Found problems: 85335
CIME I 2018, 15
A positive integer $n$ is said to be $m$-free if $n \leq m!$ and $\gcd(i,n)=1$ for each $i=1,2,...,m$. Let $\mathcal{S}_k$ denote the sum of the squares of all the $k$-free integers. Find the remainder when $\mathcal{S}_7-\mathcal{S}_6$ is divided by $1000$.
[i]Proposed by [b]FedeX333X[/b][/i]
2011 Miklós Schweitzer, 10
Let $X_0, \xi_{i, j}, \epsilon_k$ (i, j, k ∈ N) be independent, non-negative integer random variables. Suppose that $\xi_{i, j}$ (i, j ∈ N) have the same distribution, $\epsilon_k$ (k ∈ N) also have the same distribution.
$\mathbb{E}(\xi_{1,1})=1$ , $\mathbb{E}(X_0^l)<\infty$ , $\mathbb{E}(\xi_{1,1}^l)<\infty$ , $\mathbb{E}(\epsilon_1^l)<\infty$ for some $l\in\mathbb{N}$
Consider the random variable $X_n := \epsilon_n + \sum_{j=1}^{X_{n-1}} \xi_{n,j}$ (n ∈ N) , where $\sum_{j=1}^0 \xi_{n,j} :=0$
Introduce the sequence $M_n := X_n-X_{n-1}-\mathbb{E}(\epsilon_n)$ (n ∈ N)
Prove that there is a polynomial P of degree $\leq l/2$ such that $\mathbb{E}(M_n^l) = P_l(n)$ (n ∈ N).
2005 Alexandru Myller, 1
Let $ x,y,z $ be numbers distinct from $ -1 $ that verify the equation
$$ \frac{1}{1+a} +\frac{1}{1+b} +\frac{1}{1+c} =\frac{3}{2} . $$
Prove that if $ abc=1, $ then $ a $ or $ b $ or $ c $ is equal to $ 1. $
1989 All Soviet Union Mathematical Olympiad, 492
$ABC$ is a triangle. $A' , B' , C'$ are points on the segments $BC, CA, AB$ respectively. $\angle B' A' C' = \angle A$ , $\frac{AC'}{C'B} = \frac{BA' }{A' C} = \frac{CB'}{B'A}$. Show that $ABC$ and $A'B'C'$ are similar.
2022 Iran Team Selection Test, 7
Suppose that $n$ is a positive integer number. Consider a regular polygon with $2n$ sides such that one of its largest diagonals is parallel to the $x$-axis. Find the smallest integer $d$ such that there is a polynomial $P$ of degree $d$ whose graph intersects all sides of the polygon on points other than vertices.
Proposed by Mohammad Ahmadi
2015 JHMT, 1
Clyde is making a Pacman sticker to put on his laptop. A Pacman sticker is a circular sticker of radius $3$ inches with a sector of $120^o$ cut out. What is the perimeter of the Pacman sticker in inches?
2009 Today's Calculation Of Integral, 492
Find the volume formed by the revolution of the region satisfying $ 0\leq y\leq (x \minus{} p)(q \minus{} x)\ (0 < p < q)$ in the coordinate plane about the $ y$ -axis.
You are not allowed to use the formula: $ V \equal{} \boxed{\int_a^b 2\pi x|f(x)|\ dx\ (a < b)}$ here.
2016 EGMO, 1
Let $n$ be an odd positive integer, and let $x_1,x_2,\cdots ,x_n$ be non-negative real numbers. Show that \[ \min_{i=1,\ldots,n} (x_i^2+x_{i+1}^2) \leq \max_{j=1,\ldots,n} (2x_jx_{j+1}) \]where $x_{n+1}=x_1$.
2024 Pan-American Girls’ Mathematical Olympiad, 6
Let $ABC$ be a triangle, and let $a$, $b$, and $c$ be the lengths of the sides opposite vertices $A$, $B$, and $C$, respectively. Let $R$ be its circumradius and $r$ its inradius. Suppose that $b + c = 2a$ and $R = 3r$.
The excircle relative to vertex $A$ intersects the circumcircle of $ABC$ at points $P$ and $Q$. Let $U$ be the midpoint of side $BC$, and let $I$ be the incenter of $ABC$.
Prove that $U$ is the centroid of triangle $QIP$.
2011 Middle European Mathematical Olympiad, 3
In a plane the circles $\mathcal K_1$ and $\mathcal K_2$ with centers $I_1$ and $I_2$, respectively, intersect in two points $A$ and $B$. Assume that $\angle I_1AI_2$ is obtuse. The tangent to $\mathcal K_1$ in $A$ intersects $\mathcal K_2$ again in $C$ and the tangent to $\mathcal K_2$ in $A$ intersects $\mathcal K_1$ again in $D$. Let $\mathcal K_3$ be the circumcircle of the triangle $BCD$. Let $E$ be the midpoint of that arc $CD$ of $\mathcal K_3$ that contains $B$. The lines $AC$ and $AD$ intersect $\mathcal K_3$ again in $K$ and $L$, respectively. Prove that the line $AE$ is perpendicular to $KL$.
2009 Argentina Team Selection Test, 4
Find all positive integers $ n$ such that $ 20^n \minus{} 13^n \minus{} 7^n$ is divisible by $ 309$.
1981 All Soviet Union Mathematical Olympiad, 309
Three equilateral triangles $ABC, CDE, EHK$ (the vertices are mentioned counterclockwise) are lying in the plane so, that the vectors $\overrightarrow{AD}$ and $\overrightarrow{DK}$ are equal. Prove that the triangle $BHD$ is also equilateral
MBMT Guts Rounds, 2015.8
The school store is running out of supplies, but it still has five items: one pencil (costing $\$1$), one pen (costing $\$1$), one folder (costing $\$2$), one pack of paper (costing $\$3$), and one binder (costing $\$4$). If you have $\$10$, in how many ways can you spend your money? (You don't have to spend all of your money, or any of it.)
2014 Contests, 4
Let $m\ge 3$ and $n$ be positive integers such that $n>m(m-2)$. Find the largest positive integer $d$ such that $d\mid n!$ and $k\nmid d$ for all $k\in\{m,m+1,\ldots,n\}$.
2006 Romania National Olympiad, 3
We have in the plane the system of points $A_1,A_2,\ldots,A_n$ and $B_1,B_2,\ldots,B_n$, which have different centers of mass. Prove that there is a point $P$ such that \[ PA_1 + PA_2 + \ldots+ PA_n = PB_1 + PB_2 + \ldots + PB_n . \]
2021 New Zealand MO, 4
Find all triples $(x, p, n)$ of non-negative integers such that $p$ is prime and $2x(x + 5) = p^n + 3(x - 1)$.
1998 Brazil Team Selection Test, Problem 4
Let $L$ be a circle with center $O$ and tangent to sides $AB$ and $AC$ of a triangle $ABC$ in points $E$ and $F$, respectively. Let the perpendicular from $O$ to $BC$ meet $EF$ at $D$. Prove that $A,D$ and $M$ are collinear, where $M$ is the midpoint of $BC$.
2012 District Olympiad, 2
Let $a, b$ and $c$ be positive real numbers such that $$a^2+ab+ac-bc = 0.$$
a) Show that if two of the numbers $a, b$ and $c$ are equal, then at least one of the numbers $a, b$ and $c$ is irrational.
b) Show that there exist infinitely many triples $(m, n, p)$ of positive integers such that $$m^2 + mn + mp -np = 0.$$
2017 IOM, 5
Let $x $ and $y $ be positive integers such that
$[x+2,y+2]-[x+1,y+1]=[x+1,y+1]-[x,y]$.Prove that one of the two numbers $x $ and $y $ divide the other.
(Here $[a,b] $ denote the least common multiple of $a $ and $b $).
Proposed by Dusan Djukic.
2005 Flanders Junior Olympiad, 4
(a) Be M an internal point of the convex quadrilateral ABCD. Prove that $|MA|+|MB| < |AD|+|DC|+|CB|$.
(b) Be M an internal point of the triangle ABC. Note $k=\min(|MA|,|MB|,|MC|)$. Prove $k+|MA|+|MB|+|MC|<|AB|+|BC|+|CA|$.
2012 Kosovo Team Selection Test, 1
A student had $18$ papers. He seleced some of these papers, then he cut each of them in $18$ pieces.He took these pieces and selected some of them, which he again cut in $18$ pieces each.The student took this procedure untill he got tired .After a time he counted the pieces and got $2012$ pieces .Prove that the student was wrong during the counting.
2010 AMC 10, 25
Let $ a>0$, and let $ P(x)$ be a polynomial with integer coefficients such that
\[ P(1)\equal{}P(3)\equal{}P(5)\equal{}P(7)\equal{}a\text{, and}\]
\[ P(2)\equal{}P(4)\equal{}P(6)\equal{}P(8)\equal{}\minus{}a\text{.}\]
What is the smallest possible value of $ a$?
$ \textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!$
2007 Kazakhstan National Olympiad, 4
Several identical square sheets of paper are laid out on a rectangular table so that their sides are parallel to the edges of the table (sheets may overlap). Prove that you can stick a few pins in such a way that each sheet will be attached to the table exactly by one pin.
1995 IMO Shortlist, 1
Let $ A,B,C,D$ be four distinct points on a line, in that order. The circles with diameters $ AC$ and $ BD$ intersect at $ X$ and $ Y$. The line $ XY$ meets $ BC$ at $ Z$. Let $ P$ be a point on the line $ XY$ other than $ Z$. The line $ CP$ intersects the circle with diameter $ AC$ at $ C$ and $ M$, and the line $ BP$ intersects the circle with diameter $ BD$ at $ B$ and $ N$. Prove that the lines $ AM,DN,XY$ are concurrent.
2015 Ukraine Team Selection Test, 9
The set $M$ consists of $n$ points on the plane and satisfies the conditions:
$\bullet$ there are $7$ points in the set $M$, which are vertices of a convex heptagon,
$\bullet$ for arbitrary five points with $M$, which are vertices of a convex pentagon, there is a point that also belongs to $M$ and lies inside this pentagon.
Find the smallest possible value that $n$ can take .