Found problems: 85335
2021 Science ON Seniors, 2
Find all pairs $(p,q)$ of prime numbers such that
$$p^q-4~|~q^p-1.$$
[i](Vlad Robu)[/i]
2010 Baltic Way, 19
For which $k$ do there exist $k$ pairwise distinct primes $p_1,p_2,\ldots ,p_k$ such that
\[p_1^2+p_2^2+\ldots +p_k^2=2010? \]
2019 MIG, 12
Calculate the product $\tfrac13 \times \tfrac24 \times \tfrac35 \times \cdots \times \tfrac{18}{20} \times \tfrac{19}{21}$.
$\textbf{(A) }\dfrac{1}{210}\qquad\textbf{(B) }\dfrac{1}{190}\qquad\textbf{(C) }\dfrac{1}{21}\qquad\textbf{(D) }\dfrac{1}{20}\qquad\textbf{(E) }\dfrac{1}{10}$
1953 Moscow Mathematical Olympiad, 245
A quadrilateral is circumscribed around a circle. Its diagonals intersect at the center of the circle. Prove that the quadrilateral is a rhombus.
2014 ASDAN Math Tournament, 3
Compute the perimeter of the triangle that has area $3-\sqrt{3}$ and angles $45^\circ$, $60^\circ$, and $75^\circ$.
2003 China Team Selection Test, 3
Let $x_0+\sqrt{2003}y_0$ be the minimum positive integer root of Pell function $x^2-2003y^2=1$. Find all the positive integer solutions $(x,y)$ of the equation, such that $x_0$ is divisible by any prime factor of $x$.
1949-56 Chisinau City MO, 8
Prove that the remainder of dividing the sum of two squares of integers by $4$ is different from $3$.
IV Soros Olympiad 1997 - 98 (Russia), 9.9
Find an odd natural number not exceeding $1000$ if you know that the sum of the last digits of all its divisors (including $1$ and the number itself) is $33$.
2005 ISI B.Math Entrance Exam, 7
Let $M$ be a point in the triangle $ABC$ such that
\[\text{area}(ABM)=2 \cdot \text{area}(ACM)\]
Show that the locus of all such points is a straight line.
1996 Estonia National Olympiad, 2
For which positive $x$ does the expression $x^{1000}+x^{900}+x^{90}+x^6+\frac{1996}{x}$ attain the smallest value?
1958 Polish MO Finals, 1
Prove that the product of three consecutive natural numbers, the middle of which is the cube of a natural number, is divisible by $ 504 $ .
2000 Belarus Team Selection Test, 5.1
Let $AM$ and $AL$ be the median and bisector of a triangle $ABC$ ($M,L \in BC$).
If $BC = a, AM = m_a, AL = l_a$, prove the inequalities:
(a) $a\tan \frac{a}{2} \le 2m_a \le a \cot \frac{a}{2} $ if $a < \frac{\pi}{2}$ and $a\tan \frac{a}{2} \ge 2m_a \ge a \cot \frac{a}{2} $ if $a > \frac{\pi}{2}$
(b) $2l_a \le a\cot \frac{a}{2} $.
2018 Israel Olympic Revenge, 3
Let $ABC$ be a triangle with circumcircle $\omega$ and circumcenter $O$. The tangent line to from $A$ to $\omega$ intersects $BC$ at $K$. The tangent line to from $B$ to $\omega$ intersects $AC$ at $L$. Let $M,N$ be the midpoints of $AK,BL$ respectively. The line $MN$ is named by $\alpha$. The feet of perpendicular from $A,B,C$ to the edges of $\triangle ABC$ are named by $D,E,F$ respectively. The perpendicular bisectors of $EF,DF,DE$ intersect $\alpha$ at $X,Y,Z$ respectively. Let $AD,BE,CF$ intersect $\omega$ again at $D',E',F'$ respectively. If $H$ is the orthocenter of $ABC$, prove that the lines $XD',YE',ZF',OH$ are concurrent.
2017 NMTC Junior, 2
If $x,y,z,p,q,r$ are real numbers such that \[\frac{1}{x+p}+\frac{1}{y+p}+\frac{1}{z+p}=\frac{1}{p}\]\[\frac{1}{x+q}+\frac{1}{y+q}+\frac{1}{z+q}=\frac{1}{q}\]\[\frac{1}{x+r}+\frac{1}{y+r}+\frac{1}{z+r}=\frac{1}{r}.\]Find the numerical value of $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}$.
2011 Sharygin Geometry Olympiad, 6
In triangle $ABC$ $AA_0$ and $BB_0$ are medians, $AA_1$ and $BB_1$ are altitudes. The circumcircles of triangles $CA_0B_0$ and $CA_1B_1$ meet again in point $M_c$. Points $M_a, M_b$ are defined similarly. Prove that points $M_a, M_b, M_c$ are collinear and lines $AM_a, BM_b, CM_c$ are parallel.
2021 IMO Shortlist, N8
Find all positive integers $n$ for which there exists a polynomial $P(x) \in \mathbb{Z}[x]$ such that for every positive integer $m\geq 1$, the numbers $P^m(1), \ldots, P^m(n)$ leave exactly $\lceil n/2^m\rceil$ distinct remainders when divided by $n$. (Here, $P^m$ means $P$ applied $m$ times.)
[i]Proposed by Carl Schildkraut, USA[/i]
1991 French Mathematical Olympiad, Problem 2
For each $n\in\mathbb N$, the function $f_n$ is defined on real numbers $x\ge n$ by
$$f_n(x)=\sqrt{x-n}+\sqrt{x-n+1}+\ldots+\sqrt{x+n}-(2n+1)\sqrt x.$$(a) If $n$ is fixed, prove that $\lim_{x\to+\infty}f_n(x)=0$.
(b) Find the limit of $f_n(n)$ as $n\to+\infty$.
2021 Indonesia MO, 4
Let $x,y$ and $z$ be positive reals such that $x + y + z = 3$. Prove that
\[ 2 \sqrt{x + \sqrt{y}} + 2 \sqrt{y + \sqrt{z}} + 2 \sqrt{z + \sqrt{x}} \le \sqrt{8 + x - y} + \sqrt{8 + y - z} + \sqrt{8 + z - x} \]
2023 Junior Balkan Mathematical Olympiad, 4
Let $ABC$ be an acute triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to $BC$ and let $M$ be the midpoint of $OD$. The points $O_b$ and $O_c$ are the circumcenters of triangles $AOC$ and $AOB$, respectively. If $AO=AD$, prove that points $A$, $O_b$, $M$ and $O_c$ are concyclic.
[i]Marin Hristov and Bozhidar Dimitrov, Bulgaria[/i]
2002 Romania Team Selection Test, 1
Let $(a_n)_{n\ge 1}$ be a sequence of positive integers defined as $a_1,a_2>0$ and $a_{n+1}$ is the least prime divisor of $a_{n-1}+a_{n}$, for all $n\ge 2$.
Prove that a real number $x$ whose decimals are digits of the numbers $a_1,a_2,\ldots a_n,\ldots $ written in order, is a rational number.
[i]Laurentiu Panaitopol[/i]
2020 Caucasus Mathematical Olympiad, 7
A regular triangle $ABC$ is given. Points $K$ and $N$ lie in the segment $AB$, a point $L$ lies in the segment $AC$, and a point $M$ lies in the segment $BC$ so that $CL=AK$, $CM=BN$, $ML=KN$. Prove that $KL \parallel MN$.
2011 Hanoi Open Mathematics Competitions, 6
Find all pairs $(x, y)$ of real numbers satisfying the system :
$\begin{cases} x + y = 2 \\
x^4 - y^4 = 5x - 3y \end{cases}$
STEMS 2024 Math Cat A, P1
Let $n$ be a positive integer and $S = \{ m \mid 2^n \le m < 2^{n+1} \}$. We call a pair of non-negative integers $(a, b)$ [i]fancy[/i] if $a + b$ is in $S$ and is a palindrome in binary. Find the number of [i]fancy[/i] pairs $(a, b)$.
2021 Argentina National Olympiad Level 2, 1
You have two blackboards $A$ and $B$. You have to write on them some of the integers greater than or equal to $2$ and less than or equal to $20$ in such a way that each number on blackboard $A$ is co-prime with each number on blackboard $B.$ Determine the maximum possible value of multiplying the number of numbers written in $A$ by the number of numbers written in $B$.
1949 Moscow Mathematical Olympiad, 159
Consider a closed broken line of perimeter $1$ on a plane. Prove that a disc of radius $\frac14$ can cover this line.