Olympiad problems

2005 China Girls Math Olympiad, 6

An integer $ n$ is called good if there are $ n \geq 3$ lattice points $ P_1, P_2, \ldots, P_n$ in the coordinate plane satisfying the following conditions: If line segment $ P_iP_j$ has a rational length, then there is $ P_k$ such that both line segments $ P_iP_k$ and $ P_jP_k$ have irrational lengths; and if line segment $ P_iP_j$ has an irrational length, then there is $ P_k$ such that both line segments $ P_iP_k$ and $ P_jP_k$ have rational lengths. (1) Determine the minimum good number. (2) Determine if 2005 is a good number. (A point in the coordinate plane is a lattice point if both of its coordinate are integers.)

1987 IMO Longlists, 72

Tags: geometry , rectangle , combinatorics

Is it possible to cover a rectangle of dimensions $m \times n$ with bricks that have the trimino angular shape (an arrangement of three unit squares forming the letter $\text L$) if: [b](a)[/b] $m \times n = 1985 \times 1987;$ [b](b)[/b] $m \times n = 1987 \times 1989 \quad ?$

2003 National Olympiad First Round, 6

Tags: floor function

How many $0$s are there at the end of the decimal representation of $2000!$? $ \textbf{(A)}\ 222 \qquad\textbf{(B)}\ 499 \qquad\textbf{(C)}\ 625 \qquad\textbf{(D)}\ 999 \qquad\textbf{(E)}\ \text{None of the preceding} $

2021 Saint Petersburg Mathematical Olympiad, 6

Tags: graph theory , combinatorics

A school has $450$ students. Each student has at least $100$ friends among the others and among any $200$ students, there are always two that are friends. Prove that $302$ students can be sent on a kayak trip such that each of the $151$ two seater kayaks contain people who are friends. [i]D. Karpov[/i]

2009 IMC, 4

Tags: modular arithmetic , polynomial , algebra

Let $p$ be a prime number and $\mathbf{W}\subseteq \mathbb{F}_p[x]$ be the smallest set satisfying the following : [list] (a) $x+1\in \mathbf{W}$ and $x^{p-2}+x^{p-3}+\cdots +x^2+2x+1\in \mathbf{W}$ (b) For $\gamma_1,\gamma_2$ in $\mathbf{W}$, we also have $\gamma(x)\in \mathbf{W}$, where $\gamma(x)$ is the remainder $(\gamma_1\circ \gamma_2)(x)\pmod {x^p-x}$.[/list] How many polynomials are in $\mathbf{W}?$

2023 Denmark MO - Mohr Contest, 4

Tags: geometry , equal segments , regular

In the $9$-gon $ABCDEFGHI$, all sides have equal lengths and all angles are equal. Prove that $|AB| + |AC| = |AE|$. [img]https://cdn.artofproblemsolving.com/attachments/6/2/8c82e8a87bf8a557baaf6ac72b3d18d2ba3965.png[/img]

2021 BMT, 3

Tags: number theory

How many distinct sums can be made from adding together exactly 8 numbers that are chosen from the set $\{ 1,4,7,10 \}$, where each number in the set is chosen at least once? (For example, one possible sum is $1+1+1+4+7+7+10+10=41$.)

1999 North Macedonia National Olympiad, 2

Tags: combinatorics

We are given $13$ apparently equal balls, all but one having the same weight (the remaining one has a different weight). Is it posible to determine the ball with the different weight in $3$ weighings?

1998 USAMO, 1

Tags: modular arithmetic , number theory

Suppose that the set $\{1,2,\cdots, 1998\}$ has been partitioned into disjoint pairs $\{a_i,b_i\}$ ($1\leq i\leq 999$) so that for all $i$, $|a_i-b_i|$ equals $1$ or $6$. Prove that the sum \[ |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}| \] ends in the digit $9$.

2011 Balkan MO Shortlist, N1

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Given an odd number $n >1$, let \begin{align*} S =\{ k \mid 1 \le k < n , \gcd(k,n) =1 \} \end{align*} and let \begin{align*} T = \{ k \mid k \in S , \gcd(k+1,n) =1 \} \end{align*} For each $k \in S$, let $r_k$ be the remainder left by $\frac{k^{|S|}-1}{n}$ upon division by $n$. Prove \begin{align*} \prod _{k \in T} \left( r_k - r_{n-k} \right) \equiv |S| ^{|T|} \pmod{n} \end{align*}

1993 Baltic Way, 12

Tags: combinatorics

There are $13$ cities in a certain kingdom. Between some pairs of the cities a two-way direct bus, train or plane connections are established. What is the least possible number of connections to be established so that choosing any two means of transportation one can go from any city to any other without using the third kind of vehicle?

2014 ASDAN Math Tournament, 24

Tags:

It's pouring down rain, and the amount of rain hitting point $(x,y)$ is given by $$f(x,y)=|x^3+2x^2y-5xy^2-6y^3|.$$ If you start at the origin $(0,0)$, find all the possibilities for $m$ such that $y=mx$ is a straight line along which you could walk without any rain falling on you.

2016 Ukraine Team Selection Test, 5

Tags: combinatorics , combinatorial geometry

Let $ABC$ be an equilateral triangle of side $1$. There are three grasshoppers sitting in $A$, $B$, $C$. At any point of time for any two grasshoppers separated by a distance $d$ one of them can jump over other one so that distance between them becomes $2kd$, $k,d$ are nonfixed positive integers. Let $M$, $N$ be points on rays $AB$, $AC$ such that $AM=AN=l$, $l$ is fixed positive integer. In a finite number of jumps all of grasshoppers end up sitting inside the triangle $AMN$. Find, in terms of $l$, the number of final positions of the grasshoppers. (Grasshoppers can leave the triangle $AMN$ during their jumps.)

2011 Brazil National Olympiad, 6

Tags: algorithm , modular arithmetic , inequalities

Let $a_{1}, a_{2}, a_{3}, ... a_{2011}$ be nonnegative reals with sum $\frac{2011}{2}$, prove : $|\prod_{cyc} (a_{n} - a_{n+1})| = |(a_{1} - a_{2})(a_{2} - a_{3})...(a_{2011}-a_{1})| \le \frac{3 \sqrt3}{16}.$

2024 AIME, 13

Tags:

Let $p$ be the least prime number for which there exists a positive integer $n$ such that $n^{4}+1$ is divisible by $p^{2}$. Find the least positive integer $m$ such that $m^{4}+1$ is divisible by $p^{2}$.

2000 Saint Petersburg Mathematical Olympiad, 9.3

Tags: polynomial , integer polynomial , number theory , existence

Let $P(x)=x^{2000}-x^{1000}+1$. Do there exist distinct positive integers $a_1,\dots,a_{2001}$ such that $a_ia_j|P(a_i)P(a_j)$ for all $i\neq j$? [I]Proposed by A. Baranov[/i]

2014 Bundeswettbewerb Mathematik, 3

Tags: function , combinatorics

A line $g$ is given in a plane. $n$ distinct points are chosen arbitrarily from $g$ and are named as $A_1, A_2, \ldots, A_n$. For each pair of points $A_i,A_j$, a semicircle is drawn with $A_i$ and $A_j$ as its endpoints. All semicircles lie on the same side of $g$. Determine the maximum number of points (which are not lying in $g$) of intersection of semicircles as a function of $n$.

2005 MOP Homework, 6

Tags: number theory

Let $p$ be a prime number, and let $0 \le a_1<a_2<...<a_m<p$ and $0 \le b_1<b_2<...<b_n<p$ be arbitrary integers. Denote by $k$ the number of different remainders of $a_i+b_j$, $1 \le i \le m$ and $1 \le j \le n$, modulo $p$. Prove that (i) if $m+n>p$, then $k=p$ (ii) if $m+n \le p$, then $k \ge m+n-1$

2005 AMC 12/AHSME, 17

Tags: number theory , greatest common divisor , logarithm

How many distinct four-tuples $ (a,b,c,d)$ of rational numbers are there with $ a \log_{10} 2 \plus{} b \log_{10} 3 \plus{} c \log_{10} 5 \plus{} d \log_{10} 7 \equal{} 2005$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 2004\qquad \textbf{(E)}\ \text{infinitely many}$

1999 National High School Mathematics League, 6

Tags: parabola , analytic geometry , geometry , conic

Points $A(1,2)$, a line that passes $(5,-2)$ intersects the parabola $y^2=4x$ at two points $B,C$. Then, $\triangle ABC$ is $\text{(A)}$ an acute triangle $\text{(B)}$ an obtuse triangle $\text{(C)}$ a right triangle $\text{(D)}$ not sure

2021 Auckland Mathematical Olympiad, 4

Tags: power , number theory , divide , divisible

Prove that there exist two powers of $7$ whose difference is divisible by $2021$.

2003 JHMMC 8, 31

Tags:

The ages of Mr. and Mrs. Fibonacci are both two-digit numbers. If Mr. Fibonacci’s age can be formed by reversing the digits of Mrs. Fibonacci’s age, find the smallest possible positive difference between their ages.

2002 Austrian-Polish Competition, 8

Tags: trigonometry , limit , inequalities , algebra

Determine the number of real solutions of the system \[\left\{ \begin{aligned}\cos x_{1}&= x_{2}\\ &\cdots \\ \cos x_{n-1}&= x_{n}\\ \cos x_{n}&= x_{1}\\ \end{aligned}\right.\]

2020 Junior Balkan Team Selection Tests-Serbia, 2#

Tags: number theory

Solve in positive integers $x^{100}-y^{100}=100!$

2019 Junior Balkan Team Selection Tests - Romania, 4

Tags: game strategy , game , winning strategy , combinatorics

Ana and Bogdan play the following turn based game: Ana starts with a pile of $n$ ($n \ge 3$) stones. At his turn each player has to split one pile. The winner is the player who can make at his turn all the piles to have at most two stones. Depending on $n$, determine which player has a winning strategy.