Found problems: 85335
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 $8\times 8$.
2023 Harvard-MIT Mathematics Tournament, 4
The cells of a $5\times5$ grid are each colored red, white, or blue. Sam starts at the bottom-left cell of the grid and walks to the top-right cell by taking steps one cell either up or to the right. Thus, he passes through $9$ cells on his path, including the start and end cells. Compute the number of colorings for which Sam is guaranteed to pass through a total of exactly $3$ red cells, exactly $3$ white cells, and exactly $3$ blue cells no matter which route he takes.
2006 Iran MO (3rd Round), 2
Find all real polynomials that \[p(x+p(x))=p(x)+p(p(x))\]
2009 May Olympiad, 2
Let $ABCD$ be a convex quadrilateral such that the triangle $ABD$ is equilateral and the triangle $BCD$ is isosceles, with $\angle C = 90^o$. If $E$ is the midpoint of the side $AD$, determine the measure of the angle $\angle CED$.
2005 Estonia National Olympiad, 2
Let $a, b$, and $n$ be integers such that $a + b$ is divisible by $n$ and $a^2 + b^2$ is divisible by $n^2$. Prove that $a^m + b^m$ is divisible by $n^m$ for all positive integers $m$.
2012 Princeton University Math Competition, A4 / B6
How many (possibly empty) sets of lattice points $\{P_1, P_2, ... , P_M\}$, where each point $P_i =(x_i, y_i)$ for $x_i
, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$, satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$ ? An infinite slope, e.g. $P_i$ is vertically above $P_j$ , does not count as positive.
2014 ELMO Shortlist, 11
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be a point inside $ABC$, so let the points $D, E, F$ be on $BC, AC, AB$ respectively so that the Miquel point of $DEF$ with respect to $ABC$ is $P$. Let the reflections of $D, E, F$ over the midpoints of the sides that they lie on be $R, S, T$. Let the Miquel point of $RST$ with respect to the triangle $ABC$ be $Q$. Show that $OP = OQ$.
[i]Proposed by Yang Liu[/i]
1977 IMO Longlists, 28
Let $n$ be an integer greater than $1$. Define
\[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\]
where $[z]$ denotes the largest integer less than or equal to $z$. Prove that
\[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]
1999 USAMTS Problems, 2
Let $N=111...1222...2$, where there are $1999$ digits of $1$ followed by $1999$ digits of $2$. Express $N$ as the product of four integers, each of them greater than $1$.
2018 CMIMC Number Theory, 2
Find all integers $n$ for which $(n-1)\cdot 2^n + 1$ is a perfect square.