Found problems: 85335
1994 AMC 12/AHSME, 30
When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of $S$. The smallest possible value of $S$ is
$ \textbf{(A)}\ 333 \qquad\textbf{(B)}\ 335 \qquad\textbf{(C)}\ 337 \qquad\textbf{(D)}\ 339 \qquad\textbf{(E)}\ 341 $
2007 Silk Road, 2
Let $\omega$ be the incircle of triangle $ABC$ touches $BC$ at point $K$ . Draw a circle passing through points $B$ and $C$ , and touching $\omega$ at the point $S$ . Prove that $S K$ passes through the center of the exscribed circle of triangle $A B C$ , tangent to side $B C$ .
2007 Middle European Mathematical Olympiad, 1
Let $ a,b,c,d$ be positive real numbers with $ a\plus{}b\plus{}c\plus{}d \equal{} 4$.
Prove that
\[ a^{2}bc\plus{}b^{2}cd\plus{}c^{2}da\plus{}d^{2}ab\leq 4.\]
2009 Bosnia Herzegovina Team Selection Test, 3
$a_{1},a_{2},\dots,a_{100}$ are real numbers such that:\[
a_{1}\geq a_{2}\geq\dots\geq a_{100}\geq0\]
\[
a_{1}^{2}+a_{2}^{2}\geq100\]
\[
a_{3}^{2}+a_{4}^{2}+\dots+a_{100}^{2}\geq100\]
What is the minimum value of sum $a_{1}+a_{2}+\dots+a_{100}.$
2008 Brazil National Olympiad, 1
Let $ ABCD$ be a cyclic quadrilateral and $ r$ and $ s$ the lines obtained reflecting $ AB$ with respect to the internal bisectors of $ \angle CAD$ and $ \angle CBD$, respectively. If $ P$ is the intersection of $ r$ and $ s$ and $ O$ is the center of the circumscribed circle of $ ABCD$, prove that $ OP$ is perpendicular to $ CD$.
2005 Postal Coaching, 9
In how many ways can $n$ identical balls be distributed to nine persons $A,B,C,D,E,F,G,H,I$ so that the number of balls recieved by $A$ is the same as the total number of balls recieved by $B,C,D,E$ together,.
2003 AMC 10, 2
Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $ \$4$ per pair and each T-shirt costs $ \$5$ more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $ \$2366$, how many members are in the League?
$ \textbf{(A)}\ 77 \qquad
\textbf{(B)}\ 91 \qquad
\textbf{(C)}\ 143 \qquad
\textbf{(D)}\ 182 \qquad
\textbf{(E)}\ 286$
LMT Theme Rounds, 5
Pixar Prison, for Pixar villains, is shaped like a 600 foot by 1000 foot rectangle with a 300 foot by 500 foot rectangle removed from it, as shown below. The warden separates the prison into three congruent polygonal sections for villains from The Incredibles, Finding Nemo, and Cars. What is the perimeter of each of these sections?
[asy]
draw((0,0)--(0,6)--(10,6)--(10,0)--(8,0)--(8,3)--(3,3)--(3,0)--(0,0));
label("600", (1,3.5));
label("1000", (5.5,6.5));
label("300", (4,1.5));
label("500", (5.5,3.5));
label("300", (1.5,-0.5));
[/asy]
[i]Proposed by Peter Rowley
2011 Miklós Schweitzer, 9
Let $x: [0, \infty) \to\Bbb R$ be a differentiable function. Prove that if for all t>1 $$x'(t)=-x^3(t)+\frac{t-1}{t}x^3(t-1)$$ then $\lim_{t\to\infty} x(t) = 0$
2018 Canada National Olympiad, 3
Two positive integers $a$ and $b$ are prime-related if $a = pb$ or $b = pa$ for some prime $p$. Find all positive integers $n$, such that $n$ has at least three divisors, and all the divisors can be arranged without repetition in a circle so that any two adjacent divisors are prime-related.
Note that $1$ and $n$ are included as divisors.
2024 AMC 12/AHSME, 20
Points $P$ and $Q$ are chosen uniformly and independently at random on sides $\overline {AB}$ and $\overline{AC},$ respectively, of equilateral triangle $\triangle ABC.$ Which of the following intervals contains the probability that the area of $\triangle APQ$ is less than half the area of $\triangle ABC?$
$\textbf{(A) } \left[\frac 38, \frac 12\right] \qquad \textbf{(B) } \left(\frac 12, \frac 23\right] \qquad \textbf{(C) } \left(\frac 23, \frac 34\right] \qquad \textbf{(D) } \left(\frac 34, \frac 78\right] \qquad \textbf{(E) } \left(\frac 78, 1\right]$
2017 Miklós Schweitzer, 1
Can one divide a square into finitely many triangles such that no two triangles share a side? (The triangles have pairwise disjoint interiors and their union is the square.)
2016 Purple Comet Problems, 14
Find the number of positive integers $n$ such that a regular polygon with $n$ sides has internal angles with measures equal to an integer number of degrees.
2007 AIME Problems, 5
The graph of the equation $9x+223y=2007$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?
2019 Mexico National Olympiad, 5
Let $a > b$ be relatively prime positive integers. A grashopper stands at point $0$ in a number line. Each minute, the grashopper jumps according to the following rules:
[list]
[*] If the current minute is a multiple of $a$ and not a multiple of $b$, it jumps $a$ units forward.
[*] If the current minute is a multiple of $b$ and not a multiple of $a$, it jumps $b$ units backward.
[*] If the current minute is both a multiple of $b$ and a multiple of $a$, it jumps $a - b$ units forward.
[*] If the current minute is neither a multiple of $a$ nor a multiple of $b$, it doesn't move.
[/list]
Find all positions on the number line that the grasshopper will eventually reach.
2005 Thailand Mathematical Olympiad, 10
What is the remainder when $\sum_{k=1}^{2005}k^{2005\cdot 2^{2005}}$ is divided by $2^{2005}$?
1979 Romania Team Selection Tests, 5.
In how many ways can we fill the cells of a $m\times n$ board with $+1$ and $-1$ such that the product of numbers on each line and on each column are all equal to $-1$?
2020 ASDAN Math Tournament, 4
There are $2$ ways to write $2020$ as a sum of $2$ squares: $2020 = a^2 + b^2$ and $2020 = c^2 + d^2$, where $a$, $b$, $c$, and $d$ are distinct positive integers with $a < b$ and $c < d$. Compute $a+b+c+d$.
2018 Regional Olympiad of Mexico Northeast, 5
A $300\times 300$ board is arbitrarily filled with $2\times 1$ dominoes with no overflow, underflow, or overlap. (Tokens can be placed vertically or horizontally.)
Decide if it is possible to paint the tiles with three different colors, so that the following conditions are met:
$\bullet$ Each token is painted in one and only one of the colors.
$\bullet$ The same number of tiles are painted in each color.
$\bullet$ No piece is a neighbor of more than two pieces of the same color.
Note: Two dominoes are [i]neighbors [/i]if they share an edge.
2019 PUMaC Team Round, 7
For all sets $A$ of complex numbers, let $P(A)$ be the product of the elements of $A$. Let $S_z = \{1, 2, 9, 99, 999, \frac{1}{z},\frac{1}{z^2}\}$, let $T_z$ be the set of nonempty subsets of $S_z$ (including $S_z$), and let $f(z) = 1 + \sum_{s\in T_z} P(s)$. Suppose $f(z) = 6125000$ for some complex number $z$. Compute the product of all possible values of $z$.
2008 Brazil National Olympiad, 3
Let $ x,y,z$ real numbers such that $ x \plus{} y \plus{} z \equal{} xy \plus{} yz \plus{} zx$. Find the minimum value of
\[ {x \over x^2 \plus{} 1} \plus{} {y\over y^2 \plus{} 1} \plus{} {z\over z^2 \plus{} 1}\]
2006 Italy TST, 2
Let $ABC$ be a triangle, let $H$ be the orthocentre and $L,M,N$ the midpoints of the sides $AB, BC, CA$ respectively. Prove that
\[HL^{2} + HM^{2} + HN^{2} < AL^{2} + BM^{2} + CN^{2}\]
if and only if $ABC$ is acute-angled.
2008 India Regional Mathematical Olympiad, 1
Let $ ABC$ be an acute angled triangle; let $ D,F$ be the midpoints of $ BC,AB$ respectively. Let the perpendicular from $ F$ to $ AC$ and the perpendicular from $ B$ ti $ BC$ meet in $ N$: Prove that $ ND$ is the circumradius of $ ABC$.
[15 points out of 100 for the 6 problems]
2018 MIG, 1
Evaluate $1 + 2 + 4 + 7$
$\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$
1965 All Russian Mathematical Olympiad, 070
Prove that the sum of the lengths of the polyhedron edges exceeds its tripled diameter (distance between two farest vertices).