Found problems: 85335
2019 Purple Comet Problems, 11
Let $m > n$ be positive integers such that $3(3mn - 2)^2 - 2(3m -3n)^2 = 2019$. Find $3m + n$.
2011 IMC, 3
Let $p$ be a prime number. Call a positive integer $n$ interesting if
\[x^n-1=(x^p-x+1)f(x)+pg(x)\]
for some polynomials $f$ and $g$ with integer coefficients.
a) Prove that the number $p^p-1$ is interesting.
b) For which $p$ is $p^p-1$ the minimal interesting number?
2023 Czech and Slovak Olympiad III A., 5
In triangle $ABC$ let $N, M, P$ be the midpoints of the sides $BC, CA, AB$ and $G$ be the centroid of this triangle. Let the circle circumscribed to $BGP$ intersect the line $MP$ in point $K$, $P \neq K$, and the circle circumscribed to $CGN$ intersect the line $MN$ in point $L$, $N \neq L$. Prove that $ \angle BAK = \angle CAL $.
2010 AMC 8, 8
As Emily is riding her bike on a long straight road, she spots Ermenson skating in the same direction $1/2$ mile in front of her. After she passes him, she can see him in her rear mirror until he is $1/2$ mile behind her. Emily rides at a constant rate of $12$ miles per hour. Ermenson skates at a constant rate of $8$ miles per hour. For how many minutes can Emily see Ermenson?
$ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 16 $
2017 USA Team Selection Test, 1
In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is[i] color-identifiable[/i] if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$.
For all positive integers $n$ and $t$, determine the maximum integer $g(n, t)$ such that: In any sports league with exactly $n$ distinct colors present over all teams, one can always find a color-identifiable set of size at least $g(n, t)$.
2005 Taiwan TST Round 2, 4
A quadrilateral $PQRS$ has an inscribed circle, the points of tangencies with sides $PQ$, $QR$, $RS$, $SP$ being $A$, $B$, $C$, $D$, respectively. Let the midpoints of $AB$, $BC$, $CD$, $DA$ be $E$, $F$, $G$, $H$, respectively. Prove that the angle between segments $PR$ and $QS$ is equal to the angle between segments $EG$ and $FH$.
2009 Mediterranean Mathematics Olympiad, 2
Let $ABC$ be a triangle with $90^\circ \ne \angle A \ne 135^\circ$. Let $D$ and $E$ be external points to the triangle $ABC$ such that $DAB$ and $EAC$ are isoscele triangles with right angles at $D$ and $E$. Let $F = BE \cap CD$, and let $M$ and $N$ be the midpoints of $BC$ and $DE$, respectively.
Prove that, if three of the points $A$, $F$, $M$, $N$ are collinear, then all four are collinear.
2019 APMO, 4
Consider a $2018 \times 2019$ board with integers in each unit square. Two unit squares are said to be neighbours if they share a common edge. In each turn, you choose some unit squares. Then for each chosen unit square the average of all its neighbours is calculated. Finally, after these calculations are done, the number in each chosen unit square is replaced by the corresponding average.
Is it always possible to make the numbers in all squares become the same after finitely many turns?
2007 All-Russian Olympiad, 1
Unitary quadratic trinomials $ f(x)$ and $ g(x)$ satisfy the following interesting condition: $ f(g(x)) \equal{} 0$ and $ g(f(x)) \equal{} 0$ do not have real roots. Prove that at least one of equations $ f(f(x)) \equal{} 0$ and $ g(g(x)) \equal{} 0$ does not have real roots too.
[i]S. Berlov [/i]
2025 Belarusian National Olympiad, 8.5
Ten monkeys have 60 bananas. Each monkey has at least one banana and any two monkeys have different amounts of bananas.
Prove that any six monkeys can distribute their bananas between others such that all 4 remaining monkeys have the same amount of bananas.
[i]A. Voidelevich[/i]
2023 Iran Team Selection Test, 6
$ABC$ is an acute triangle with orthocenter $H$. Point $P$ is in triangle $BHC$ that $\angle HPC = 3 \angle HBC $ and $\angle HPB =3 \angle HCB $. Reflection of point $P$ through $BH,CH$ is $X,Y$. if $S$ is the center of circumcircle of $AXY$ , Prove that:
$$\angle BAS = \angle CAP$$
[i]Proposed by Pouria Mahmoudkhan Shirazi [/i]
1999 Ukraine Team Selection Test, 1
A triangle $ABC$ is given. Points $E,F,G$ are arbitrarily selected on the sides $AB,BC,CA$, respectively, such that $AF\perp EG$ and the quadrilateral $AEFG$ is cyclic. Find the locus of the intersection point of $AF$ and $EG$.
2014 HMNT, 10
Let $z$ be a complex number and k a positive integer such that $z^k$ is a positive real number other than $1$. Let $f(n)$ denote the real part of the complex number $z^n$. Assume the parabola $p(n) = an^2 +bn+c$ intersects $f(n)$ four times, at $n = 0, 1, 2, 3$. Assuming the smallest possible value of $k$, find the largest possible value of $a$.
1992 AMC 8, 24
Four circles of radius $3$ are arranged as shown. Their centers are the vertices of a square. The area of the shaded region is closest to
[asy]
fill((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle,lightgray);
fill(arc((3,3),(0,3),(3,0),CCW)--(3,3)--cycle,white);
fill(arc((3,-3),(3,0),(0,-3),CCW)--(3,-3)--cycle,white);
fill(arc((-3,-3),(0,-3),(-3,0),CCW)--(-3,-3)--cycle,white);
fill(arc((-3,3),(-3,0),(0,3),CCW)--(-3,3)--cycle,white);
draw(circle((3,3),3));
draw(circle((3,-3),3));
draw(circle((-3,-3),3));
draw(circle((-3,3),3));
draw((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle);
[/asy]
$\text{(A)}\ 7.7 \qquad \text{(B)}\ 12.1 \qquad \text{(C)}\ 17.2 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 27$
1987 Czech and Slovak Olympiad III A, 3
Let $f:(0,\infty)\to(0,\infty)$ be a function satisfying $f\bigl(xf(y)\bigr)+f\bigl(yf(x)\bigr)=2xy$ for all $x,y>0$. Show that $f(x) = x$ for all positive $x$.
2000 AMC 10, 11
Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
$ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 180\qquad \textbf{(E)}\ 231$
2019 USEMO, 6
Let $ABC$ be an acute scalene triangle with circumcenter $O$ and altitudes $\overline{AD}$, $\overline{BE}$, $\overline{CF}$. Let $X$, $Y$, $Z$ be the midpoints of $\overline{AD}$, $\overline{BE}$, $\overline{CF}$. Lines $AD$ and $YZ$ intersect at $P$, lines $BE$ and $ZX$ intersect at $Q$, and lines $CF$ and $XY$ intersect at $R$.
Suppose that lines $YZ$ and $BC$ intersect at $A'$, and lines $QR$ and $EF$ intersect at $D'$. Prove that the perpendiculars from $A$, $B$, $C$, $O$, to the lines $QR$, $RP$, $PQ$, $A'D'$, respectively, are concurrent.
[i]Ankan Bhattacharya[/i]
2015 Mexico National Olympiad, 1
Let $ABC$ be an acuted-angle triangle and let $H$ be it's orthocenter. Let $PQ$ be a segment through $H$ such that $P$ lies on $AB$ and $Q$ lies on $AC$ and such that $ \angle PHB= \angle CHQ$. Finally, in the circumcircle of $\triangle ABC$, consider $M$ such that $M$ is the mid point of the arc $BC$ that doesn't contain $A$. Prove that $MP=MQ$
Proposed by Eduardo Velasco/Marco Figueroa
1998 USAMO, 6
Let $n \geq 5$ be an integer. Find the largest integer $k$ (as a function of $n$) such that there exists a convex $n$-gon $A_{1}A_{2}\dots A_{n}$ for which exactly $k$ of the quadrilaterals $A_{i}A_{i+1}A_{i+2}A_{i+3}$ have an inscribed circle. (Here $A_{n+j} = A_{j}$.)
2017 AMC 10, 14
An integer $N$ is selected at random in the range $1\le N \le 2020.$ What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$?
$\textbf{(A)} \text{ }\frac{1}{5} \qquad \textbf{(B)} \text{ }\frac{2}{5} \qquad \textbf{(C)} \text{ }\frac{3}{5} \qquad \textbf{(D)} \text{ }\frac{4}{5} \qquad \textbf{(E)} \text{ 1}$
2021 Serbia Team Selection Test, P6
Let $S=\{1,2, \ldots ,10^{10}\}$. Find all functions $f:S \rightarrow S$, such that $$f(x+1)=f(f(x))+1 \pmod {10^{10}}$$ for each $x \in S$ (assume $f(10^{10}+1)=f(1)$).
2014 Contests, 1
In a bag there are $1007$ black and $1007$ white balls, which are randomly numbered $1$ to $2014$. In every step we draw one ball and put it on the table; also if we want to, we may choose two different colored balls from the table and put them in a different bag. If we do that we earn points equal to the absolute value of their differences. How many points can we guarantee to earn after $2014$ steps?
2019 Dutch IMO TST, 1
In each of the different grades of a high school there are an odd number of pupils. Each pupil has a best friend (who possibly is in a different grade). Everyone is the best friend of their best friend. In the upcoming school trip, every pupil goes to either Rome or Paris. Show that the pupils can be distributed over the two destinations in such a way that
(i) every student goes to the same destination as their best friend;
(ii) for each grade the absolute difference between the number of pupils that are going to Rome and that of those who are going to Paris is equal to $1$.
2012 China Second Round Olympiad, 3
Let $P_0 ,P_1 ,P_2 , ... ,P_n$ be $n+1$ points in the plane. Let $d$($d>0$) denote the minimal value of all the distances between any two points. Prove that
\[|P_0P_1|\cdot |P_0P_2|\cdot ... \cdot |P_0P_n|>(\frac{d}{3})^n\sqrt{(n+1)!}.\]
2010 Saudi Arabia BMO TST, 3
Let $a > 0$ be a real number and let $f : R \to R$ be a function satisfying $$f(x_1) + f(x_2) \ge a f(x_1 + x_2), \forall x_1 ,x_2 \in R.$$ Prove that $$f(x_1) + f(x_2) +(x_3) \ge \frac{3a^2}{a+2} f(x_1+ x_2 + x_3), \forall x_1 ,x_2,x_3 \in R$$.