Olympiad problems

2010 AMC 10, 22

Tags:

Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible? $ \textbf{(A)}\ 1930\qquad\textbf{(B)}\ 1931\qquad\textbf{(C)}\ 1932\qquad\textbf{(D)}\ 1933\qquad\textbf{(E)}\ 1934$

1988 Czech And Slovak Olympiad IIIA, 1

Let $f$ be a representation of the set $M = \{1, 2,..., 1988\}$ into $M$. For any natural $n$, let $x_1 = f(1)$, $x_{n+1} = f(x_n)$. Find out if there exists $m$ such that $x_{2m} = x_m$.

2007 Germany Team Selection Test, 3

Tags: ratio , geometry , projective geometry , homothety

Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.

2017 Purple Comet Problems, 26

Tags: geometry

The incircle of $\vartriangle ABC$ is tangent to sides $\overline{BC}, \overline{AC}$, and $\overline{AB}$ at $D, E$, and $F$, respectively. Point $G$ is the intersection of lines $AC$ and $DF$ as shown. The sides of $\vartriangle ABC$ have lengths $AB = 73, BC = 123$, and $AC = 120$. Find the length $EG$. [img]https://cdn.artofproblemsolving.com/attachments/d/a/aede28071a1a6b94bbe3ad8e1e104822b89439.png[/img]

1991 Federal Competition For Advanced Students, 2

Tags: algebra

Solve in real numbers the equation: $ \frac{1}{x}\plus{}\frac{1}{x\plus{}2}\minus{}\frac{1}{x\plus{}4}\minus{}\frac{1}{x\plus{}6}\minus{}\frac{1}{x\plus{}8}\minus{}\frac{1}{x\plus{}10}\plus{}\frac{1}{x\plus{}12}\plus{}\frac{1}{x\plus{}14}\equal{}0.$

2023 Israel TST, P3

Tags: functional equation , algebra , function

Find all functions $f:\mathbb{Z}\to \mathbb{Z}_{>0}$ for which \[f(x+f(y))^2+f(y+f(x))^2=f(f(x)+f(y))^2+1\] holds for any $x,y\in \mathbb{Z}$.

2012 Miklós Schweitzer, 11

Tags: probability , real analysis

Let $X_1,X_2,..$ be independent random variables with the same distribution, and let $S_n=X_1+X_2+...+X_n, n=1,2,...$. For what real numbers $c$ is the following statement true: $$P\left(\left| \frac{S_{2n}}{2n}- c \right| \leqslant \left| \frac{S_n}{n}-c\right| \right)\geqslant \frac{1}{2}$$

2010 Dutch IMO TST, 3

Tags: number theory , perfect square

Let $n\ge 2$ be a positive integer and $p $ a prime such that $n|p-1$ and $p | n^3-1$. Show $ 4p-3$ is a square.

2009 Today's Calculation Of Integral, 426

Tags: calculus , integration , algebra , polynomial , calculus computations

Consider the polynomial $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$, with degree less than or equal to 2. When $ f$ varies with subject to the constrain $ f(0) \equal{} 0,\ f(2) \equal{} 2$, find the minimum value of $ S\equal{}\int_0^2 |f'(x)|\ dx$.

Ukraine Correspondence MO - geometry, 2008.7

Tags: equal angles , equal segments , geometry

On the sides $AC$ and $AB$ of the triangle $ABC$, the points $D$ and $E$ were chosen such that $\angle ABD =\angle CBD$ and $3 \angle ACE = 2\angle BCE$. Let $H$ be the point of intersection of $BD$ and $CE$, and $CD = DE = CH$. Find the angles of triangle $ABC$.

PEN A Problems, 39

Tags: divisibility theory

Let $n$ be a positive integer. Prove that the following two statements are equivalent. [list][*] $n$ is not divisible by $4$ [*] There exist $a, b \in \mathbb{Z}$ such that $a^{2}+b^{2}+1$ is divisible by $n$. [/list]

2023 India EGMO TST, P3

Tags: combinatorics

Let $N \geqslant 3$ be an integer. In the country of Sibyl, there are $N^2$ towns arranged as the vertices of an $N \times N$ grid, with each pair of towns corresponding to an adjacent pair of vertices on the grid connected by a road. Several automated drones are given the instruction to traverse a rectangular path starting and ending at the same town, following the roads of the country. It turned out that each road was traversed at least once by some drone. Determine the minimum number of drones that must be operating. [i]Proposed by Sutanay Bhattacharya and Anant Mudgal[/i]

2016 Kurschak Competition, 1

Tags: combinatorics , subset

Let $1\le k\le n$ be integers. At most how many $k$-element subsets can we select from $\{1,2,\dots,n\}$ such that for any two selected subsets, one of the subsets consists of the $k$ smallest elements of their union?

2020-21 IOQM India, 9

Tags: geometry , angle bisector

Let A$BC$ be a triangle with $AB = 5, AC = 4, BC = 6$. The internal angle bisector of $C$ intersects the side $AB$ at $D$. Points $M$ and $N$ are taken on sides $BC$ and $AC$, respectively, such that $DM\parallel AC$ and $DN \parallel BC$. If $(MN)^2 =\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers then what is the sum of the digits of $|p - q|$?

2011 Morocco National Olympiad, 3

Tags:

Problem 3 (MAR CP 1992) : From the digits $1,2,...,9$, we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order as follows : $123456789$, $123456798$, ..., $987654321$. What is the $100000th$ number ?

1973 USAMO, 3

Tags: probability , symmetry

Three distinct vertices are chosen at random from the vertices of a given regular polygon of $ (2n\plus{}1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points?

2017 Korea National Olympiad, problem 5

Tags: number theory , polynomial , cubic , divisibility

Given a prime $p$, show that there exist two integers $a, b$ which satisfies the following. For all integers $m$, $m^3+ 2017am+b$ is not a multiple of $p$.

2011 Singapore MO Open, 3

Tags: inequalities , trigonometry

Let $x,y,z>0$ such that $\frac1x+\frac1y+\frac1z<\frac{1}{xyz}$. Show that \[\frac{2x}{\sqrt{1+x^2}}+\frac{2y}{\sqrt{1+y^2}}+\frac{2z}{\sqrt{1+z^2}}<3.\]

2025 Abelkonkurransen Finale, 2a

Tags: number theory

A teacher asks each of eleven pupils to write a positive integer with at most four digits, each on a separate yellow sticky note. Show that if all the numbers are different, the teacher can always submit two or more of the eleven stickers so that the average of the numbers on the selected notes are not an integer.

2018 Miklós Schweitzer, 4

Tags: combinatorics

Let $P$ be a finite set of points in the plane. Assume that the distance between any two points is an integer. Prove that $P$ can be colored by three colors such that the distance between any two points of the same color is an even number.

2009 Germany Team Selection Test, 1

Tags: geometry

Let $ I$ be the incircle centre of triangle $ ABC$ and $ \omega$ be a circle within the same triangle with centre $ I.$ The perpendicular rays from $ I$ on the sides $ \overline{BC}, \overline{CA}$ and $ \overline{AB}$ meets $ \omega$ in $ A', B'$ and $ C'.$ Show that the three lines $ AA', BB'$ and $ CC'$ have a common point.

2007 Pre-Preparation Course Examination, 16

Tags: induction , number theory

Prove that $2^{2^{n}}+2^{2^{{n-1}}}+1$ has at least $n$ distinct prime divisors.

2015 Thailand TSTST, 2

Tags: combinatorics

Let $C$ be the set of all 100-digit numbers consisting of only the digits $1$ and $2$. Given a number in $C$, we may transform the number by considering any $10$ consecutive digits $x_0x_1x_2 \dots x_9$ and transform it into $x_5x_6\dots x_9x_0x_1\dots x_4$. We say that two numbers in $C$ are similar if one of them can be reached from the other by performing finitely many such transformations. Let $D$ be a subset of $C$ such that any two numbers in $D$ are not similar. Determine the maximum possible size of $D$.

2023 Israel National Olympiad, P7

Tags: number theory , digit , guessing game

Ana and Banana are playing a game. Initially, Ana secretly picks a number $1\leq A\leq 10^6$. In each subsequent turn of the game, Banana may pick a positive integer $B$, and Ana will reveal to him the most common digit in the product $A\cdot B$ (written in decimal notation). In the case when at least two digits are tied for being the most common, Ana will reveal all of them to Banana. For example, if $A\cdot B=2022$, Ana will tell Banana that the digit $2$ is the most common, while if $A\cdot B=5783783$, Ana will reveal that $3, 7$ and $8$ are the most common. Banana's goal is to determine with certainty the number $A$ after some number of turns. Does he have a winning strategy?

2020 Israel Olympic Revenge, P1

Tags: functional equation , algebra , function , symmetry

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x,y\in \mathbb{R}$ one has \[f(f(x)+y)=f(x+f(y))\] and in addition the set $f^{-1}(a)=\{b\in \mathbb{R}\mid f(b)=a\}$ is a finite set for all $a\in \mathbb{R}$.