Found problems: 85335
2020 Miklós Schweitzer, 11
Given a real number $p>1$, a continuous function $h\colon [0,\infty)\to [0,\infty)$, and a smooth vector field $Y\colon \mathbb{R}^n \to \mathbb{R}^n$ with $\mathrm{div}~Y=0$, prove the following inequality
\[\int_{\mathbb{R}^n}h(|x|)|x|^{p}\leq \int_{\mathbb{R}^{n}}h(|x|)|x+Y(x)|^{p}.\]
I Soros Olympiad 1994-95 (Rus + Ukr), 9.4
Use a compass and a ruler to construct a triangle, given the intersection point of its median, the orthocenter, and one from the vertices.
2018 International Olympic Revenge, 4
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{R}$ such that
\[
f(x)^2-f(y)^2=f(x+y)\cdot f(x-y),
\]
for all $x,y\in \mathbb{Q}$.
[i]Proposed by Portugal.[/i]
MOAA Accuracy Rounds, 2021.7
Jeffrey rolls fair three six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
2017 QEDMO 15th, 10
Let $\ell$ be a straight line and $P \notin \ell$ be a point in the plane. On $\ell$ are, in this arrangement, points $A_1, A_2,...$ such that the radii of the incircles of all triangles $P A_iA_{i + 1}$ are equal. Let $k \in N$. Show that the radius of the incircle of the triangle $P A_j A_{j + k}$ does not depend on the choice of $j \in N$ .
2013 Online Math Open Problems, 24
For a permutation $\pi$ of the integers from 1 to 10, define
\[ S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)), \]
where $\pi (i)$ denotes the $i$th element of the permutation. Suppose that $M$ is the maximum possible value of $S(\pi)$ over all permutations $\pi$ of the integers from 1 to 10. Determine the number of permutations $\pi$ for which $S(\pi) = M$.
[i]Ray Li[/i]
2011 Bogdan Stan, 3
Find all Riemann integrable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ which have the property that, for all nonconstant and continuous functions $ g:\mathbb{R}\longrightarrow\mathbb{R}, $ and all real numbers $ a,b $ such that $ a<b, $ the following equality holds.
$$ \int_a^b \left( f\circ g \right) (x)dx=\int_a^b \left( g\circ f \right) (x)dx $$
[i]Cosmin Nițu[/i]
2023 CMIMC Algebra/NT, 7
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Compute $\displaystyle \sum_{i=1}^{\phi(2023)} \dfrac{\gcd(i,\phi(2023))}{\phi(2023)}$.
[i]Proposed by Giacomo Rizzo[/i]
2019 District Olympiad, 1
Find the functions $f: \mathbb{R} \to (0, \infty)$ which satisfy $$2^{-x-y} \le \frac{f(x)f(y)}{(x^2+1)(y^2+1)} \le \frac{f(x+y)}{(x+y)^2+1},$$ for all $x,y \in \mathbb{R}.$
2005 Greece Junior Math Olympiad, 1
We are given a trapezoid $ABCD$ with $AB \parallel CD$, $CD=2AB$ and $DB \perp BC$. Let $E$ be the intersection of lines $DA$ and $CB$, and $M$ be the midpoint of $DC$.
(a) Prove that $ABMD$ is a rhombus.
(b) Prove that triangle $CDE$ is isosceles.
(c) If $AM$ and $BD$ meet at $O$, and $OE$ and $AB$ meet at $N,$ prove that the line $DN$ bisects segment $EB$.
2007 Baltic Way, 19
Let $r$ and $k$ be positive integers such that all prime divisors of $r$ are greater than $50$.
A positive integer, whose decimal representation (without leading zeroes) has at least $k$ digits, will be called [i]nice[/i] if every sequence of $k$ consecutive digits of this decimal representation forms a number (possibly with leading zeroes) which is a multiple of $r$.
Prove that if there exist infinitely many nice numbers, then the number $10^k-1$ is nice as well.
2019 India PRMO, 22
In parallelogram $ABCD$, $AC=10$ and $BD=28$. The points $K$ and $L$ in the plane of $ABCD$ move in such a way that $AK=BD$ and $BL=AC$. Let $M$ and $N$ be the midpoints of $CK$ and $DL$, respectively. What is the maximum walue of $\cot^2 (\tfrac{\angle BMD}{2})+\tan^2(\tfrac{\angle ANC}{2})$ ?
2013 Princeton University Math Competition, 11
If two points are selected at random on a fixed circle and the chord between the two points is drawn, what is the probability that its length exceeds the radius of the circle?
1997 Denmark MO - Mohr Contest, 1
Let $n =123456789101112 ... 998999$ be the natural number where is obtained by writing the natural numbers from $1$ to $999$ one after the other. What is the $1997$-th digit number in $n$?
2018 ELMO Shortlist, 4
Elmo calls a monic polynomial with real coefficients [i]tasty[/i] if all of its coefficients are in the range $[-1,1]$. A monic polynomial $P$ with real coefficients and complex roots $\chi_1,\cdots,\chi_m$ (counted with multiplicity) is given to Elmo, and he discovers that there does not exist a monic polynomial $Q$ with real coefficients such that $PQ$ is tasty. Find all possible values of $\max\left(|\chi_1|,\cdots,|\chi_m|\right)$.
[i]Proposed by Carl Schildkraut[/i]
1995 AMC 12/AHSME, 6
The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked $x$?
[asy]
defaultpen(linewidth(0.7));
path p=origin--(0,1)--(1,1)--(1,2)--(2,2)--(2,3);
draw(p^^(2,3)--(4,3)^^shift(2,0)*p^^(2,0)--origin);
draw(shift(1,0)*p, dashed);
label("$x$", (0.3,0.5), E);
label("$A$", (1.3,0.5), E);
label("$B$", (1.3,1.5), E);
label("$C$", (2.3,1.5), E);
label("$D$", (2.3,2.5), E);
label("$E$", (3.3,2.5), E);[/asy]
$
\mathbf{(A)}\; A\qquad
\mathbf{(B)}\; B\qquad
\mathbf{(C)}\; C\qquad
\mathbf{(D)}\; D\qquad
\mathbf{(E)}\; E$
2021 Serbia National Math Olympiad, 3
In a triangle $ABC$, let $AB$ be the shortest side. Points $X$ and $Y$ are given on the circumcircle of $\triangle ABC$ such that $CX=AX+BX$ and $CY=AY+BY$. Prove that $\measuredangle XCY<60^{o}$.
2011 China Second Round Olympiad, 2
Find the range of the function $f(x)=\frac{\sqrt{x^2+1}}{x-1}$.
1977 Putnam, B4
Let $C$ be a continuous closed curve in the plane which does not cross itself and let $Q$ be a point inside $C$. Show that there exists points $P_1$ and $P_2$ on $C$ such that $Q$ is the midpoint of the line segment $P_1P_2.$
2005 Today's Calculation Of Integral, 35
Determine the value of $a,b$ for which $\int_0^1 (\sqrt{1-x}-ax-b)^2 dx$ is minimized.
2024 AMC 8 -, 5
Aaliyah rolls two standard 6-sided dice. She notices that the product of the two numbers rolled is a multiple of 6. Which of the following integers [i]cannot[/i] be the sum of the two numbers?
$\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9$
2007 Nicolae Coculescu, 4
Let be three nonnegative integers $ m,n,p $ and three real numbers $ x,y,z $ such that $ 2^mx+2^ny+2^pz\ge 0. $ Prove:
$$ 2^m\left( 2^x-1 \right)+2^n\left( 2^y-1 \right)+2^p\left( 2^z-1 \right)\ge 0 $$
[i]Cristinel Mortici[/i]
2000 Switzerland Team Selection Test, 6
Positive real numbers $x,y,z$ have the sum $1$. Prove that $\sqrt{7x+3}+ \sqrt{7y+3}+\sqrt{7z+3} \le 7$.
Can number $7$ on the right hand side be replaced with a smaller constant?
1999 Austrian-Polish Competition, 4
Three lines $k, l, m$ are drawn through a point $P$ inside a triangle $ABC$ such that $k$ meets $AB$ at $A_1$ and $AC$ at $A_2 \ne A_1$ and $PA_1 = PA_2$, $l $ meets $BC$ at $B_1$ and $BA$ at $B_2 \ne B_1$ and $PB_1 = PB_2$, $m$ meets $CA$ at $C_1$ and $CB$ at $C_2\ne C_1$ and $PC_1=PC_2$. Prove that the lines $k,l,m$ are uniquely determined by these conditions. Find point $P$ for which the triangles $AA_1A_2, BB_1B_2, CC_1C_2$ have the same area and show that this point is unique.
2003 AIME Problems, 1
The product $N$ of three positive integers is $6$ times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N.$