Found problems: 85335
1991 Federal Competition For Advanced Students, P2, 3
$ (a)$ Prove that $ 91$ divides $ n^{37}\minus{}n$ for all integers $ n$.
$ (b)$ Find the largest $ k$ that divides $ n^{37}\minus{}n$ for all integers $ n$.
1989 IMO Shortlist, 29
155 birds $ P_1, \ldots, P_{155}$ are sitting down on the boundary of a circle $ C.$ Two birds $ P_i, P_j$ are mutually visible if the angle at centre $ m(\cdot)$ of their positions $ m(P_iP_j) \leq 10^{\circ}.$ Find the smallest number of mutually visible pairs of birds, i.e. minimal set of pairs $ \{x,y\}$ of mutually visible pairs of birds with $ x,y \in \{P_1, \ldots, P_{155}\}.$ One assumes that a position (point) on $ C$ can be occupied simultaneously by several birds, e.g. all possible birds.
2016 CCA Math Bonanza, L2.4
What is the largest integer that must divide $n^5-5n^3+4n$ for all integers $n$?
[i]2016 CCA Math Bonanza Lightning #2.4[/i]
2022 Romania National Olympiad, P4
Let $a<b<c<d$ be positive integers which satisfy $ad=bc.$ Prove that $2a+\sqrt{a}+\sqrt{d}<b+c+1.$
[i]Marius Mînea[/i]
2015 BAMO, 3
Let $k$ be a positive integer. Prove that there exist integers $x$ and $y$, neither of which is divisible by $3$, such that
$x^2+2y^2 = 3^k$.
2020 Saint Petersburg Mathematical Olympiad, 2.
For the triple $(a,b,c)$ of positive integers we say it is interesting if $c^2+1\mid (a^2+1)(b^2+1)$ but none of the $a^2+1, b^2+1$ are divisible by $c^2+1$.
Let $(a,b,c)$ be an interesting triple, prove that there are positive integers $u,v$ such that $(u,v,c)$ is interesting and $uv<c^3$.
2024 Iran Team Selection Test, 10
Let $\{a_n\}$ be a sequence of natural numbers such that each prime number greater than $1402$ divides a member of that. Prove that the set of prime divisors of members of sequence $\{b_n\}$ which $b_n=a_1a_2...a_n-1$ , is infinite.
[i]Proposed by Navid Safaei[/i]
2004 Baltic Way, 13
The $25$ member states of the European Union set up a committee with the following rules:
1) the committee should meet daily;
2) at each meeting, at least one member should be represented;
3) at any two different meetings, a different set of member states should be represented;
4) at $n^{th}$ meeting, for every $k<n$, the set of states represented should include at least one state that was represented at the $k^{th}$ meeting.
For how many days can the committee have its meetings?
2003 China Team Selection Test, 2
Can we find positive reals $a_1, a_2, \dots, a_{2002}$ such that for any positive integer $k$, with $1 \leq k \leq 2002$, every complex root $z$ of the following polynomial $f(x)$ satisfies the condition $|\text{Im } z| \leq |\text{Re } z|$,
\[f(x)=a_{k+2001}x^{2001}+a_{k+2000}x^{2000}+ \cdots + a_{k+1}x+a_k,\] where $a_{2002+i}=a_i$, for $i=1,2, \dots, 2001$.
2017 Taiwan TST Round 3, 2
Prove that there exists a polynomial with integer coefficients satisfying the following conditions:
(a)$f(x)=0$ has no rational root.
(b) For any positive integer $n$, there always exists an integer $m$ such that $n\mid f(m)$.
2017 Morocco TST-, 4
Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP \equal{} EQ$.
2017 CMI B.Sc. Entrance Exam, 2
Let $L$ be the line of intersection of the planes $~x+y=0~$ and $~y+z=0$.
[b](a)[/b] Write the vector equation of $L$, i.e. find $(a,b,c)$ and $(p,q,r)$ such that $$L=\{(a,b,c)+\lambda(p,q,r)~~\vert~\lambda\in\mathbb{R}\}$$
[b](b)[/b] Find the equation of a plane obtained by $x+y=0$ about $L$ by $45^{\circ}$.
2018 PUMaC Algebra B, 2
For what value of $n$ is $\frac{1}{2\cdot5}+\frac{1}{5\cdot8}+\frac{1}{8\cdot 11}+\frac{1}{n(n+3)}=\frac{25}{154}$?
2018 Azerbaijan JBMO TST, 3
Determine the integers $x$ such that $2^x + x^2 + 25$ is the cube of a prime number
MBMT Guts Rounds, 2015.5
In the diagram below, the larger square has side length $6$. Find the area of the smaller square.
MOAA Team Rounds, 2023.2
Let $ABCD$ be a square with side length $6$. Let $E$ be a point on the perimeter of $ABCD$ such that the area of $\triangle{AEB}$ is $\frac{1}{6}$ the area of $ABCD$. Find the maximum possible value of $CE^2$.
[i]Proposed by Anthony Yang[/i]
2002 All-Russian Olympiad, 4
Prove that there exist infinitely many natural numbers $ n$ such that the numerator of $ 1 \plus{} \frac {1}{2} \plus{} \frac {1}{3} \plus{} \frac {1}{4} \plus{} ... \plus{} \frac {1}{n}$ in the lowest terms is not a power of a prime number.
2020 BMT Fall, 22
Three lights are placed horizontally on a line on the ceiling. All the lights are initially off. Every second, Neil picks one of the three lights uniformly at random to switch: if it is off, he switches it on; if it is on, he switches it off. When a light is switched, any lights directly to the left or right of that light also get turned on (if they were off) or off (if they were on). The expected number of lights that are on after Neil has flipped switches three times can be expressed in the form $m/
n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
2021 JHMT HS, 1
In the diagram below, a triangular array of three congruent squares is configured such that the top row has one square and the bottom row has two squares. The top square lies on the two squares immediately below it. Suppose that the area of the triangle whose vertices are the centers of the three squares is $100.$ Find the area of one of the squares.
[asy]
unitsize(1.25cm);
draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0));
draw((1,0)--(2,0)--(2,1)--(1,1));
draw((1.5,1)--(1.5,2)--(0.5,2)--(0.5,1));
draw((0.5,0.5)--(1.5,0.5)--(1,1.5)--(0.5,0.5),dashed);
[/asy]
2020 Portugal MO, 3
Given a subset of $\{1,2,...,n\}$, we define its [i]alternating sum [/i] in the following way: we order the elements of the subset in descending order and, starting with the largest, we alternately add and subtract the successive numbers. For example, the [i]alternating sum[/i] of the set $\{1,3,4,6,8\}$ is $8-6+4-3+1 = 4$. Determines the sum of the alternating sums of all subsets of $\{1,2,...,10\}$ with an odd number of elements.
2010 AMC 10, 14
The average of the numbers $ 1,2,3,...,98,99$, and $ x$ is $ 100x$. What is $ x$?
$ \textbf{(A)}\ \frac{49}{101} \qquad\textbf{(B)}\ \frac{50}{101} \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ \frac{51}{101} \qquad\textbf{(E)}\ \frac{50}{99}$
2011 Kazakhstan National Olympiad, 5
On the table lay a pencil, sharpened at one end. The student can rotate the pencil around one of its ends at $45^{\circ}$ clockwise or counterclockwise. Can the student, after a few turns of the pencil, go back to the starting position so that the sharpened end and the not sharpened are reversed?
2007 Junior Balkan Team Selection Tests - Romania, 4
We call a set of points [i]free[/i] if there is no equilateral triangle with the vertices among the points of the set. Prove that every set of $n$ points in the plane contains a [i]free[/i] subset with at least $\sqrt{n}$ elements.
2022 Kyiv City MO Round 2, Problem 4
Let $ABCD$ be the cyclic quadrilateral. Suppose that there exists some line $l$ parallel to $BD$ which is tangent to the inscribed circles of triangles $ABC, CDA$. Show that $l$ passes through the incenter of $BCD$ or through the incenter of $DAB$.
[i](Proposed by Fedir Yudin)[/i]
2011 Kyrgyzstan National Olympiad, 6
[b]a)[/b] Among the $21$ pairwise distances between the $7$ points of the plane, prove that one and the same number occurs not more than $12$ times.
[b]b)[/b] Find a maximum number of times may meet the same number among the $15$ pairwise distances between $6$ points of the plane.