosceles triangles.

2024 SEEMOUS, P4

Tags: matrix , linear algebra

Let $n\in\mathbb{N}$, $n\geq 2$. Find all values of $k\in\mathbb{N}$, $k\geq 1$, for which the following statement holds: $$\text{"If }A\in\mathcal{M}_n(\mathbb{C})\text{ is such that }A^kA^*=A\text{, then }A=A^*\text{."}$$ (here, $A^*$ denotes the conjugate transpose of $A$).

1962 Vietnam National Olympiad, 3

Tags: 3d geometry , geometry , tetrahedron

Let $ ABCD$ is a tetrahedron. Denote by $ A'$, $ B'$ the feet of the perpendiculars from $ A$ and $ B$, respectively to the opposite faces. Show that $ AA'$ and $ BB'$ intersect if and only if $ AB$ is perpendicular to $ CD$. Do they intersect if $ AC \equal{} AD \equal{} BC \equal{} BD$?

1963 AMC 12/AHSME, 35

Tags: area of a triangle , heron's formula , modular arithmetic , perimeter , geometry

The lengths of the sides of a triangle are integers, and its area is also an integer. One side is $21$ and the perimeter is $48$. The shortest side is: $\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 10\qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$

2010 Contests, 1

Tags: geometry

A square with side length $2$ cm is placed next to a square with side length $6$ cm, as shown in the diagram. Find the shaded area, in cm$^2$. [img]https://cdn.artofproblemsolving.com/attachments/5/7/ceb4912a6e73ca751113b2b5c92cbfdbb6e0d1.png[/img]