Found problems: 85335
2024 CMIMC Team, 7
In the national math league, there are $7$ teams. Their season is a round robin format, where each team plays other. Find the number of ways the games could go such that they have equal number of wins.
[i]Proposed by Ishin Shah[/i]
2011 ELMO Shortlist, 1
Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be points on segments $AB$, $BC$, $CD$, $DA$, respectively, and let $P$ be the intersection of $EG$ and $FH$. Given that quadrilaterals $HAEP$, $EBFP$, $FCGP$, $GDHP$ all have inscribed circles, prove that $ABCD$ also has an inscribed circle.
[i]Evan O'Dorney.[/i]
2015 JBMO Shortlist, A1
Let x; y; z be real numbers, satisfying the relations
$x \ge 20$
$y \ge 40$
$z \ge 1675$
x + y + z = 2015
Find the greatest value of the product P = $xy z$
2009 Princeton University Math Competition, 8
Consider $\triangle ABC$ and a point $M$ in its interior so that $\angle MAB = 10^\circ$, $\angle MBA = 20^\circ$, $\angle MCA = 30^\circ$ and $\angle MAC = 40^\circ$. What is $\angle MBC$?
2024 Iran MO (3rd Round), 4
For a given positive integer number $n$ find all subsets $\{r_0,r_1,\cdots,r_n\}\subset \mathbb{N}$ such that
$$
n^n+n^{n-1}+\cdots+1 | n^{r_n}+\cdots+ n^{r_0}.
$$
Proposed by [i]Shayan Tayefeh[/i]
2021 Brazil EGMO TST, 4
The [i][b]duchess[/b][/i] is a chess piece such that the duchess attacks all the cells in two of the four diagonals which she is contained(the directions of the attack can vary to two different duchesses). Determine the greatest integer $n$, such that we can put $n$ duchesses in a table $8\times 8$ and none duchess attacks other duchess.
Note: The attack diagonals can be "outside" the table; for instance, a duchess on the top-leftmost cell we can choose attack or not the main diagonal of the table $8\times 8$.
LMT Team Rounds 2021+, 4
Find the least positive integer ending in $7$ with exactly $12$ positive divisors.
2003 Croatia Team Selection Test, 1
Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.
2013 Harvard-MIT Mathematics Tournament, 6
Let triangle $ABC$ satisfy $2BC = AB+AC$ and have incenter $I$ and circumcircle $\omega$. Let $D$ be the intersection of $AI$ and $\omega$ (with $A, D$ distinct). Prove that $I$ is the midpoint of $AD$.
2002 Putnam, 1
Shanille O'Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly $50$ of her first $100$ shots?
1985 Traian Lălescu, 1.4
Let $ ABCD $ be a convex quadrilateral, and $ P $ be a point that isn't found on any of the lines formed by the sides of the quadrilateral. Prove that the centers of mass of the triangles $ PAB, PBC, PCD $ and $ PDA, $ form a parallelogram, and calculate the legths of its sides in terms of its diagonals.
1989 All Soviet Union Mathematical Olympiad, 492
$ABC$ is a triangle. $A' , B' , C'$ are points on the segments $BC, CA, AB$ respectively. $\angle B' A' C' = \angle A$ , $\frac{AC'}{C'B} = \frac{BA' }{A' C} = \frac{CB'}{B'A}$. Show that $ABC$ and $A'B'C'$ are similar.
1972 AMC 12/AHSME, 2
If a dealer could get his goods for $8\%$ less while keeping his selling price fixed, his profit, based on cost, would be increased to $(x+10)\%$ from his present profit of $x\%$, which is
$\textbf{(A) }12\%\qquad\textbf{(B) }15\%\qquad\textbf{(C) }30\%\qquad\textbf{(D) }50\%\qquad \textbf{(E) }75\%$
1998 Gauss, 21
Ten points are spaced equally around a circle. How many different chords can be formed by joining
any 2 of these points? (A chord is a straight line joining two points on the circumference of a circle.)
$\textbf{(A)}\ 9 \qquad \textbf{(B)}\ 45 \qquad \textbf{(C)}\ 17 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 55$
1987 Tournament Of Towns, (156) 7
Three triangles (blue, green and red) have a common interior point $M$. Prove that it is possible to choose one vertex from each triangle so that point $M$ is located inside the triangle formed by these selected vertices.
(Imre Barani, Hungary)
2015 China Western Mathematical Olympiad, 1
Let the integer $n \ge 2$ , and $x_1,x_2,\cdots,x_n $ be real numbers such that $\sum_{k=1}^nx_k$ be integer . $d_k=\underset{m\in {Z}}{\min}\left|x_k-m\right| $, $1\leq k\leq n$ .Find the maximum value of $\sum_{k=1}^nd_k$.
2010 Indonesia TST, 1
The integers $ 1,2,\dots,20$ are written on the blackboard. Consider the following operation as one step: [i]choose two integers $ a$ and $ b$ such that $ a\minus{}b \ge 2$ and replace them with $ a\minus{}1$ and $ b\plus{}1$[/i]. Please, determine the maximum number of steps that can be done.
[i]Yudi Satria, Jakarta[/i]
2021 AMC 12/AHSME Spring, 24
Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$. Circle $\Omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$. If $QR=3\sqrt3$ and $\angle QPR=60^\circ$, then the area of $\triangle PQR$ is $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. What is $a+b+c$?
$\textbf{(A) }110 \qquad \textbf{(B) }114 \qquad \textbf{(C) }118 \qquad \textbf{(D) }122\qquad \textbf{(E) }126$
2016 South African National Olympiad, 3
The inscribed circle of triangle $ABC$, with centre $I$, touches sides $BC$, $CA$ and $AB$ at $D$, $E$ and
$F$, respectively. Let $P$ be a point, on the same side of $FE$ as $A$, for which $\angle PFE = \angle BCA$
and $\angle PEF = \angle ABC$. Prove that $P$, $I$ and $D$ lie on a straight line.
2015 Regional Competition For Advanced Students, 2
Let $x$, $y$, and $z$ be positive real numbers with $x+y+z = 3$. Prove that at least one of the three numbers
$$x(x+y-z)$$
$$y(y+z-x)$$
$$z(z+x-y)$$
is less or equal $1$.
(Karl Czakler)
2009 Austria Beginners' Competition, 1
A positive integer number is written in red on each side of a square. The product of the two red numbers on the adjacent sides is written in green for each corner point. The sum of the green numbers is $40$. Which values are possible for the sum of the red numbers?
(G. Kirchner, University of Innsbruck)
2019 Jozsef Wildt International Math Competition, W. 45
Consider the complex numbers $a_1, a_2,\cdots , a_n$, $n \geq 2$. Which have the following properties:
[list]
[*] $|a_i|=1$ $\forall$ $i=1,2,\cdots , n$
[*] $\sum \limits_{k=1}^n arg(a_k)\leq \pi$
[/list]
Show that the inequality$$\left(n^2\cot \left(\frac{\pi}{2n}\right)\right)^{-1}\left |\sum \limits_{k=0}^n(-1)^k\left[3n^2-(8k+5)n+4k(k+1)\sigma_k\right]\right |\geq \sqrt{\left(1+\frac{1}{n}\right)^2\cot^2 \left(\frac{\pi}{2n}\right)}+16\left |\sum \limits_{k=0}^n(-1)^k\sigma_k\right |$$where $\sigma_0=1$, $\sigma_k=\sum \limits_{1\leq i_1\leq i_2\leq \cdots \leq i_k\leq n}a_{i_1}a_{i_2}\cdots a_{i_k}$, $\forall$ $k=1,2,\cdots , n$
2009 Belarus Team Selection Test, 3
Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which
\[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\]
Find the number of elements of the set $A_n$.
[i]Proposed by Vidan Govedarica, Serbia[/i]
2021 China Team Selection Test, 1
Let $ n(\ge2) $ be a positive integer. Find the minimum $ m $, so that there exists $x_{ij}(1\le i ,j\le n)$ satisfying:
(1)For every $1\le i ,j\le n, x_{ij}=max\{x_{i1},x_{i2},...,x_{ij}\} $ or $ x_{ij}=max\{x_{1j},x_{2j},...,x_{ij}\}.$
(2)For every $1\le i \le n$, there are at most $m$ indices $k$ with $x_{ik}=max\{x_{i1},x_{i2},...,x_{ik}\}.$
(3)For every $1\le j \le n$, there are at most $m$ indices $k$ with $x_{kj}=max\{x_{1j},x_{2j},...,x_{kj}\}.$
2020 Jozsef Wildt International Math Competition, W38
Let $(a_n)_{n\in\mathbb N}$ be a sequence, given by the recurrence:
$$ma_{n+1}+(m-2)a_n-a_{n-1}=0$$
where $m\in\mathbb R$ is a parameter and the first two terms of $a_n$ are fixed known real numbers. Find $m\in\mathbb R$, so that
$$\lim_{n\to\infty}a_n=0$$
[i]Proposed by Laurențiu Modan[/i]