Olympiad problems

2020 Iranian Combinatorics Olympiad, 7

Tags: coin , combinatorics

Seyed has 998 white coins, a red coin, and an unusual coin with one red side and one white side. He can not see the color of the coins instead he has a scanner which checks if all of the coin sides touching the scanner glass are white. Is there any algorithm to find the red coin by using the scanner at most 17 times? [i]Proposed by Seyed Reza Hosseini[/i]

2017 Olympic Revenge, 1

Tags: number theory

Prove that does not exist positive integers $a$, $b$ and $k$ such that $4abk-a-b$ is a perfect square.

2011 Postal Coaching, 3

Tags: algebra , function

Let $f : \mathbb{N} \longrightarrow \mathbb{N}$ be a function such that $(x + y)f (x) \le x^2 + f (xy) + 110$, for all $x, y$ in $\mathbb{N}$. Determine the minimum and maximum values of $f (23) + f (2011)$.

2019 Czech-Austrian-Polish-Slovak Match, 4

Tags: function , algebra

Given a real number $\alpha$, find all pairs $(f,g)$ of functions $f,g :\mathbb{R} \to \mathbb{R}$ such that $$xf(x+y)+\alpha \cdot yf(x-y)=g(x)+g(y) \;\;\;\;\;\;\;\;\;\;\; ,\forall x,y \in \mathbb{R}.$$

1983 Canada National Olympiad, 5

Tags: combinatorics

The geometric mean (G.M.) of $k$ positive integers $a_1$, $a_2$, $\dots$, $a_k$ is defined to be the (positive) $k$-th root of their product. For example, the G.M. of 3, 4, 18 is 6. Show that the G.M. of a set $S$ of $n$ positive numbers is equal to the G.M. of the G.M.'s of all non-empty subsets of $S$.

2006 Brazil National Olympiad, 3

Tags: function , algebra , induction , functional equation

Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that \[f(xf(y)+f(x)) = 2f(x)+xy\] for every reals $x,y$.

2019 Switzerland Team Selection Test, 7

Tags: number theory

Prove that for all positive integers $n$ there are positive integers $a,b$ such that $$n\mid 4a^2+9b^2-1.$$

MathLinks Contest 2nd, 4.2

Tags: geometry

Given is a finite set of points $M$ and an equilateral triangle $\Delta$ in the plane. It is known that for any subset $M' \subset M$, which has no more than $9$ points, can be covered by two translations of the triangle $\Delta$. Prove that the entire set $M$ can be covered by two translations of $\Delta$.

2018 Thailand Mathematical Olympiad, 7

Tags: coloring , combinatorics

We color each number in the set $S = \{1, 2, ..., 61\}$ with one of $25$ given colors, where it is not necessary that every color gets used. Let $m$ be the number of non-empty subsets of $S$ such that every number in the subset has the same color. What is the minimum possible value of $m$?

1973 IMO Shortlist, 15

Tags: trigonometric identities , algebra , trigonometry , polynomial

Prove that for all $n \in \mathbb N$ the following is true: \[2^n \prod_{k=1}^n \sin \frac{k \pi}{2n+1} = \sqrt{2n+1}\]

1993 Nordic, 4

Tags: number theory , sum of digits , positive integer

Denote by $T(n)$ the sum of the digits of the decimal representation of a positive integer $n$. a) Find an integer $N$, for which $T(k \cdot N)$ is even for all $k, 1 \le k \le 1992, $ but $T(1993 \cdot N)$ is odd. b) Show that no positive integer $N$ exists such that $T(k \cdot N)$ is even for all positive integers $k$.

2005 National Olympiad First Round, 34

Tags:

How many triples $(x,y,z)$ of positive integers are there such that $xyz=510510$ and $x^2y+y^2z+z^2x = xy^2 + yz^2 + zx^2$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \text{None of above} $

2013 All-Russian Olympiad, 2

Tags: combinatorics , geometric transformation , geometry , rotation , invariant

Circle is divided into $n$ arcs by $n$ marked points on the circle. After that circle rotate an angle $ 2\pi k/n $ (for some positive integer $ k $), marked points moved to $n$ [i] new points [/i], dividing the circle into $ n $ [i] new arcs[/i]. Prove that there is a new arc that lies entirely in the one of the old arсs. (It is believed that the endpoints of arcs belong to it.) [i]I. Mitrophanov[/i]

1989 IMO Longlists, 38

Tags: algebra , induction

A sequence of real numbers $ x_0, x_1, x_2, \ldots$ is defined as follows: $ x_0 \equal{} 1989$ and for each $ n \geq 1$ \[ x_n \equal{} \minus{} \frac{1989}{n} \sum^{n\minus{}1}_{k\equal{}0} x_k.\] Calculate the value of $ \sum^{1989}_{n\equal{}0} 2^n x_n.$

1976 All Soviet Union Mathematical Olympiad, 221

Tags: blackboard , combinatorics , game strategy

A row of $1000$ numbers is written on the blackboard. We write a new row, below the first according to the rule: We write under every number $a$ the natural number, indicating how many times the number $a$ is encountered in the first line. Then we write down the third line: under every number $b$ -- the natural number, indicating how many times the number $b$ is encountered in the second line, and so on. a) Prove that there is a line that coincides with the preceding one. b) Prove that the eleventh line coincides with the twelfth. c) Give an example of the initial line such, that the tenth row differs from the eleventh.

2018 Argentina National Olympiad, 5

Tags: combinatorial geometry , coloring , combinatorics

In the plane you have $2018$ points between which there are not three on the same line. These points are colored with $30$ colors so that no two colors have the same number of points. All triangles are formed with their three vertices of different colors. Determine the number of points for each of the $30$ colors so that the total number of triangles with the three vertices of different colors is as large as possible.

2007 Balkan MO Shortlist, N2

Tags: function , number theory

Prove that there are no distinct positive integers $x$ and $y$ such that $x^{2007} + y! = y^{2007} + x! $

XMO (China) 2-15 - geometry, 2.1

Tags: perpendicular , geometry

As shown in the figure, $BQ$ is a diameter of the circumcircle of $ABC$, and $D$ is the midpoint of arc $BC$ (excluding point $A$) . The bisector of the exterior angle of $\angle BAC$ intersects and the extension of $BC$ at point $E$. The ray $EQ$ intersects $\odot (ABC)$ at point $P$. Point $S$ lies on $PQ$ so that $SA = SP$. Point $T$ lies on $BC$ such that $TB = TD$. Prove that $TS \perp SE$. [img]https://cdn.artofproblemsolving.com/attachments/c/4/01460565e70b32b29cddb65d92e041bea40b25.png[/img]

2001 Moldova Team Selection Test, 6

Tags: geometry

Find the smallest possible area of a convex pentagon whose vertexes are lattice points in a plane.

2023 LMT Fall, 11

Tags: geometry

Let $LEX INGT_1ONMAT_2H$ be a regular $13$-gon. Find $\angle LMT_1$, in degrees. [i]Proposed by Edwin Zhao[/i]

2013 Stanford Mathematics Tournament, 20

Tags: expected value , probability , ceiling function

Ben is throwing darts at a circular target with diameter 10. Ben never misses the target when he throws a dart, but he is equally likely to hit any point on the target. Ben gets $\lceil 5-x \rceil$ points for having the dart land $x$ units away from the center of the target. What is the expected number of points that Ben can earn from throwing a single dart? (Note that $\lceil y \rceil$ denotes the smallest integer greater than or equal to $y$.)

2015 Kyoto University Entry Examination, 2

Tags: geometry

2. Find the minimum area of quadrilateral satisfy two condition as follows, (a) At least two interior angles are right angles. (b) A circle radius of $1$ inscribed.

2007 AIME Problems, 12

Tags: geometry , rotation

In isosceles triangle $ABC$, $A$ is located at the origin and $B$ is located at $(20, 0)$. Point $C$ is in the first quadrant with $AC = BC$ and $\angle BAC = 75^\circ$. If $\triangle ABC$ is rotated counterclockwise about point $A$ until the image of $C$ lies on the positive y-axis, the area of the region common to the original and the rotated triangle is in the form $p\sqrt{2}+q\sqrt{3}+r\sqrt{6}+s$ where $p$, $q$, $r$, $s$ are integers. Find $(p-q+r-s)/2$.

2024 IFYM, Sozopol, 3

Tags: geometry

Let $ X $ be an arbitrary point on the side $ BC $ of triangle $ ABC $. The point $ M $ on the ray $ AB^\to $ beyond $ B $, the point $ N $ on the ray $ AC^\to $ beyond $ C $, and the point $ K $ inside $ ABC $ are such that $ \angle BMX = \angle CNX = \angle KBC = \angle KCB $. The line through $ A $, parallel to $ BC $, intersects the line $ KX $ at point $ P $. Prove that the points $ A $, $ P $, $ M $, $ N $ lie on a circle.

1995 China Team Selection Test, 2

Tags: geometric transformation , locus problems , geometry , tangent

Given a fixed acute angle $\theta$ and a pair of internally tangent circles, let the line $l$ which passes through the point of tangency, $A$, cut the larger circle again at $B$ ($l$ does not pass through the centers of the circles). Let $M$ be a point on the major arc $AB$ of the larger circle, $N$ the point where $AM$ intersects the smaller circle, and $P$ the point on ray $MB$ such that $\angle MPN = \theta$. Find the locus of $P$ as $M$ moves on major arc $AB$ of the larger circle.