Found problems: 85335
1994 India Regional Mathematical Olympiad, 8
If $a,b,c$ are positive real numbers such that $a+b+c = 1$, prove that \[ (1+a)(1+b)(1+c) \geq 8 (1-a)(1-b)(1-c) . \]
1999 Baltic Way, 16
Find the smallest positive integer $k$ which is representable in the form $k=19^n-5^m$ for some positive integers $m$ and $n$.
2005 Singapore Senior Math Olympiad, 4
Is there integer $n$ such that $n!$ begins with $2005$ ?
2014 Costa Rica - Final Round, 6
$n$ people are in the plane, so that the closest person is unique and each one shoot this closest person with a squirt gun. If $n$ is odd, prove that there exists at least one person that nobody shot. If $n$ is even, will there always be a person who escape? Justify that.
2022 AMC 8 -, 18
The midpoints of the four sides of a rectangle are $(-3, 0), (2, 0), (5, 4)$ and $(0, 4)$. What is the area of the rectangle?
$\textbf{(A)} ~20\qquad\textbf{(B)} ~25\qquad\textbf{(C)} ~40\qquad\textbf{(D)} ~50\qquad\textbf{(E)} ~80\qquad$
2014 CHMMC (Fall), 1
For $a_1,..., a_5 \in R$, $$\frac{a_1}{k^2 + 1}+ ... +\frac{a_5}{k^2 + 5}=\frac{1}{k^2}$$ for all $k \in \{2, 3, 4, 5, 6\}$. Calculate $$\frac{a_1}{2}+... +\frac{a_5}{6}.$$
2017 Romania EGMO TST, P4
In $p{}$ of the vertices of the regular polygon $A_0A_1\ldots A_{2016}$ we write the number $1{}$ and in the remaining ones we write the number $-1.{}$ Let $x_i{}$ be the number written on the vertex $A_i{}.$ A vertex is [i]good[/i] if \[x_i+x_{i+1}+\cdots+x_j>0\quad\text{and}\quad x_i+x_{i-1}+\cdots+x_k>0,\]for any integers $j{}$ and $k{}$ such that $k\leqslant i\leqslant j.$ Note that the indices are taken modulo $2017.$ Determine the greatest possible value of $p{}$ such that, regardless of numbering, there always exists a good vertex.
2015 Taiwan TST Round 3, 2
Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q,$ respectively. The point $R$ is chosen on the line $P Q$ so that $BR = MR$. Prove that $BR\parallel AC$.
(Here we always assume that an angle bisector is a ray.)
[i]Proposed by Sergey Berlov, Russia[/i]
2006 Estonia National Olympiad, 1
Calculate the sum $$\frac{1}{1+2^{-2006}}+...+ \frac{1}{1+2^{-1}}+ \frac{1}{1+2^{0}}+ \frac{1}{1+2^{1}}+...+ \frac{1}{1+2^{2006}}$$
1999 Gauss, 5
Which one of the following gives an odd integer?
$\textbf{(A)}\ 6^2 \qquad \textbf{(B)}\ 23-17 \qquad \textbf{(C)}\ 9\times24 \qquad \textbf{(D)}\ 96\div8 \qquad \textbf{(E)}\ 9\times41$
2016 Auckland Mathematical Olympiad, 1
How many $3 \times 5$ rectangular pieces of cardboard can be cut from a $17 \times 22$ rectangular piece of cardboard, when the amount of waste is minimised?
2020 Peru EGMO TST, 6
A table $110\times 110$ is given, we define the distance between two cells $A$ and $B$ as the least quantity of moves to move a chess king from the cell $A$ to cell $B$. We marked $n$ cells on the table $110\times 110$ such that the distance between any two cells is not equal to $15$. Determine the greatest value of $n$.
2010 AMC 8, 13
The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shorter side is $30\%$ of the perimeter. What is the length of the longest side?
$ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 $
OIFMAT II 2012, 3
In the interior of an equilateral triangle $ ABC $ a point $ P $ is chosen such that $ PA ^2 = PB ^2 + PC ^2 $. Find the measure of $ \angle BPC $.
2011 Morocco National Olympiad, 1
Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$, and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$.
2019 Thailand TST, 2
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.
2010 AIME Problems, 15
In triangle $ ABC$, $ AC \equal{} 13, BC \equal{} 14,$ and $ AB\equal{}15$. Points $ M$ and $ D$ lie on $ AC$ with $ AM\equal{}MC$ and $ \angle ABD \equal{} \angle DBC$. Points $ N$ and $ E$ lie on $ AB$ with $ AN\equal{}NB$ and $ \angle ACE \equal{} \angle ECB$. Let $ P$ be the point, other than $ A$, of intersection of the circumcircles of $ \triangle AMN$ and $ \triangle ADE$. Ray $ AP$ meets $ BC$ at $ Q$. The ratio $ \frac{BQ}{CQ}$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\minus{}n$.
2012 ELMO Problems, 1
In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$.
[i]Ray Li.[/i]
2013 IMO Shortlist, G5
Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.
2005 Portugal MO, 1
In line for a SuperRockPop concert were 2005 people. With the aim of offering $3$ tickets for the "backstage", the first person in line was asked to shout "Super", ` the second "Rock", ` the third "Pop", ` the fourth "Super", ` the fifth "Rock", ` the sixth "Pop" and so on. Anyone who said "Rock" or "Pop" was eliminated. This process was repeated, always starting from the first person in the new line, until only $3$ people remained. What positions were these people in at the beginning?
2012 Indonesia MO, 3
Given an acute triangle $ABC$ with $AB>AC$ that has circumcenter $O$. Line $BO$ and $CO$ meet the bisector of $\angle BAC$ at $P$ and $Q$, respectively. Moreover, line $BQ$ and $CP$ meet at $R$. Show that $AR$ is perpendicular to $BC$.
[i]Proposer: Soewono and Fajar Yuliawan[/i]
2007 AMC 8, 2
Six-hundred fifty students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti?
[asy]
size(200);
defaultpen(linewidth(0.7));
defaultpen(fontsize(8));
draw(origin--(0,250));
int i;
for(i=0; i<6; i=i+1) {
draw((0,50*i)--(5,50*i));
}
filldraw((25,0)--(75,0)--(75,150)--(25,150)--cycle, gray, black);
filldraw((75,0)--(125,0)--(125,100)--(75,100)--cycle, gray, black);
filldraw((125,0)--(175,0)--(175,150)--(125,150)--cycle, gray, black);
filldraw((225,0)--(175,0)--(175,250)--(225,250)--cycle, gray, black);
label("$50$", (0,50), W);
label("$100$", (0,100), W);
label("$150$", (0,150), W);
label("$200$", (0,200), W);
label("$250$", (0,250), W);
label(rotate(90)*"Lasagna", (50,0), S);
label(rotate(90)*"Manicotti", (100,0), S);
label(rotate(90)*"Ravioli", (150,0), S);
label(rotate(90)*"Spaghetti", (200,0), S);
label(rotate(90)*"$\mbox{Number of People}$", (-40,140), W);[/asy]
$\textbf{(A)} \: \frac25\qquad \textbf{(B)} \: \frac12\qquad \textbf{(C)} \: \frac54\qquad \textbf{(D)} \: \frac53\qquad \textbf{(E)} \: \frac52$
1993 Miklós Schweitzer, 7
Let H be a Hilbert space over the field of real numbers $\Bbb R$. Find all $f: H \to \Bbb R$ continuous functions for which
$$f(x + y + \pi z) + f(x + \sqrt{2} z) + f(y + \sqrt{2} z) + f (\pi z)$$
$$= f(x + y + \sqrt{2} z) + f (x + \pi z) + f (y + \pi z) + f(\sqrt{2} z)$$
is satisfied for any $x , y , z \in H$.
2012 QEDMO 11th, 4
The fields of an $n\times n$ chess board are colored black and white, such that in every small $2\times 2$-square both colors should be the same number. How many there possibilities are for this?
2013 Estonia Team Selection Test, 1
Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\
a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$