Olympiad problems

2014 China National Olympiad, 1

Tags: geometry , incenter , ratio , symmetry , circumcircle , trigonometry , angle bisector

Let $ABC$ be a triangle with $AB>AC$. Let $D$ be the foot of the internal angle bisector of $A$. Points $F$ and $E$ are on $AC,AB$ respectively such that $B,C,F,E$ are concyclic. Prove that the circumcentre of $DEF$ is the incentre of $ABC$ if and only if $BE+CF=BC$.

1994 Swedish Mathematical Competition, 4

Tags: number theory , diophantine , diophantine equation

Find all integers $m, n$ such that $2n^3 - m^3 = mn^2 + 11$.

2007 Purple Comet Problems, 3

Tags:

A bowl contained $10\%$ blue candies and $25\%$ red candies. A bag containing three quarters red candies and one quarter blue candies was added to the bowl. Now the bowl is $16\%$ blue candies. What percentage of the candies in the bowl are now red?

2002 Mid-Michigan MO, 10-12

Tags: algebra , geometry , combinatorics , number theory

[b]p1.[/b] Find all integer solutions of the equation $a^2 - b^2 = 2002$. [b]p2.[/b] Prove that the disks drawn on the sides of a convex quadrilateral as on diameters cover this quadrilateral. [b]p3.[/b] $30$ students from one school came to Mathematical Olympiad. In how many different ways is it possible to place them in four rooms? [b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

2001 AIME Problems, 13

Tags: number theory , relatively prime

In quadrilateral $ABCD$, $\angle{BAD}\cong\angle{ADC}$ and $\angle{ABD}\cong\angle{BCD}$, $AB=8$, $BD=10$, and $BC=6$. The length $CD$ may be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Kyiv City MO 1984-93 - geometry, 1993.9.3

Tags: equilateral , fixed , geometry

The circle divides each side of an equilateral triangle into three equal parts. Prove that the sum of the squares of the distances from any point of this circle to the vertices of the triangle is constant.

2000 Slovenia National Olympiad, Problem 2

Tags: functional equation , fe

Find all functions $f:\mathbb R\to\mathbb R$ such that for all $x,y\in\mathbb R$, $$f(x-f(y))=1-x-y.$$

2023 Sharygin Geometry Olympiad, 4

Tags: geometry , incenter

Points $D$ and $E$ lie on the lateral sides $AB$ and $BC$ respectively of an isosceles triangle $ABC$ in such a way that $\angle BED = 3\angle BDE$. Let $D'$ be the reflection of $D$ about $AC$. Prove that the line $D'E$ passes through the incenter of $ABC$.

1974 Putnam, A4

Tags: expected value , coin tosses

An unbiased coin is tossed $n$ times. What is the expected value of $|H-T|$, where $H$ is the number of heads and $T$ is the number of tails?

1997 Tournament Of Towns, (563) 4

Tags: geometry , combinatorial geometry , combinatorics , regular

(a) Several identical napkins, each in the shape of a regular hexagon, are put on a table (the napkins may overlap). Each napkin has one side which is parallel to a fixed line. Is it always possible to hammer a few nails into the table so that each napkin is nailed with exactly one nail? (b) The same question for regular pentagons. (A Kanel)