Found problems: 85335
II Soros Olympiad 1995 - 96 (Russia), 9.1
Solve the equation
$$(x+1)^2-5(x+1) \sqrt{x}+4x=0$$
2011 NIMO Summer Contest, 12
In triangle $ABC$, $AB = 100$, $BC = 120$, and $CA = 140$. Points $D$ and $F$ lie on $\overline{BC}$ and $\overline{AB}$, respectively, such that $BD = 90$ and $AF = 60$. Point $E$ is an arbitrary point on $\overline{AC}$. Denote the intersection of $\overline{BE}$ and $\overline{CF}$ as $K$, the intersection of $\overline{AD}$ and $\overline{CF}$ as $L$, and the intersection of $\overline{AD}$ and $\overline{BE}$ as $M$. If $[KLM] = [AME] + [BKF] + [CLD]$, where $[X]$ denotes the area of region $X$, compute $CE$.
[i]Proposed by Lewis Chen
[/i]
2020 Princeton University Math Competition, A3
Let $n$ be a positive integer, and let $F$ be a family of subsets of $\{1, 2, ... , 2^n\}$ such that for any non-empty $ A\in F$ there exists $B \in F$ so that $|A| = |B| + 1$ and $B \subset A$. Suppose that F contains all $(2^n - 1)$-element subsets of $\{1, 2, ... , 2^n\}$ Determine the minimal possible value of $|F|$.
2003 USA Team Selection Test, 1
For a pair of integers $a$ and $b$, with $0 < a < b < 1000$, set $S\subseteq \{ 1, 2, \dots , 2003\}$ is called a [i]skipping set[/i] for $(a, b)$ if for any pair of elements $s_1, s_2 \in S$, $|s_1 - s_2|\not\in \{ a, b\}$. Let $f(a, b)$ be the maximum size of a skipping set for $(a, b)$. Determine the maximum and minimum values of $f$.
2015 AIME Problems, 8
Let $a$ and $b$ be positive integers satisfying $\frac{ab+1}{a+b}<\frac{3}{2}$. The maximum possible value of $\frac{a^3b^3+1}{a^3+b^3}$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2014 Romania National Olympiad, 1
Find all primes $p$ and $q$, with $p \le q$, so that $$p (2q + 1) + q (2p + 1) = 2 (p^2 + q^2).$$
2020 Dürer Math Competition (First Round), P1
a) Is it possible that the sum of all the positive divisors of two different natural numbers are equal?
b) Is it possible that the product of all the positive divisors of two different natural numbers are equal?
2013 Brazil Team Selection Test, 3
Let $x$ and $y$ be positive integers. If ${x^{2^n}}-1$ is divisible by $2^ny+1$ for every positive integer $n$, prove that $x=1$.
1990 AMC 8, 3
What fraction of the square is shaded?
[asy]
draw((0,0)--(0,3)--(3,3)--(3,0)--cycle);
draw((0,2)--(2,2)--(2,0)); draw((0,1)--(1,1)--(1,0)); draw((0,0)--(3,3));
fill((0,0)--(0,1)--(1,1)--cycle,grey);
fill((1,0)--(1,1)--(2,2)--(2,0)--cycle,grey);
fill((0,2)--(2,2)--(3,3)--(0,3)--cycle,grey);[/asy]
$ \text{(A)}\ \frac{1}{3}\qquad\text{(B)}\ \frac{2}{5}\qquad\text{(C)}\ \frac{5}{12}\qquad\text{(D)}\ \frac{3}{7}\qquad\text{(E)}\ \frac{1}{2} $
1994 Austrian-Polish Competition, 3
A rectangular building consists of $30$ square rooms situated like the cells of a $2 \times 15$ board. In each room there are three doors, each of which leads to another room (not necessarily different). How many ways are there to distribute the doors between the rooms so that it is possible to get from any room to any other one without leaving the building?
2019 Singapore Junior Math Olympiad, 3
Find all positive integers $m, n$ such that $\frac{2m-1}{n}$ and $\frac{2n-1}{m}$ are both integers.
2002 National High School Mathematics League, 11
If $\log_4 (x+2y)+\log_4 (x-2y)=1$, then the minumum value of $|x|-|y|$ is________.
2018 Regional Olympiad of Mexico Northwest, 3
Let $ABC$ be an acute triangle orthocenter angle $H$. Let $\omega_1$ be the circle tangent to $BC$ at $B$ and passing through $H$ and $\omega_2$ the circle tangent to $BC$ at $C$ and passing through through $H$. A line $\ell$ passing through $H$ intersects the circles $\omega_1$ and $\omega_2$ at points $D$ and $E$, respectively (with $D$ and $E$ other than $H$). Lines $BD$ and $CE$ intersect at $F$, the lines $\ell$ and $AF$ intersect at $X$ and the circles $\omega_1$ and $\omega_2$ intersect at the points $P$ and $H$. Prove that the points $A, H, P$ and $X$ are still on the same circle.
2007 Moldova National Olympiad, 11.5
Real numbers $a_{1},a_{2},\dots,a_{n}$ satisfy $a_{i}\geq\frac{1}{i}$, for all $i=\overline{1,n}$. Prove the inequality:
\[\left(a_{1}+1\right)\left(a_{2}+\frac{1}{2}\right)\cdot\dots\cdot\left(a_{n}+\frac{1}{n}\right)\geq\frac{2^{n}}{(n+1)!}(1+a_{1}+2a_{2}+\dots+na_{n}).\]
2005 MOP Homework, 4
Consider an infinite array of integers. Assume that each integer is equal to the sum of the integers immediately above and immediately to the left. Assume that there exists a row $R_0$ such that all the number in the row are positive. Denote by $R_1$ the row below row $R_0$, by $R_2$ the row below row $R_1$, and so on. Show that for each positive integer $n$, row $R_n$ cannot contain more than $n$ zeros.
2009 Today's Calculation Of Integral, 456
Find $ \lim_{n\to\infty} \frac{\pi}{n}\left\{\frac{1}{\sin \frac{\pi (n\plus{}1)}{4n}}\plus{}\frac{1}{\sin \frac{\pi (n\plus{}2)}{4n}}\plus{}\cdots \plus{}\frac{1}{\sin \frac{\pi (n\plus{}n)}{4n}}\right\}$
2007 Tournament Of Towns, 5
Two players in turns color the squares of a $4 \times 4$ grid, one square at the time. Player loses if after his move a square of $2\times2$ is colored completely. Which of the players has the winning strategy, First or Second?
[i](4 points)[/i]
2021-2022 OMMC, 3
Evan has $10$ cards numbered $1$ through $10$. He chooses some of the cards and takes the product of the numbers on them. When the product is divided by $3$, the remainder is $1$. Find the maximum number of cards he could have chose.
[i]Proposed by Evan Chang [/i]
2010 IMC, 3
Denote by $S_n$ the group of permutations of the sequence $(1,2,\dots,n).$ Suppose that $G$ is a subgroup of $S_n,$ such that for every $\pi\in G\setminus\{e\}$ there exists a unique $k\in \{1,2,\dots,n\}$ for which $\pi(k)=k.$ (Here $e$ is the unit element of the group $S_n.$) Show that this $k$ is the same for all $\pi \in G\setminus \{e\}.$
2019 Dutch BxMO TST, 5
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?
2010 Purple Comet Problems, 1
Let $x$ satisfy $(6x + 7) + (8x + 9) = (10 + 11x) + (12 + 13x).$ There are relatively prime positive integers so that $x = -\tfrac{m}{n}$. Find $m + n.$
2022 MIG, 7
Consider the rectangular strip of length $12$ below, divided into three rectangles. The distance between the centers of two of the rectangles is $4$. What is the length of the other rectangle?
[asy]
size(120);
draw((0,0)--(12,0)--(12,1)--(0,1)--cycle);
draw((8,1)--(8,0));
draw((3,1)--(3,0));
dot((1.5,0.5));
dot((5.5,0.5));
draw((1.5,0.5)--(5.5,0.5));
[/asy]
$\textbf{(A) }2.5\qquad\textbf{(B) }3\qquad\textbf{(C) }3.5\qquad\textbf{(D) }4\qquad\textbf{(E) }4.5$
1980 All Soviet Union Mathematical Olympiad, 299
Let the edges of rectangular parallelepiped be $x,y$ and $z$ ($x<y<z$). Let
$$p=4(x+y+z), s=2(xy+yz+zx) \,\,\, and \,\,\, d=\sqrt{x^2+y^2+z^2}$$ be its perimeter, surface area and diagonal length, respectively. Prove that $$x < \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )\,\,\, and \,\,\, z > \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )$$
1959 Putnam, A2
"Let $\omega^3 = 1, \omega \neq 1$. Show that$z_1, z_2, -\omega z_1, -\omega^2z_2$ are the vertices of an equilateral triangle."
2007 CentroAmerican, 3
Consider a circle $S$, and a point $P$ outside it. The tangent lines from $P$ meet $S$ at $A$ and $B$, respectively. Let $M$ be the midpoint of $AB$. The perpendicular bisector of $AM$ meets $S$ in a point $C$ lying inside the triangle $ABP$. $AC$ intersects $PM$ at $G$, and $PM$ meets $S$ in a point $D$ lying outside the triangle $ABP$. If $BD$ is parallel to $AC$, show that $G$ is the centroid of the triangle $ABP$.
[i]Arnoldo Aguilar (El Salvador)[/i]