Found problems: 85335
1998 USAMO, 1
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
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
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
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
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
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
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
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
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
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
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
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
Prove that there exist two powers of $7$ whose difference is divisible by $2021$.
2003 JHMMC 8, 31
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
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#
Solve in positive integers $x^{100}-y^{100}=100!$
2019 Junior Balkan Team Selection Tests - Romania, 4
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.
1989 AMC 8, 6
If the markings on the number line are equally spaced, what is the number $\text{y}$?
[asy]
draw((-4,0)--(26,0),Arrows);
for(int a=0; a<6; ++a)
{
draw((4a,-1)--(4a,1));
}
label("0",(0,-1),S); label("20",(20,-1),S); label("y",(12,-1),S);
[/asy]
$\text{(A)}\ 3 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 16$
1990 All Soviet Union Mathematical Olympiad, 534
Given $2n$ genuine coins and $2n$ fake coins. The fake coins look the same as genuine coins but weigh less (but all fake coins have the same weight). Show how to identify each coin as genuine or fake using a balance at most $3n$ times.
2014 China Girls Math Olympiad, 1
In the figure of [url]http://www.artofproblemsolving.com/Forum/download/file.php?id=50643&mode=view[/url]
$\odot O_1$ and $\odot O_2$ intersect at two points $A$, $B$.
The extension of $O_1A$ meets $\odot O_2$ at $C$, and the extension of $O_2A$ meets $\odot O_1$ at $D$,
and through $B$ draw $BE \parallel O_2A$ intersecting $\odot O_1$ again at $E$.
If $DE \parallel O_1A$, prove that $DC \perp CO_2$.
2023 CMIMC Integration Bee, 10
\[\int_{\frac 1{\sqrt 3}}^{\sqrt 3} \frac{\arctan(x)\log^2(x)}{x}\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
1959 AMC 12/AHSME, 26
The base of an isosceles triangle is $\sqrt 2$. The medians to the leg intersect each other at right angles. The area of the triangle is:
$ \textbf{(A)}\ 1.5 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 2.5\qquad\textbf{(D)}\ 3.5\qquad\textbf{(E)}\ 4 $
2018 Math Prize for Girls Problems, 2
How many ordered pairs of integers $(x, y)$ satisfy $2 |y| \le x \le 40\,$?
2008 Bulgaria Team Selection Test, 1
For each positive integer $n$, denote by $a_{n}$ the first digit of $2^{n}$ (base ten). Is the number $0.a_{1}a_{2}a_{3}\cdots$ rational?
2023 Korea Summer Program Practice Test, P4
In a country there are infinitely many towns and for every pair of towns there is one road connecting them. Initially there are $n$ coin in each city. Every day traveller Hong starts from one town and moves on to another, but if Hong goes from town $A$ to $B$ on the $k$-th day, he has to send $k$ coins from $B$ to $A$, and he can no longer use the road connecting $A$ and $B$. Prove that Hong can't travel for more than $n+2n^\frac{2}{3}$ days.