Found problems: 85335
2013 AMC 10, 20
The number $2013$ is expressed in the form \[2013=\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!},\] where $a_1\ge a_2\ge\cdots\ge a_m$ and $b_1\ge b_2\ge\cdots\ge b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is $|a_1-b_1|$?
${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $
2015 Purple Comet Problems, 20
The diagram below shows an $8$x$7$ rectangle with a 3-4-5 right triangle drawn in each corner. The lower two triangles have their sides of length 4 along the bottom edge of the rectangle, while the upper two
triangles have their sides of length 3 along the top edge of the rectangle. A circle is tangent to the hypotenuse of each triangle. The diameter of the circle is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find m + n.
For diagram go to http://www.purplecomet.org/welcome/practice, go to the 2015 middle school contest questions, and then go to #20
2022 Tuymaada Olympiad, 2
Given are integers $a, b, c$ and an odd prime $p.$ Prove that $p$ divides $x^2 + y^2 + ax + by + c$ for some integers $x$ and $y.$
[i](A. Golovanov )[/i]
1999 All-Russian Olympiad Regional Round, 8.4
There are $40$ identical gas cylinders, gas pressure values in which we are unknown and may be evil. It is allowed to connect any cylinders with each other in an amount not exceeding a given natural number $k$, and then separate them; while the pressure gas in the connected cylinders is set equal to the arithmetic average of the pressures in them before the connection. At what minimum $k$ is there a way to equalize the pressures in all $40$ cylinders, regardless of initial pressure distribution in the cylinders?
2012 District Olympiad, 3
Let $a, b$, and $c$ be positive real numbers. Find the largest integer $n$ such that $$\frac{1}{ax + b + c}
+\frac{1}{a + bx + c}+\frac{1}{a + b + cx} \ge \frac{n}{a + b + c},$$
for all $ x \in [0, 1]$ .
2015 Junior Balkan Team Selection Tests - Romania, 5
Let $ABCD$ be a convex quadrilateral with non perpendicular diagonals and with the sides $AB$ and $CD$ non parallel . Denote by $O$ the intersection of the diagonals , $H_1$ the orthocenter of the triangle $AOB$ and $H_2$ the orthocenter of the triangle $COD$ . Also denote with $M$ the midpoint of the side $AB$ and with $N$ the midpoint of the side $CD$ . Prove that $H_1H_2$ and $MN$ are parallel if and only if $AC=BD$
2003 Balkan MO, 4
A rectangle $ABCD$ has side lengths $AB = m$, $AD = n$, with $m$ and $n$ relatively prime and both odd. It is divided into unit squares and the diagonal AC intersects the sides of the unit squares at the points $A_1 = A, A_2, A_3, \ldots , A_k = C$. Show that \[ A_1A_2 - A_2A_3 + A_3A_4 - \cdots + A_{k-1}A_k = {\sqrt{m^2+n^2}\over mn}. \]
2024 ELMO Shortlist, G8
Let $ABC$ be a triangle, and let $D$ be a point on the internal angle bisector of $BAC$. Let $x$ be the ellipse with foci $B$ and $C$ passing through $D$, $y$ be the ellipse with foci $A$ and $C$ passing through $D$, and $z$ be the ellipse with foci $A$ and $B$ passing through $D$. Ellipses $x$ and $z$ intersect at distinct points $D$ and $E$, and ellipses $x$ and $y$ intersect at distinct points $D$ and $F$. Prove that $AD$ bisects angle $EAF$.
[i]Andrew Carratu[/i]
2006 Federal Competition For Advanced Students, Part 2, 1
Let $ N$ be a positive integer. How many non-negative integers $ n \le N$ are there that have an integer multiple, that only uses the digits $ 2$ and $ 6$ in decimal representation?
2009 Stanford Mathematics Tournament, 8
Simplify $\sum_{k=1}^{n}\frac{k^2(k - n)}{n^4}$
2007-2008 SDML (Middle School), 7
Each of the first $150$ positive integers is painted on a different marble, and the $150$ marbles are placed in a bag. If $n$ marbles are chosen (without replacement) from the bag, what is the smallest value of $n$ such that we are guaranteed to choose three marbles with consecutive numbers?
2023 UMD Math Competition Part I, #7
Suppose $S = \{1, 2, 3, x\}$ is a set with four distinct real numbers for which the difference between the largest and smallest values of $S$ is equal to the sum of elements of $S.$ What is the value of $x?$
$$
\mathrm a. ~ {-1}\qquad \mathrm b.~{-3/2}\qquad \mathrm c. ~{-2} \qquad \mathrm d. ~{-2/3} \qquad \mathrm e. ~{-3}
$$
2016 Indonesia TST, 4
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
2017 Romania National Olympiad, 2
Let be a natural number $ n $ and $ 2n $ real numbers $ b_1,b_2,\ldots ,b_n,a_1<a_2<\cdots <a_n. $ Show that
[b]a)[/b] if $ b_1,b_2,\ldots ,b_n>0, $ then there exists a polynomial $ f\in\mathbb{R}[X] $ irreducible in $ \mathbb{R}[X] $ such that $$ f\left( a_i \right) =b_i,\quad\forall i\in\{ 1,2,\ldots ,n \} . $$
[b]b)[/b] there exists a polynom $ g\in\mathbb{R} [X] $ of degree at least $ 1 $ which has only real roots and such that
$$ g\left( a_i \right) =b_i,\quad\forall i\in\{ 1,2,\ldots ,n \} . $$
2006 China Team Selection Test, 2
Prove that for any given positive integer $m$ and $n$, there is always a positive integer $k$ so that $2^k-m$ has at least $n$ different prime divisors.
2019 CCA Math Bonanza, I8
If $a!+\left(a+2\right)!$ divides $\left(a+4\right)!$ for some nonnegative integer $a$, what are all possible values of $a$?
[i]2019 CCA Math Bonanza Individual Round #8[/i]
2019 Romanian Master of Mathematics Shortlist, original P4
Let there be an equilateral triangle $ABC$ and a point $P$ in its plane such that $AP<BP<CP.$ Suppose that the lengths of segments $AP,BP$ and $CP$ uniquely determine the side of $ABC$. Prove that $P$ lies on the circumcircle of triangle $ABC.$
1995 IMO, 2
Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc \equal{} 1$. Prove that
\[ \frac {1}{a^{3}\left(b \plus{} c\right)} \plus{} \frac {1}{b^{3}\left(c \plus{} a\right)} \plus{} \frac {1}{c^{3}\left(a \plus{} b\right)}\geq \frac {3}{2}.
\]
1989 All Soviet Union Mathematical Olympiad, 491
Eight pawns are placed on a chessboard, so that there is one in each row and column. Show that an even number of the pawns are on black squares.
2016 JBMO TST - Turkey, 6
Prove that
\[ (x^4+y)(y^4+z)(z^4+x) \geq (x+y^2)(y+z^2)(z+x^2) \]
for all positive real numbers $x, y, z$ satisfying $xyz \geq 1$.
VMEO IV 2015, 10.2
Given a triangle $ABC$ with obtuse $\angle A$ and attitude $AH$ with $H \in BC$. Let $E,F$ on $CA$, $AB$ satisfying $\angle BEH = \angle C$ and $\angle CFH = \angle B$. Let $BE$ cut $CF$ at $D$. Prove that $DE = DF$.
2000 Singapore Senior Math Olympiad, 2
Prove that there exist no positive integers $m$ and $n$ such that $m > 5$ and $(m - 1)! + 1 = m^n$.
2018 JHMT, 6
$\vartriangle ABC$ is inscribed in a unit circle. The three angle bisectors of $A$,$B$,$C$ are extended to intersect the circle at $A_1$,$B_1$,$C_1$, respectively. Find $$\frac{AA_1 \cos \frac{A}{2} + BB_1 \cos \frac{B}{2} + CC_1 \cos \frac{C}{2}}{\sin A + \sin B + \sin C}.$$
1901 Eotvos Mathematical Competition, 3
Let $a$ and $b$ be two natural numbers whose greatest common divisor is $d$. Prove that exactly $d$ of the numbers $$a, 2a, 3a, ..., (b-1)a, ba$$ is divisible by $b$.
2005 Korea Junior Math Olympiad, 6
For two different prime numbers $p, q$, defi ne $S_{p,q} = \{p,q,pq\}$. If two elements in $S_{p,q}$ are numbers in the form of $x^2 + 2005y^2, (x, y \in Z)$, prove that all three elements in $S_{p,q}$ are in such form.