Olympiad problems

Ukrainian TYM Qualifying - geometry, 2018.17

Using a compass and a ruler, construct a triangle $ABC$ given the sides $b, c$ and the segment $AI$, where$ I$ is the center of the inscribed circle of this triangle.

2008 Switzerland - Final Round, 9

Tags: combinatorics

There are 7 lines in the plane. A point is called a [i]good[/i] point if it is contained on at least three of these seven lines. What is the maximum number of [i]good[/i] points?

2007 Germany Team Selection Test, 3

Tags: geometry

A point $ P$ in the interior of triangle $ ABC$ satisfies \[ \angle BPC \minus{} \angle BAC \equal{} \angle CPA \minus{} \angle CBA \equal{} \angle APB \minus{} \angle ACB.\] Prove that \[ \bar{PA} \cdot \bar{BC} \equal{} \bar{PB} \cdot \bar{AC} \equal{} \bar{PC} \cdot \bar{AB}.\]

2014 Purple Comet Problems, 22

Tags: vector , algorithm

For positive integers $m$ and $n$, let $r(m, n)$ be the remainder when $m$ is divided by $n$. Find the smallest positive integer $m$ such that \[r(m, 1) + r(m, 2) + r(m, 3) +\cdots+ r(m, 10) = 4.\]

2022 HMNT, 25

Tags:

In convex quadrilateral $ABCD$ with $AB = 11$ and $CD = 13,$ there is a point $P$ for which $\triangle{ADP}$ and $\triangle{BCP}$ are congruent equilateral triangles. Compute the side length of these triangles.

Novosibirsk Oral Geo Oly VIII, 2019.3

Tags: ratio , paper folding , folding , geometry , square

A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$. In what ratio does the fold line divide side $BC$?

2004 Romania National Olympiad, 2

Tags: algebra , polynomial , function , superior algebra

Let $f \in \mathbb Z[X]$. For an $n \in \mathbb N$, $n \geq 2$, we define $f_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z$ through $f_n \left( \widehat x \right) = \widehat{f \left( x \right)}$, for all $x \in \mathbb Z$. (a) Prove that $f_n$ is well defined. (b) Find all polynomials $f \in \mathbb Z[X]$ such that for all $n \in \mathbb N$, $n \geq 2$, the function $f_n$ is surjective. [i]Bogdan Enescu[/i]

2010 Chile National Olympiad, 4

Tags: number theory , algebra

Let $m, n$ integers such that satisfy $$m + n\sqrt2 = \left(1 +\sqrt2\right)^{2010} .$$ Find the remainder that is obtained when dividing $n$ by $5$.

2016 China Northern MO, 5

Tags: algebra

$a_1=2,a_{n+1}=\frac{2^{n+1}a_n}{(n+\frac{1}{2})a_n+2^n}(n\in\mathbb{Z}_+)$ [b](a)[/b] Find $a_n$. [b](b)[/b] Let $b_n=\frac{n^3+2n^2+2n+2}{n(n+1)(n^2+1)a_n}$. Find $S_n=\sum_{i=1}^nb_i$.

2015 CCA Math Bonanza, L1.3

Tags:

Daniel can hack a finite cylindrical log into $3$ pieces in $6$ minutes. How long would it take him to cut it into $9$ pieces, assuming each cut takes Daniel the same amount of time? [i]2015 CCA Math Bonanza Lightning Round #1.3[/i]

2012 Indonesia TST, 4

Tags: quadratic , floor function , search , modular arithmetic , induction , pigeonhole principle , number theory

Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$. Remark: "Natural numbers" is the set of positive integers.

2018 India IMO Training Camp, 1

Tags: number theory

For a natural number $k>1$, define $S_k$ to be the set of all triplets $(n,a,b)$ of natural numbers, with $n$ odd and $\gcd (a,b)=1$, such that $a+b=k$ and $n$ divides $a^n+b^n$. Find all values of $k$ for which $S_k$ is finite.

2021 Saudi Arabia Training Tests, 24

Tags: geometry , excenter

Let $ABC$ be triangle with $M$ is the midpoint of $BC$ and $X, Y$ are excenters with respect to angle $B,C$. Prove that $MX$, $MY$ intersect $AB$, $AC$ at four points that are vertices of circumscribed quadrilateral.

2021 ELMO Problems, 3

Tags: combinatorics

Each cell of a $100\times 100$ grid is colored with one of $101$ colors. A cell is [i]diverse[/i] if, among the $199$ cells in its row or column, every color appears at least once. Determine the maximum possible number of diverse cells.

2012 IMO Shortlist, C7

Tags: combinatorics

There are given $2^{500}$ points on a circle labeled $1,2,\ldots ,2^{500}$ in some order. Prove that one can choose $100$ pairwise disjoint chords joining some of theses points so that the $100$ sums of the pairs of numbers at the endpoints of the chosen chord are equal.

1955 Putnam, A2

Tags:

$A_1 ~A_2~ \ldots ~A_n$ is a regular polygon inscribed in a circle of radius $r$ and center $O.$ $P$ is a point on line $OA_1$ extended beyond $A_1.$ Show that \[ \prod^n_{i=1} ~ \overline{PA}_{~i} = \overline{OP}^{~n} - r^n. \]

2019 Iranian Geometry Olympiad, 3

Tags: geometry , combinatorics

There are $n>2$ lines on the plane in general position; Meaning any two of them meet, but no three are concurrent. All their intersection points are marked, and then all the lines are removed, but the marked points are remained. It is not known which marked point belongs to which two lines. Is it possible to know which line belongs where, and restore them all? [i]Proposed by Boris Frenkin - Russia[/i]

2011 Today's Calculation Of Integral, 682

Tags: calculus , integration , trigonometry , geometry , function , calculus computations

On the $x$-$y$ plane, 3 half-lines $y=0,\ (x\geq 0),\ y=x\tan \theta \ (x\geq 0),\ y=-\sqrt{3}x\ (x\leq 0)$ intersect with the circle with the center the origin $O$, radius $r\geq 1$ at $A,\ B,\ C$ respectively. Note that $\frac{\pi}{6}\leq \theta \leq \frac{\pi}{3}$. If the area of quadrilateral $OABC$ is one third of the area of the regular hexagon which inscribed in a circle with radius 1, then evaluate $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} r^2d\theta .$ [i]2011 Waseda University of Education entrance exam/Science[/i]

2013 IMO Shortlist, G5

Tags: geometry , complex number , hexagon

Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.

2014 Dutch IMO TST, 2

Tags: number theory

The sets $A$ and $B$ are subsets of the positive integers. The sum of any two distinct elements of $A$ is an element of $B$. The quotient of any two distinct elements of $B$ (where we divide the largest by the smallest of the two) is an element of $A$. Determine the maximum number of elements in $A\cup B$.

2019 Saudi Arabia JBMO TST, 4

Tags: combinatorics

Given is a grid 11x11 with 121 cells. Four of them are colored in black, the rest are white. We have to cut a completely white rectangle (it could be a square and the rectangle must have its sides parralel to the lines of the grid), so that this rectangle has maximal possible area. What largest area of this rectangle we can guarantee? (We can cut this rectangle for every placement of the black squares)

1969 IMO Longlists, 41

Tags: linear recurrence , algebra , recurrence relation , system of equations

$(MON 2)$ Given reals $x_0, x_1, \alpha, \beta$, find an expression for the solution of the system \[x_{n+2} -\alpha x_{n+1} -\beta x_n = 0, \qquad n= 0, 1, 2, \ldots\]

2010 Oral Moscow Geometry Olympiad, 2

Tags: geometry , cyclic quadrilateral , perpendicular , parallel

Quadrangle $ABCD$ is inscribed in a circle. The perpendicular from the vertex $C$ on the bisector of $\angle ABD$ intersects the line $AB$ at the point $C_1$. The perpendicular from the vertex $B$ on the bisector of $\angle ACD$ intersects the line $CD$ at the point $B_1$. Prove that $B_1C_1 \parallel AD$.

2002 Germany Team Selection Test, 1

Tags: quadratic , number theory

Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \in \{1, 2, 3, \ldots, 1000\}$ and which are divisible by 121.

2016 China Team Selection Test, 6

Tags: combinatorial number theory , number theory , combinatorics

Let $m,n$ be naturals satisfying $n \geq m \geq 2$ and let $S$ be a set consisting of $n$ naturals. Prove that $S$ has at least $2^{n-m+1}$ distinct subsets, each whose sum is divisible by $m$. (The zero set counts as a subset).