Found problems: 85335
LMT Team Rounds 2021+, B8
Find the number of arithmetic sequences $a_1,a_2,a_3$ of three nonzero integers such that the sum of the terms in the sequence is equal to the product of the terms in the sequence.
[i]Proposed by Sammy Charney[/i]
1959 Putnam, B2
Let $c$ be a positive real number. Prove that $c$ can be expressed in infinitely many ways as a sum of infinitely many distinct terms selected from the sequence $\left( \frac{1}{10n} \right)_{n\in \mathbb{N}}$
1977 IMO Longlists, 54
If $0 \leq a \leq b \leq c \leq d,$ prove that
\[a^bb^cc^dd^a \geq b^ac^bd^ca^d.\]
2003 ITAMO, 2
A museum has the shape of a $n \times n$ square divided into $n^2$ rooms of the shape of a unit square $(n>1)$. Between every two adjacent rooms (i.e. sharing a wall) there is a door. A night guardian wants to organize an inspection journey through the museum according to the following rules. He starts from some room and, whenever he enters a room, he stays there for exactly one minute and then proceeds to another room. He is allowed to enter a room more than once, but at the end of his journey he must have spent exactly $k$ minutes in every room. Find all $n$ and $k$ for which it is possible to organize such a journey.
2002 National High School Mathematics League, 3
Function $f(x)=\frac{x}{1-2^x}-\frac{x}{2}$ is
$\text{(A)}$ an even function, not an odd function.
$\text{(B)}$ an odd function, not an even function.
$\text{(C)}$ an even function, also an odd function.
$\text{(D)}$ neither an even function, nor an odd function.
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 $1$?
$\textbf{(A)} \text{ }\frac{1}{5} \qquad \textbf{(B)} \text{ }\frac{2}{5} \qquad \textbf{(C)} \text{ }\frac{3}{5} \qquad \textbf{(D)} \text{ }\frac{4}{5} \qquad \textbf{(E)} \text{ 1}$