Found problems: 85335
2018 South East Mathematical Olympiad, 4
Does there exist a set $A\subseteq\mathbb{N}^*$ such that for any positive integer $n$, $A\cap\{n,2n,3n,...,15n\}$ contains exactly one element? Please prove your conclusion.
2021 Indonesia TST, C
In a country, there are $2018$ cities, some of which are connected by roads. Each city is connected to at least three other cities. It is possible to travel from any city to any other city using one or more roads. For each pair of cities, consider the shortest route between these two cities. What is the greatest number of roads that can be on such a shortest route?
2021 Canadian Junior Mathematical Olympiad, 5
A function $f$ from the positive integers to the positive integers is called [i]Canadian[/i] if it satisfies $$\gcd\left(f(f(x)), f(x+y)\right)=\gcd(x, y)$$ for all pairs of positive integers $x$ and $y$.
Find all positive integers $m$ such that $f(m)=m$ for all Canadian functions $f$.
2005 Peru MO (ONEM), 3
Let $A,B,C,D$, be four different points on a line $\ell$, so that $AB=BC=CD$. In one of the semiplanes determined by the line $\ell$, the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points of the plane be such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the angle $\angle MBN$.
2004 VTRMC, Problem 7
Let $\{a_n\}$ be a sequence of positive real numbers such that $\lim_{n\to\infty}a_n=0$. Prove that $\sum^\infty_{n=1}\left|1-\frac{a_{n+1}}{a_n}\right|$ is divergent.
2017 Swedish Mathematical Competition, 2
Let $p$ be a prime number. Find all pairs of coprime positive integers $(m,n)$ such that $$
\frac{p+m}{p+n}=\frac{m}{n}+\frac{1}{p^2}.$$
1990 Iran MO (2nd round), 2
Let $\alpha$ be a root of the equation $x^3-5x+3=0$ and let $f(x)$ be a polynomial with rational coefficients. Prove that if $f(\alpha)$ be the root of equation above, then $f(f(\alpha))$ is a root, too.
2020 USMCA, 16
How many paths from $(0, 0)$ to $(2020, 2020)$, consisting of unit steps up and to the right, pass through at most one point with both coordinates even, other than $(0,0)$ and $(2020,2020)$?
2023 ELMO Shortlist, N2
Determine the greatest positive integer \(n\) for which there exists a sequence of distinct positive integers \(s_1\), \(s_2\), \(\ldots\), \(s_n\) satisfying \[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\]
[i]Proposed by Holden Mui[/i]
2018 EGMO, 1
Let $ABC$ be a triangle with $CA=CB$ and $\angle{ACB}=120^\circ$, and let $M$ be the midpoint of $AB$. Let $P$ be a variable point of the circumcircle of $ABC$, and let $Q$ be the point on the segment $CP$ such that $QP = 2QC$. It is given that the line through $P$ and perpendicular to $AB$ intersects the line $MQ$ at a unique point $N$.
Prove that there exists a fixed circle such that $N$ lies on this circle for all possible positions of $P$.
2017 CMIMC Individual Finals, 2
Define
\[f(h,t) =
\begin{cases}
8h & h = t \\
(h-t)^2 & h \neq t.
\end{cases}\]
Cody plays a game with a fair coin, where he begins by flipping it once. At each turn in the game, if he has flipped $h$ heads and $t$ tails and $h + t < 6$, he can choose either to stop and receive $f(h,t)$ dollars or he can flip the coin again; if $h + t = 6$ then the game ends and he receives $f(h,t)$ dollars. If Cody plays to maximize expectancy, how much money, in dollars, can he expect to win from this game?
2006 AMC 12/AHSME, 16
Regular hexagon $ ABCDEF$ has vertices $ A$ and $ C$ at $ (0,0)$ and $ (7,1)$, respectively. What is its area?
$ \textbf{(A) } 20\sqrt {3} \qquad \textbf{(B) } 22\sqrt {3} \qquad \textbf{(C) } 25\sqrt {3} \qquad \textbf{(D) } 27\sqrt {3} \qquad \textbf{(E) } 50$
2012 ELMO Shortlist, 4
Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$. Show that there exists a positive integer $n$ such that $a_n=b_n$.
[i]David Yang.[/i]
2008 Dutch Mathematical Olympiad, 4
Three circles $C_1,C_2,C_3$, with radii $1, 2, 3$ respectively, are externally tangent.
In the area enclosed by these circles, there is a circle $C_4$ which is externally tangent to all three circles.
Find the radius of $C_4$.
[asy]
unitsize(0.4 cm);
pair[] O;
real[] r;
O[1] = (-12/5,16/5);
r[1] = 1;
O[2] = (0,5);
r[2] = 2;
O[3] = (0,0);
r[3] = 3;
O[4] = (-132/115, 351/115);
r[4] = 6/23;
draw(Circle(O[1],r[1]));
draw(Circle(O[2],r[2]));
draw(Circle(O[3],r[3]));
draw(Circle(O[4],r[4]));
label("$C_1$", O[1]);
label("$C_2$", O[2]);
label("$C_3$", O[3]);
[/asy]
2006 China Team Selection Test, 2
Given three positive real numbers $ x$, $ y$, $ z$ such that $ x \plus{} y \plus{} z \equal{} 1$, prove that
$ \frac {xy}{\sqrt {xy \plus{} yz}} \plus{} \frac {yz}{\sqrt {yz \plus{} zx}} \plus{} \frac {zx}{\sqrt {zx \plus{} xy}} \le \frac {\sqrt {2}}{2}$.
Indonesia Regional MO OSP SMA - geometry, 2019.1
Given cube $ ABCD.EFGH $ with $ AB = 4 $ and $ P $ midpoint of the side $ EFGH $. If $ M $ is the midpoint of $ PH $, find the length of segment $ AM $.
2007 AIME Problems, 5
The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C=\frac{5}{9}(F-32)$. An integer Fahrenheit temperature is converted to Celsius and rounded to the nearest integer; the resulting integer Celsius temperature is converted back to Fahrenheit and rounded to the nearest integer. For how many integer Fahrenheit temperatures $T$ with $32 \leq T \leq 1000$ does the original temperature equal the final temperature?
2007 Puerto Rico Team Selection Test, 4
Just wondering: what exactly is Power of a Point?
2016 Macedonia JBMO TST, 4
Let $x$, $y$, and $z$ be positive real numbers. Prove that
$\sqrt {\frac {xy}{x^2 + y^2 + 2z^2}} + \sqrt {\frac {yz}{y^2 + z^2 + 2x^2}}+\sqrt {\frac {zx}{z^2 + x^2 + 2y^2}} \le \frac{3}{2}$.
When does equality hold?
Today's calculation of integrals, 850
Evaluate
\[\int_0^{\pi} \{(1-x\sin 2x)e^{\cos ^2 x}+(1+x\sin 2x)e^{\sin ^ 2 x}\}\ dx.\]
1973 IMO Shortlist, 8
Prove that there are exactly $\binom{k}{[k/2]}$ arrays $a_1, a_2, \ldots , a_{k+1}$ of nonnegative integers such that $a_1 = 0$ and $|a_i-a_{i+1}| = 1$ for $i = 1, 2, \ldots , k.$
2019 Taiwan TST Round 2, 1
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$
2011 Benelux, 1
An ordered pair of integers $(m,n)$ with $1<m<n$ is said to be a [i]Benelux couple[/i] if the following two conditions hold: $m$ has the same prime divisors as $n$, and $m+1$ has the same prime divisors as $n+1$.
(a) Find three Benelux couples $(m,n)$ with $m\leqslant 14$.
(b) Prove that there are infinitely many Benelux couples
India EGMO 2023 TST, 6
Let $ABC$ be an isosceles triangle with $AB = AC$. Suppose $P,Q,R$ are points on segments $AC, AB, BC$ respectively such that $AP = QB$, $\angle PBC = 90^\circ - \angle BAC$ and $RP = RQ$. Let $O_1, O_2$ be the circumcenters of $\triangle APQ$ and $\triangle CRP$. Prove that $BR = O_1O_2$.
[i]Proposed by Atul Shatavart Nadig[/i]
2018 India PRMO, 5
Let $ABCD$ be a trapezium in which $AB //CD$ and $AD \perp AB$. Suppose $ABCD$ has an incircle which touches $AB$ at $Q$ and $CD$ at $P$. Given that $PC = 36$ and $QB = 49$, find $PQ$.