Found problems: 85335
2006 Iran Team Selection Test, 5
Let $ABC$ be an acute angle triangle.
Suppose that $D,E,F$ are the feet of perpendicluar lines from $A,B,C$ to $BC,CA,AB$.
Let $P,Q,R$ be the feet of perpendicular lines from $A,B,C$ to $EF,FD,DE$.
Prove that
\[ 2(PQ+QR+RP)\geq DE+EF+FD \]
2022 Kyiv City MO Round 1, Problem 4
In some magic country, there are banknotes only of values $3$, $25$, $80$ hryvnyas. Businessman Victor ate in one restaurant of this country for $2024$ days in a row, and each day (except the first) he spent exactly $1$ hryvnya more than the day before (without any change). Could he have spent exactly $1000000$ banknotes?
[i](Proposed by Oleksii Masalitin)[/i]
1990 Greece National Olympiad, 2
Let $ACBD$ be a asquare and $K,L,M,N$ be points of $AB,BC,CD,DA$ respectively. If $O$ is the center of the square , prove that the expression $$ \overrightarrow{OK}\cdot \overrightarrow{OL}+\overrightarrow{OL}\cdot\overrightarrow{OM}+\overrightarrow{OM}\cdot\overrightarrow{ON}+\overrightarrow{ON}\cdot\overrightarrow{OK}$$
is independent of positions of $K,L,M,N$, (i.e. is constant )
2011 Today's Calculation Of Integral, 678
Evaluate
\[\int_0^{\pi} \left(1+\sum_{k=1}^n k\cos kx\right)^2dx\ \ (n=1,\ 2,\ \cdots).\]
[i]2011 Doshisya University entrance exam/Life Medical Sciences[/i]
2011 Belarus Team Selection Test, 2
Find the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots , s_n\}$ consisting of $n$ distinct positive integers such that
\[ \left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.\]
[i]Proposed by Daniel Brown, Canada[/i]
1993 USAMO, 1
For each integer $\, n \geq 2, \,$ determine, with proof, which of the two positive real numbers $\, a \,$ and $\, b \,$ satisfying \[ a^n = a + 1, \hspace{.3in} b^{2n} = b + 3a \] is larger.
2002 IMO Shortlist, 6
Let $n\geq3$ be a positive integer. Let $C_1,C_2,C_3,\ldots,C_n$ be unit circles in the plane, with centres $O_1,O_2,O_3,\ldots,O_n$ respectively. If no line meets more than two of the circles, prove that \[ \sum\limits^{}_{1\leq i<j\leq n}{1\over O_iO_j}\leq{(n-1)\pi\over 4}. \]
2013 AMC 12/AHSME, 17
Let $a,b,$ and $c$ be real numbers such that \begin{align*}
a+b+c &= 2, \text{ and} \\
a^2+b^2+c^2&= 12
\end{align*}
What is the difference between the maximum and minimum possible values of $c$?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ \frac{10}{3}\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ \frac{16}{3}\qquad\textbf{(E)}\ \frac{20}{3} $
2010 Balkan MO, 4
For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$.
Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.
2022 JBMO Shortlist, A1
Find all pairs of positive integers $(a, b)$ such that $$11ab \le a^3 - b^3 \le 12ab.$$
2006 Cuba MO, 4
Let $f : Z_+ \to Z_+$ such that:
a) $f(n + 1) > f(n)$ for all $n \in Z_+$
b) $f(n + f(m)) = f(n) + m + 1$ for all $n,m \in Z_+$
Find $f(2006)$.
2013 Today's Calculation Of Integral, 894
Let $a$ be non zero real number. Find the area of the figure enclosed by the line $y=ax$, the curve $y=x\ln (x+1).$
PEN O Problems, 35
Let $ n \ge 3$ be a prime number and $ a_{1} < a_{2} < \cdots < a_{n}$ be integers. Prove that $ a_{1}, \cdots,a_{n}$ is an arithmetic progression if and only if there exists a partition of $ \{0, 1, 2, \cdots \}$ into sets $ A_{1},A_{2},\cdots,A_{n}$ such that
\[ a_{1} \plus{} A_{1} \equal{} a_{2} \plus{} A_{2} \equal{} \cdots \equal{} a_{n} \plus{} A_{n},\]
where $ x \plus{} A$ denotes the set $ \{x \plus{} a \vert a \in A \}$.
2012 Turkey MO (2nd round), 4
For all positive real numbers $x, y, z$, show that $ \frac{x(2x-y)}{y(2z+x)}+\frac{y(2y-z)}{z(2x+y)}+\frac{z(2z-x)}{x(2y+z)} \geq 1$ is true.
1989 AMC 12/AHSME, 30
Suppose that $7$ boys and $13$ girls line up in a row. Let $S$ be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row $GBBGGGBGBGGGBGBGGBGG$ we have $S=12$. The average value of $S$ (if all possible orders of the 20 people are considered) is closest to
$ \textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13 $
2021 AMC 10 Fall, 24
Each of the $12$ edges of a cube is labeled $0$ or $1$. Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the $6$ faces of the cube equal to $2?$
$\textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }16\qquad\textbf{(E) }20$
2018 ASDAN Math Tournament, 3
In $\vartriangle ABC$, $AC > AB$. $B$ is reflected across $\overline{AC}$ to a point $D$, and $C$ is reflected across $\overline{AD}$ to a point $E$. Suppose that $AC = 6\sqrt3 + 6$, $BC = 6$, and $\overline{BC} \parallel \overline{AE}$. Compute $AB$.
Kvant 2024, M2812
On the coordinate plane, at some points with integer coordinates, there is a pebble (a finite number of pebbles). It is allowed to make the following move: select a pair of pebbles, take some vector $\vec{a}$ with integer coordinates and then move one of the selected pebbles to vector $\vec{a}$, and the other to the opposite vector $-\vec{a}$; it is forbidden that there should be more than one pebble at one point. Is it always possible to achieve a situation in which all the pebbles lie on the same straight line in a few moves?
[i] K. Ivanov [/i]
2010 Harvard-MIT Mathematics Tournament, 10
Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. Segment $PQ$ is tangent to $\omega_1$ at $P$ and to $\omega_2$ at $Q$, and $A$ is closer to $PQ$ than $B$. Point $X$ is on $\omega_1$ such that $PX\parallel QB$, and point $Y$ is on $\omega_2$ such that $QY\parallel PB$. Given that $\angle APQ=30^\circ$ and $\angle PQA=15^\circ$, find the ratio $AX/AY$.
2014 ELMO Shortlist, 10
We are given triangles $ABC$ and $DEF$ such that $D\in BC, E\in CA, F\in AB$, $AD\perp EF, BE\perp FD, CF\perp DE$. Let the circumcenter of $DEF$ be $O$, and let the circumcircle of $DEF$ intersect $BC,CA,AB$ again at $R,S,T$ respectively. Prove that the perpendiculars to $BC,CA,AB$ through $D,E,F$ respectively intersect at a point $X$, and the lines $AR,BS,CT$ intersect at a point $Y$, such that $O,X,Y$ are collinear.
[i]Proposed by Sammy Luo[/i]
1958 Miklós Schweitzer, 9
[b]9.[/b] Show that if $f(z) = 1+a_1 z+a_2z^2+\dots$ for $\mid z \mid\leq 1$ and
$\frac{1}{2\pi}\int_{0}^{2\pi}\mid f(e^{i\phi}) \mid^{2} d\phi < \left (1+\frac{\mid a_1\mid ^2} {4} \right )^2$,
then $f(z)$ has a root in the disc $\mid z \mid \leq 1$.[b](F. 4)[/b]
2018 HMNT, 3
A square in the [i]xy[/i]-plane has area [i]A[/i], and three of its vertices have [i]x[/i]-coordinates $2,0,$ and $18$ in some order. Find the sum of all possible values of [i]A[/i].
1998 Switzerland Team Selection Test, 2
Find all nonnegative integer solutions $(x,y,z)$ of the equation
$\frac{1}{x+2}+\frac{1}{y+2}=\frac{1}{2} +\frac{1}{z+2}$
2017 Romania National Olympiad, 1
Consider the set
$$M = \left\{\frac{a}{\overline{ba}}+\frac{b}{\overline{ab}} \, | a,b\in\{1,2,3,4,5,6,7,8,9\} \right\}.$$
a) Show that the set $M$ contains no integer.
b) Find the smallest and the largest element of $M$
2010 Lithuania National Olympiad, 4
Decimal digits $a,b,c$ satisfy
\[ 37\mid (a0a0\ldots a0b0c0c\ldots 0c)_{10} \]
where there are $1001$ a's and $1001$ c's. Prove that $b=a+c$.