Olympiad problems

2021 China Second Round, 3

Tags: number theory

If $n\ge 4,\ n\in\mathbb{N^*},\ n\mid (2^n-2)$. Prove that $\frac{2^n-2}{n}$ is not a prime number.

2010 Contests, 1

Tags: combinatorics , pigeonhole principle

Given an integer number $n \geq 3$, consider $n$ distinct points on a circle, labelled $1$ through $n$. Determine the maximum number of closed chords $[ij]$, $i \neq j$, having pairwise non-empty intersections. [i]János Pach[/i]

2016 PUMaC Combinatorics A, 5

Tags: probability , expected value

Let $a_1,a_2,a_3,\ldots$ be an infinite sequence where for all positive integers $i$, $a_i$ is chosen to be a random positive integer between $1$ and $2016$, inclusive. Let $S$ be the set of all positive integers $k$ such that for all positive integers $j<k$, $a_j\neq a_k$. (So $1\in S$; $2\in S$ if and only if $a_1\neq a_2$; $3\in S$ if and only if $a_1\neq a_3$ and $a_2\neq a_3$; and so on.) In simplest form, let $\dfrac{p}{q}$ be the expected number of positive integers $m$ such that $m$ and $m+1$ are in $S$. Compute $pq$.

Kvant 2024, M2811

Tags: number theory

A sequence of positive integer numbers $a_1,...,a_{100}$ such is that $a_1=1$, and for all $n=1, 2,...,100$ number $(a_1+...+a_n) \left ( \frac{1}{a_1}+...+\frac{1}{a_n} \right )$ is integer. What is the maximum value that can take $ a_{100}$? [i] M. Turevskii [/i]

2001 AIME Problems, 14

Tags: recursion , combinatorics , pigeonhole principle

A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are possible?

2018 Hanoi Open Mathematics Competitions, 3

Tags: number theory , diophantine , inequalities

How many integers $n$ are there those satisfy the following inequality $n^4 - n^3 - 3n^2 - 3n - 17 < 0$? A. $4$ B. $6$ C. $8$ D. $10$ E. $12$

1993 National High School Mathematics League, 5

Tags:

In $\triangle ABC$, $c-a$ is equal to height on side $AC$. Then, the value of $\sin\frac{C-A}{2}+\cos\frac{C+A}{2}$ is $\text{(A)}1\qquad\text{(B)}\frac{1}{2}\qquad\text{(C)}\frac{1}{3}\qquad\text{(D)}-1$

2012 Chile National Olympiad, 3

Tags: combinatorics

A person enters the social network facebook. He befriends at least one person a day for the first $30$ days. At the end of those $30$ days, it has been exactly $45$ friends. Prove that there is a sequence of consecutive days where made exactly $14$ friends.

2014 Argentina Cono Sur TST, 1

Tags: number theory

A positive integer $N$ is written on a board. In a step, the last digit $c$ of the number on the board is erased, and after this, the remaining number $m$ is erased and replaced with $|m-3c|$ (for example, if the number $1204$ is on the board, after one step, we will replace it with $120 - 3 \cdot 4 = 108$). We repeat this until the number on the board has only one digit. Find all positive integers $N$ such that after a finite number of steps, the remaining one-digit number is $0$.

2020 China Team Selection Test, 2

Tags: equal segments , geometry , tangent circle , right angle

Given an isosceles triangle $\triangle ABC$, $AB=AC$. A line passes through $M$, the midpoint of $BC$, and intersects segment $AB$ and ray $CA$ at $D$ and $E$, respectively. Let $F$ be a point of $ME$ such that $EF=DM$, and $K$ be a point on $MD$. Let $\Gamma_1$ be the circle passes through $B,D,K$ and $\Gamma_2$ be the circle passes through $C,E,K$. $\Gamma_1$ and $\Gamma_2$ intersect again at $L \neq K$. Let $\omega_1$ and $\omega_2$ be the circumcircle of $\triangle LDE$ and $\triangle LKM$. Prove that, if $\omega_1$ and $\omega_2$ are symmetric wrt $L$, then $BF$ is perpendicular to $BC$.

2017 Vietnam Team Selection Test, 3

Tags: combinatorics

For each integer $n>0$, a permutation $a_1,a_2,\dots ,a_{2n}$ of $1,2,\dots 2n$ is called [i]beautiful[/i] if for every $1\leq i<j \leq 2n$, $a_i+a_{n+i}=2n+1$ and $a_i-a_{i+1}\not \equiv a_j-a_{j+1}$ (mod $2n+1$) (suppose that $a_{2n+1}=a_1$). a. For $n=6$, point out a [i]beautiful [/i] permutation. b. Prove that there exists a [i]beautiful[/i] permutation for every $n$.

2002 National Olympiad First Round, 32

Tags:

Which of the following is true if $S = \dfrac 1{1^2} + \dfrac 1{2^2} + \dfrac 1{3^2} + \cdots + \dfrac 1{2001^2} + \dfrac 1{2002^2}$? $ \textbf{a)}\ 1\leq S < \dfrac 43 \qquad\textbf{b)}\ \dfrac 43 \leq S < 2 \qquad\textbf{c)}\ 2 \leq S < \dfrac 73$ $\textbf{d)}\ \dfrac 73 \leq S < \dfrac 52 \qquad\textbf{e)}\ \dfrac 52 \leq S < 3 $

2011 Moldova Team Selection Test, 3

Tags: geometric transformation , rotation , vector , geometry

Let $ABCD$ be a quadrilateral and $M$ the midpoint of the segment $AB$. Outside of the quadrilateral are constructed the equilateral triangles $BCE$, $CDF$ and $DAG$. Let $P$ and $N$ be the midpoints of the segments $GF$ and $EF$. Prove that the triangle $MNP$ is equilateral.

2021 Auckland Mathematical Olympiad, 2

Tags: distance , equilateral , geometry

Given five points inside an equilateral triangle of side length $2$, show that there are two points whose distance from each other is at most $ 1$.

2015 Mathematical Talent Reward Programme, SAQ: P 5

Tags: number theory , least common multiple , greatest common divisor

Let $a$ be the smallest and $A$ the largest of $n$ distinct positive integers. Prove that the least common multiple of these numbers is greater than or equal to $n a$ and that the greatest common divisor is less than or equal to $\frac{A}{n}$

2024 Indonesia TST, 1

Tags: geometry

Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$. Prove that line $AO$ passes through the midpoint of segment $BE$.

2018-IMOC, N1

Tags: number theory , fe , functional equation

Find all functions $f:\mathbb N\to\mathbb N$ satisfying $$x+f^{f(x)}(y)\mid2(x+y)$$for all $x,y\in\mathbb N$.

2011 Korea Junior Math Olympiad, 3

Tags: number theory , coprime , perfect square

Let $x, y$ be positive integers such that $gcd(x, y) = 1$ and $x + 3y^2$ is a perfect square. Prove that $x^2 + 9y^4$ can't be a perfect square.

Kyiv City MO Seniors 2003+ geometry, 2015.10.5

Tags: equal segments , circles , geometry

Circles ${{w} _ {1}}$ and ${{w} _ {2}}$ with centers at points ${{O} _ {1}}$ and ${{ O} _ {2}}$ intersect at points $A$ and $B$, respectively. Around the triangle ${{O} _ {1}} {{O} _ {2}} B$ circumscribe a circle $w$ centered at the point $O$, which intersects the circles ${{w } _ {1}}$ and ${{w} _ {2}}$ for the second time at points $K$ and $L$, respectively. The line $OA$ intersects the circles ${{w} _ {1}}$ and ${{w} _ {2}}$ at the points $M$ and $N$, respectively. The lines $MK$ and $NL$ intersect at the point $P$. Prove that the point $P$ lies on the circle $w$ and $PM = PN$. (Vadym Mitrofanov)

2001 China Team Selection Test, 1

Tags: combinatorics

Given any odd integer $n>3$ that is not divisible by $3$, determine whether it is possible to fill an $n \times n$ grid with $n^2$ integers such that (each cell filled with a number, the number at the intersection of the $i$-th row and $j$-th column is denoted as $a_{ij}$): $\cdot$ Each row and each column contains a permutation of the numbers $0,1,2, \cdots, n-1$. $\cdot$ The pairs $(a_{ij},a_{ji})$ for $i<j$ are all distinct.

2010 Dutch IMO TST, 3

Tags: perfect square , number theory

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.

2003 AMC 12-AHSME, 24

Tags: function , algebra , analytic geometry , slope , graphing lines , system of equations , ceiling function

Positive integers $ a$, $ b$, and $ c$ are chosen so that $ a<b<c$, and the system of equations \[ 2x\plus{}y\equal{}2003\text{ and }y\equal{}|x\minus{}a|\plus{}|x\minus{}b|\plus{}|x\minus{}c| \]has exactly one solution. What is the minimum value of $ c$? $ \textbf{(A)}\ 668 \qquad \textbf{(B)}\ 669 \qquad \textbf{(C)}\ 1002 \qquad \textbf{(D)}\ 2003 \qquad \textbf{(E)}\ 2004$

2016 Saudi Arabia GMO TST, 2

Tags: algebra , polynomial

Let $c$ be a given real number. Find all polynomials $P$ with real coefficients such that: $(x + 1)P(x - 1) - (x - 1)P(x) = c$ for all $x \in R$

1957 AMC 12/AHSME, 16

Tags: algebra , analytic geometry , linear equation

Goldfish are sold at $ 15$ cents each. The rectangular coordinate graph showing the cost of $ 1$ to $ 12$ goldfish is: $ \textbf{(A)}\ \text{a straight line segment} \qquad \\ \textbf{(B)}\ \text{a set of horizontal parallel line segments}\qquad \\ \textbf{(C)}\ \text{a set of vertical parallel line segments}\qquad \\ \textbf{(D)}\ \text{a finite set of distinct points}\qquad \textbf{(E)}\ \text{a straight line}$

2002 Austria Beginners' Competition, 3

Tags: algebra , inequalities

Find all real numbers $x$ that satisfy the following inequality $|x^2-4x+1|>|x^2-4x+5|$