Olympiad problems

2010 Princeton University Math Competition, 5

Tags:

Given that $x$, $y$ are positive integers with $x(x+1)|y(y+1)$, but neither $x$ nor $x+1$ divides either of $y$ or $y+1$, and $x^2 + y^2$ as small as possible, find $x^2 + y^2$.

2010 Middle European Mathematical Olympiad, 12

Tags: number theory

We are given a positive integer $n$ which is not a power of two. Show that ther exists a positive integer $m$ with the following two properties: (a) $m$ is the product of two consecutive positive integers; (b) the decimal representation of $m$ consists of two identical blocks with $n$ digits. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 8)[/i]

2010 Contests, 3

Tags: combinatorics

A token is placed in one square of a $m\times n$ board, and is moved according to the following rules: [list] [*]In each turn, the token can be moved to a square sharing a side with the one currently occupied. [*]The token cannot be placed in a square that has already been occupied. [*]Any two consecutive moves cannot have the same direction.[/list] The game ends when the token cannot be moved. Determine the values of $m$ and $n$ for which, by placing the token in some square, all the squares of the board will have been occupied in the end of the game.

2018 Costa Rica - Final Round, A1

Tags: algebra , inequalities , min

If $x \in R-\{-7\}$, determine the smallest value of the expression $$\frac{2x^2 + 98}{(x + 7)^2}$$

2016 Mexico National Olmypiad, 6

Tags: tangent circle , geometry

Let $ABCD$ a quadrilateral inscribed in a circumference, $l_1$ the parallel to $BC$ through $A$, and $l_2$ the parallel to $AD$ through $B$. The line $DC$ intersects $l_1$ and $l_2$ at $E$ and $F$, respectively. The perpendicular to $l_1$ through $A$ intersects $BC$ at $P$, and the perpendicular to $l_2$ through $B$ cuts $AD$ at $Q$. Let $\Gamma_1$ and $\Gamma_2$ be the circumferences that pass through the vertex of triangles $ADE$ and $BFC$, respectively. Prove that $\Gamma_1$ and $\Gamma_2$ are tangent to each other if and only if $DP$ is perpendicular to $CQ$.

2000 Harvard-MIT Mathematics Tournament, 6

Tags: combinatorics

Three cards, only one of which is an ace, are placed face down on a table. You select one, but do not look at it. The dealer turns over one of the other cards, which is not the ace (if neither are, he picks one of them randomly to turn over). You get a chance to change your choice and pick either of the remaining two face-down cards. If you selected the cards so as to maximize the chance of finding the ace on the second try, what is the probability that you selected it on the (a) first try? (b) second try?

2007 Croatia Team Selection Test, 3

Tags: search , geometry

Let $ABC$ be a triangle such that $|AC|>|AB|$. Let $X$ be on line $AB$ (closer to $A$) such that $|BX|=|AC|$ and let $Y$ be on the segment $AC$ such that $|CY|=|AB|$. Intersection of lines $XY$ and bisector of $BC$ is point $P$. Prove that $\angle BPC+\angle BAC = 180^\circ$.

2018 Malaysia National Olympiad, A1

Tags: geometry , hexagon , equilateral

Hassan has a piece of paper in the shape of a hexagon. The interior angles are all $120^o$, and the side lengths are $1$, $2$, $3$, $4$, $5$, $6$, although not in that order. Initially, the paper is in the shape of an equilateral triangle, then Hassan has cut off three smaller equilateral triangle shapes, one at each corner of the paper. What is the minimum possible side length of the original triangle?

2021 South Africa National Olympiad, 3

Tags: perfect square , number theory , sum of squares , power of 2 , prime

Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$.

2001 India IMO Training Camp, 2

Tags: modular arithmetic , number theory

Let $p > 3$ be a prime. For each $k\in \{1,2, \ldots , p-1\}$, define $x_k$ to be the unique integer in $\{1, \ldots, p-1\}$ such that $kx_k\equiv 1 \pmod{p}$ and set $kx_k = 1+ pn_k$. Prove that : \[\sum_{k=1}^{p-1}kn_k \equiv \frac{p-1}{2} \pmod{p}\]

1994 China Team Selection Test, 2

Tags: algebra , polynomial , calculus , integration , gauss , number theory , prime number

Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

1979 Bundeswettbewerb Mathematik, 4

Tags: sequence , infinitely many solutions , number theory , last digit

An infinite sequence $p_1, p_2, p_3, \ldots$ of natural numbers in the decimal system has the following property: For every $i \in \mathbb{N}$ the last digit of $p_{i+1}$ is different from $9$, and by omitting this digit one obtains number $p_i$. Prove that this sequence contains infinitely many composite numbers.

1966 IMO Longlists, 5

Tags: inequalities , trigonometry , geometric inequality

Prove the inequality \[\tan \frac{\pi \sin x}{4\sin \alpha} + \tan \frac{\pi \cos x}{4\cos \alpha} >1\] for any $x, \alpha$ with $0 \leq x \leq \frac{\pi }{2}$ and $\frac{\pi}{6} < \alpha < \frac{\pi}{3}.$

2020 Romanian Master of Mathematics, 1

Tags: geometry

Let $ABC$ be a triangle with a right angle at $C$. Let $I$ be the incentre of triangle $ABC$, and let $D$ be the foot of the altitude from $C$ to $AB$. The incircle $\omega$ of triangle $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $A_1$, $B_1$, and $C_1$, respectively. Let $E$ and $F$ be the reflections of $C$ in lines $C_1A_1$ and $C_1B_1$, respectively. Let $K$ and $L$ be the reflections of $D$ in lines $C_1A_1$ and $C_1B_1$, respectively. Prove that the circumcircles of triangles $A_1EI$, $B_1FI$, and $C_1KL$ have a common point.

1997 Czech And Slovak Olympiad IIIA, 4

Tags: sequence , prime , number theory

Show that there exists an increasing sequence $a_1,a_2,a_3,...$ of natural numbers such that, for any integer $k \ge 2$, the sequence $k+a_n$ ($n \in N$) contains only finitely many primes.

2024 Yasinsky Geometry Olympiad, 5

Tags: geometry , angle bisector , median , parallelogram

Let $ AL $ be the bisector of triangle $ ABC $, $ O $ the center of its circumcircle, and $ D $ and $ E $ the midpoints of $ BL $ and $ CL $, respectively. Points $ P $ and $ Q $ are chosen on segments $ AD $ and $ AE $ such that $ APLQ $ is a parallelogram. Prove that $ PQ \perp AO $. [i]Proposed by Mykhailo Plotnikov[/i]

2003 Kazakhstan National Olympiad, 8

Tags: function , algebra

Determine all functions $f: \mathbb R \to \mathbb R$ with the property \[f(f(x)+y)=2x+f(f(y)-x), \quad \forall x,y \in \mathbb R.\]

2018 CMI B.Sc. Entrance Exam, 3

Tags: functional equation , number theory , combinatorics

Let $f$ be a function on non-negative integers defined as follows $$f(2n)=f(f(n))~~~\text{and}~~~f(2n+1)=f(2n)+1$$ [b](a)[/b] If $f(0)=0$ , find $f(n)$ for every $n$. [b](b)[/b] Show that $f(0)$ cannot equal $1$. [b](c)[/b] For what non-negative integers $k$ (if any) can $f(0)$ equal $2^k$ ?

2020 BMT Fall, 17

Tags: combinatorics

Shrek throws $5$ balls into $5$ empty bins, where each ball’s target is chosen uniformly at random. After Shrek throws the balls, the probability that there is exactly one empty bin can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

2024 BAMO, D/2

Tags: geometry , combinatorics

Sasha wants to bake $6$ cookies in his $8$ inch $\times$ $8$ inch square baking sheet. With a cookie cutter, he cuts out from the dough six circular shapes, each exactly $3$ inches in diameter. Can he place these six dough shapes on the baking sheet without the shapes touching each other? If yes, show us how. If no, explain why not. (Assume that the dough does not expand during baking.)

1949-56 Chisinau City MO, 27

Tags: right triangle , similar triangles , similar , geometry

The areas of two right-angled triangles have ratio equal to the squares of their hypotenuses. Show that these triangles are similar.

1999 IMO Shortlist, 4

Tags: modular arithmetic , combinatorics , counting , additive combinatorics , additive number theory

Let $A$ be a set of $N$ residues $\pmod{N^{2}}$. Prove that there exists a set $B$ of of $N$ residues $\pmod{N^{2}}$ such that $A + B = \{a+b|a \in A, b \in B\}$ contains at least half of all the residues $\pmod{N^{2}}$.

1996 Estonia National Olympiad, 1

Tags: prime , diophantine equation , diophantine , number theory

Let $p$ be a fixed prime. Find all pairs $(x,y)$ of positive numbers satisfying $p(x-y) = xy$.

2017 AMC 8, 20

Tags:

An integer between $1000$ and $9999$, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct? $\textbf{(A) }\frac{14}{75}\qquad\textbf{(B) }\frac{56}{225}\qquad\textbf{(C) }\frac{107}{400}\qquad\textbf{(D) }\frac{7}{25}\qquad\textbf{(E) }\frac{9}{25}$

2007 AMC 8, 14

Tags: pythagorean theorem , geometry

The base of isosceles $\triangle{ABC}$ is $24$ and its area is $60$. What is the length of one of the congruent sides? $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 18$