Found problems: 85335
2015 Princeton University Math Competition, A3/B5
Consider a random permutation of the set $\{1, 2, . . . , 2015\}$. In other words, for each $1 \le i \le 2015$, $i$ is sent to the element $a_i$ where $a_i \in \{1, 2, . . . , 2015\}$ and if $i \neq j$, then $a_i \neq a_j$. What is the expected number of ordered pairs $(a_i, a_j )$ with $i - j > 155$ and $a_i - a_j > 266$?
1963 Putnam, A2
Let $f:\mathbb{N}\rightarrow \mathbb{N}$ be a strictly increasing function such that $f(2)=2$ and $f(mn)=f(m)f(n)$
for every pair of relatively prime positive integers $m$ and $n$. Prove that $f(n)=n$ for every positive integer $n$.
2016 Portugal MO, 3
Let $[ABC]$ be an equilateral triangle on the side $1$. Determine the length of the smallest segment $[DE]$, where $D$ and $E$ are on the sides of the triangle, which divides $[ABC]$ into two figures with equal area.
2005 Indonesia MO, 2
For an arbitrary positive integer $ n$, define $ p(n)$ as the product of the digits of $ n$ (in decimal). Find all positive integers $ n$ such that $ 11p(n)\equal{}n^2\minus{}2005$.
2019 Thailand TST, 1
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
2002 Baltic Way, 3
Find all sequences $0\le a_0\le a_1\le a_2\le \ldots$ of real numbers such that
\[a_{m^2+n^2}=a_m^2+a_n^2 \]
for all integers $m,n\ge 0$.
1988 IMO Shortlist, 28
The sequence $ \{a_n\}$ of integers is defined by
\[ a_1 \equal{} 2, a_2 \equal{} 7
\]
and
\[ \minus{} \frac {1}{2} < a_{n \plus{} 1} \minus{} \frac {a^2_n}{a_{n \minus{} 1}} \leq \frac {}{}, n \geq 2.
\]
Prove that $ a_n$ is odd for all $ n > 1.$
2019 Centroamerican and Caribbean Math Olympiad, 3
Let $ABC$ be a triangle and $\Gamma$ its circumcircle. Let $D$ be the foot of the altitude from $A$ to the side $BC$, $M$ and $N$ the midpoints of $AB$ and $AC$, and $Q$ the point on $\Gamma$ diametrically opposite to $A$. Let $E$ be the midpoint of $DQ$. Show that the lines perpendicular to $EM$ and $EN$ passing through $M$ and $N$, respectively, meet on $AD$.
2011 Portugal MO, 2
The point $P$, inside the triangle $[ABC]$, lies on the perpendicular bisector of $[AB]$. $Q$ and $R$ points, exterior to the triangle, they are such that $ [BPA], [BQC]$ and $[CRA]$ are similar triangles. Shows that $[PQCR]$ is a parallelogram.
[img]https://cdn.artofproblemsolving.com/attachments/f/5/6e036b127f8a013794b8246cbb1544e7280d4a.png[/img]
2017 IMO Shortlist, A5
An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have
$$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$
Find the largest constant $K = K(n)$ such that
$$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$
holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.
2017 Junior Regional Olympiad - FBH, 4
Let $n$ and $k$ be positive integers for which we have $4$ statements:
$i)$ $n+1$ is divisible with $k$
$ii)$ $n=2k+5$
$iii)$ $n+k$ is divisible with $3$
$iv)$ $n+7k$ is prime
Determine all possible values for $n$ and $k$, if out of the $4$ statements, three of them are true and one is false
2014 AIME Problems, 14
In $\triangle ABC$, $AB=10$, $\angle A=30^\circ$, and $\angle C=45^\circ$. Let $H,D$, and $M$ be points on line $\overline{BC}$ such that $\overline{AH}\perp\overline{BC}$, $\angle BAD=\angle CAD$, and $BM=CM$. Point $N$ is the midpoint of segment $\overline{HM}$, and point $P$ is on ray $AD$ such that $\overline{PN}\perp\overline{BC}$. Then $AP^2=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2020 USMCA, 23
The sequences $a_1,a_2,\ldots$ and $b_1,b_2,\ldots$ are defined by $a_1=\frac{5}{2}\sqrt[3]{2}$, $b_1=2\sqrt[3]{4}$, and for $n\ge 1$, $a_{n+1} = a_n^2 - 2b_n$, $b_{n+1} = b_n^2 - 2a_n$. There exist real numbers $u,v$ such that
\[\lim_{n\rightarrow\infty} \frac{a_n}{ub_n^v} = 1.\]
Determine the pair $(u,v)$.
2020 Kosovo National Mathematical Olympiad, 1
Two players, Agon and Besa, choose a number from the set $\{1,2,3,4,5,6,7,8\}$, in turns, until no number is left. Then, each player sums all the numbers that he has chosen. We say that a player wins if the sum of his chosen numbers is a prime and the sum of the numbers that his opponent has chosen is composite. In the contrary, the game ends in a draw. Agon starts first. Does there exist a winning strategy for any of the players?
Novosibirsk Oral Geo Oly VII, 2022.3
Three angle bisectors were drawn in a triangle, and it turned out that the angles between them are $50^o$, $60^o$ and $70^o$. Find the angles of the original triangle.
2010 Greece Junior Math Olympiad, 3
If $a, b$ are positive real numbers with sum $3$ and the positive real numbers $x, y, z$ have product $1$, prove that: $(ax + b)(ay + b)(az + b) \ge 27$. When equality holds?
2008 ITAMO, 1
Let $ ABCDEFGHILMN$ be a regular dodecagon, let $ P$ be the intersection point of the diagonals $ AF$ and $ DH$. Let $ S$ be the circle which passes through $ A$ and $ H$, and which has the same radius of the circumcircle of the dodecagon, but is different from the circumcircle of the dodecagon. Prove that:
1. $ P$ lies on $ S$
2. the center of $ S$ lies on the diagonal $ HN$
3. the length of $ PE$ equals the length of the side of the dodecagon
1979 Putnam, A2
Establish necessary and sufficient conditions on the constant $k$ for the existence of a continuous real valued function $f(x)$ satisfying $$f(f(x))=kx^9$$ for all real $x$.
1973 IMO Shortlist, 15
Prove that for all $n \in \mathbb N$ the following is true:
\[2^n \prod_{k=1}^n \sin \frac{k \pi}{2n+1} = \sqrt{2n+1}\]
2018 Thailand TST, 2
A positive integer $n < 2017$ is given. Exactly $n$ vertices of a regular 2017-gon are colored red, and the remaining vertices are colored blue. Prove that the number of isosceles triangles whose vertices are monochromatic does not depend on the chosen coloring (but does depend on $n$.)
1985 Federal Competition For Advanced Students, P2, 6
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying: $ x^2 f(x)\plus{}f(1\minus{}x)\equal{}2x\minus{}x^4$ for all $ x \in \mathbb{R}$.
PEN Q Problems, 1
Suppose $p(x) \in \mathbb{Z}[x]$ and $P(a)P(b)=-(a-b)^2$ for some distinct $a, b \in \mathbb{Z}$. Prove that $P(a)+P(b)=0$.
2016 JBMO Shortlist, 6
Given an acute triangle ${ABC}$, erect triangles ${ABD}$ and ${ACE}$ externally, so that ${\angle ADB= \angle AEC=90^o}$ and ${\angle BAD= \angle CAE}$. Let ${{A}_{1}}\in BC,{{B}_{1}}\in AC$ and ${{C}_{1}}\in AB$ be the feet of the altitudes of the triangle ${ABC}$, and let $K$ and ${K,L}$ be the midpoints of $[ B{{C}_{1}} ]$ and ${BC_1, CB_1}$, respectively. Prove that the circumcenters of the triangles $AKL,{{A}_{1}}{{B}_{1}}{{C}_{1}}$ and ${DEA_1}$ are collinear.
(Bulgaria)
2019 IMO Shortlist, N8
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
1998 Tournament Of Towns, 3
(a) The numbers $1 , 2, 4, 8, 1 6 , 32, 64, 1 28$ are written on a blackboard.
We are allowed to erase any two numbers and write their difference instead (this is always a non-negative number). After this procedure has been repeated seven times, only a single number will remain. Could this number be $97$?
(b) The numbers $1 , 2, 22, 23 , . . . , 210$ are written on a blackboard.
We are allowed to erase any two numbers and write their difference instead (this is always a non-negative number) . After this procedure has been repeated ten times, only a single number will remain. What values could this number have?
(A.Shapovalov)