Olympiad problems

2013 EGMO, 5

Tags: geometry

Let $\Omega$ be the circumcircle of the triangle $ABC$. The circle $\omega$ is tangent to the sides $AC$ and $BC$, and it is internally tangent to the circle $\Omega$ at the point $P$. A line parallel to $AB$ intersecting the interior of triangle $ABC$ is tangent to $\omega$ at $Q$. Prove that $\angle ACP = \angle QCB$.

2008 National Olympiad First Round, 14

Tags: euler , modular arithmetic

What is the last three digits of $49^{303}\cdot 3993^{202}\cdot 39^{606}$? $ \textbf{(A)}\ 001 \qquad\textbf{(B)}\ 081 \qquad\textbf{(C)}\ 561 \qquad\textbf{(D)}\ 721 \qquad\textbf{(E)}\ 961 $

2007 Stars of Mathematics, 1

Tags: function , algebra , functional equation

Prove that there exists just one function $ f:\mathbb{N}^2\longrightarrow\mathbb{N} $ which simultaneously satisfies: $ \text{(1)}\quad f(m,n)=f(n,m),\quad\forall m,n\in\mathbb{N} $ $ \text{(2)}\quad f(n,n)=n,\quad\forall n\in\mathbb{N} $ $ \text{(3)}\quad n>m\implies (n-m)f(m,n)=nf(m,n-m), \quad\forall m,n\in\mathbb{N} $

2012 CHMMC Fall, 3

Tags: combinatorics

A particular graph has $6$ vertices, $12$ edges, and has the property that it contains no Eulerian path; a Eulerian path is a route from vertex to vertex along edges that traces each edge exactly once. Determine all the possible degrees of its vertices in no particular order. There are two solutions, and you need to get both to get credit for this problem.

2008 Sharygin Geometry Olympiad, 17

Tags: incenter , parallelogram , geometry

(A.Myakishev, 9--11) Given triangle $ ABC$ and a ruler with two marked intervals equal to $ AC$ and $ BC$. By this ruler only, find the incenter of the triangle formed by medial lines of triangle $ ABC$.

2018 Serbia Team Selection Test, 2

Tags: inequality , algebra , inequalities

Let $n$ be a fixed positive integer and let $x_1,\ldots,x_n$ be positive real numbers. Prove that $$x_1\left(1-x_1^2\right)+x_2\left(1-(x_1+x_2)^2\right)+\cdots+x_n\left(1-(x_1+...+x_n)^2\right)<\frac{2}{3}.$$

I Soros Olympiad 1994-95 (Rus + Ukr), 9.2

Tags: algebra , geometry

What can be the angle between the hour and minute hands of a clock if it is known that its value has not changed after $30$ minutes?

2001 Abels Math Contest (Norwegian MO), 3a

Tags: max , area , geometry

What is the largest possible area of a quadrilateral with sidelengths $1, 4, 7$ and $8$ ?

2013 AMC 10, 15

Tags: geometry

Two sides of a triangle have lengths $10$ and $15$. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side? $\textbf{(A) }6\qquad \textbf{(B) }8\qquad \textbf{(C) }9\qquad \textbf{(D) }12\qquad \textbf{(E) }18\qquad$

2016 Swedish Mathematical Competition, 5

Tags: algebra

Peter wants to create a new multiplication table for the four numbers $1, 2, 3, 4$ in such a way that the product of two of them is also one of them. He wants also that $(a\cdot b)\cdot c = a\cdot (b\cdot c)$ holds and that $ab \ne ac$ and $ba \ne ca$ and $b \ne c$. Peter is successful in constructing the new table. In his new table, $1\cdot 3 = 2$ and $2\cdot 2 = 4$. What is the product $3\cdot 1$ according to Peter's table?

2005 Oral Moscow Geometry Olympiad, 1

Tags: geometry , rectangle , area

The hexagon has five $90^o$ angles and one $270^o$ angle (see picture). Use a straight-line ruler to divide it into two equal-sized polygons. [img]https://cdn.artofproblemsolving.com/attachments/d/8/cdd4df68644bb8e04adbe1b265039b82a5382b.png[/img]

1999 Ukraine Team Selection Test, 11

Tags: geometry , parallelogram , hexagon , area , equilateral

Let $ABCDEF$ be a convex hexagon such that $BCEF$ is a parallelogram and $ABF$ an equilateral triangle. Given that $BC = 1, AD = 3, CD+DE = 2$, compute the area of $ABCDEF$

1967 IMO Longlists, 11

Tags: counting , triangle inequality , combinatorics , triangle

Let $n$ be a positive integer. Find the maximal number of non-congruent triangles whose sides lengths are integers $\leq n.$

1954 Moscow Mathematical Olympiad, 272

Tags: trigonometry , equation , algebra

Find all real solutions of the equation $x^2 + 2x \sin (xy) + 1 = 0$.

2017 Pan-African Shortlist, A?

Tags: algebra , integer part

Find all the real numbers $x$ such that $\frac{1}{[x]}+\frac{1}{[2x]}=\{x\}+\frac{1}{3}$ where $[x]$ denotes the integer part of $x$ and $\{x\}=x-[x]$. For example, $[2.5]=2, \{2.5\} = 0.5$ and $[-1.7]= -2, \{-1.7\} = 0.3$

2003 Tournament Of Towns, 2

Tags: induction , number theory

Prove that every positive integer can be represented in the form \[3^{u_1} \ldots 2^{v_1} + 3^{u_2} \ldots 2^{v_2} + \ldots + 3^{u_k} \ldots 2^{v_k}\] with integers $u_1, u_2, \ldots , u_k, v_1, \ldots, v_k$ such that $u_1 > u_2 >\ldots > u_k\ge 0$ and $0 \le v_1 < v_2 <\ldots < v_k$.

2011 HMNT, 10

Tags: geometry

Let $\Omega$ be a circle of radius $8$ centered at point $O$, and let $M$ be a point on $\Omega$. Let $S$ be the set of points $P$ such that $P$ is contained within $\Omega$ , or such that there exists some rectangle $ABCD$ containing $P$ whose center is on $\Omega$ with$ AB = 4$, $BC = 5$, and $BC \parallel OM$. Find the area of $S$.

2023 Assara - South Russian Girl's MO, 8

Tags: geometry , perimeter

a) Given a convex hexagon $ABCDEF$, which has a center of symmetry. Prove that the perimeter of triangle $ACE$ is greater than half the perimeter of hexagon $ABCDEF$. b) Given a convex $(2n)$-gon $P$ having a center of symmetry, its vertices are colored alternately red and blue. Let $Q$ be an $n$-gon with red vertices. Is it possible to say that the perimeter of $Q$ is certainly greater than half the perimeter $P$? Solve the problem for $n = 4$ and $n = 5$.

2005 Bulgaria Team Selection Test, 6

Tags: graph theory , ramsey theory , combinatorics

In a group of nine persons it is not possible to choose four persons such that every one knows the three others. Prove that this group of nine persons can be partitioned into four groups such that nobody knows anyone from his or her group.

2017 Canadian Mathematical Olympiad Qualification, 8

Tags: geometry

A convex quadrilateral $ABCD$ is said to be [i]dividable[/i] if for every internal point $P$, the area of $\triangle PAB$ plus the area of $\triangle PCD$ is equal to the area of $\triangle PBC$ plus the area of $\triangle PDA$. Characterize all quadrilaterals which are dividable.

2005 Purple Comet Problems, 8

Tags:

Find $x$ if\[\cfrac{1}{\cfrac{1}{\cfrac{1}{\cfrac{1}{x}+\cfrac12}+\cfrac{1}{\cfrac{1}{x}+\cfrac12}}+\cfrac{1}{\cfrac{1}{\cfrac{1}{x}+\cfrac12}+\cfrac{1}{\cfrac{1}{x}+\cfrac12}}}=\frac{x}{36}.\]

1992 India National Olympiad, 3

Tags: relatively prime , number theory

Find the remainder when $19^{92}$ is divided by 92.

2010 Math Prize for Girls Olympiad, 1

Tags: floor function , ceiling function

Let $S$ be a set of 100 integers. Suppose that for all positive integers $x$ and $y$ (possibly equal) such that $x + y$ is in $S$, either $x$ or $y$ (or both) is in $S$. Prove that the sum of the numbers in $S$ is at most 10,000.

1964 German National Olympiad, 5

Tags: geometry , angle

A triangle $ABC$ with $\beta= 45^o$ is given. There is a point $P$ on side $BC$, where $BP : PC =1 : 2$ (inner division) and $\angle APC = 60^o$. Someone claims that you can do it with elementary geometric theorems alone without using the plane trigonometry, the size of the angle $\gamma$ determine.

2013-2014 SDML (High School), 1

Tags:

In base $10$, the product $31\times33$ does not equal $1243$. In what base does $31\times33=1243$?