Found problems: 85335
2014 PUMaC Geometry B, 7
Consider quadrilateral $ABCD$. It is given that $\angle DAC=70^\circ$, $\angle BAC=40^\circ$, $\angle BDC=20^\circ$, $\angle CBD=35^\circ$. Let $P$ be the intersection of $AC$ and $BD$. Find $\angle BPC$.
2011 Saudi Arabia BMO TST, 3
Consider a triangle $ABC$. Let $A_1$ be the symmetric point of $A$ with respect to the line $BC$, $B_1$ the symmetric point of $B$ with respect to the line $CA$, and $C_1$ the symmetric point of $C$ with respect to the line $AB$. Determine the possible set of angles of triangle $ABC$ for which $A_1B_1C_1$ is equilateral.
2023 Azerbaijan National Mathematical Olympiad, 3
Find all the real roots of the system of equations:
$$ \begin{cases} x^3+y^3=19 \\ x^2+y^2+5x+5y+xy=12 \end{cases} $$
2016 Peru IMO TST, 6
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.
2015 Regional Olympiad of Mexico Center Zone, 5
In the triangle $ABC$, we have that $M$ and $N$ are points on $AB$ and $AC$, respectively, such that $BC$ is parallel to $MN$. A point $D$ is chosen inside the triangle $AMN$. Let $E$ and $F$ be the points of intersection of $MN$ with $BD$ and $CD$, respectively. Show that the line joining the centers of the circles circumscribed to the triangles $DEN$ and $DFM$ is perpendicular to $AD$.
2011 Iran Team Selection Test, 8
Let $p$ be a prime and $k$ a positive integer such that $k \le p$. We know that $f(x)$ is a polynomial in $\mathbb Z[x]$ such that for all $x \in \mathbb{Z}$ we have $p^k | f(x)$.
[b](a)[/b] Prove that there exist polynomials $A_0(x),\ldots,A_k(x)$ all in $\mathbb Z[x]$ such that
\[ f(x)=\sum_{i=0}^{k} (x^p-x)^ip^{k-i}A_i(x),\]
[b](b)[/b] Find a counter example for each prime $p$ and each $k > p$.
2009 Bosnia And Herzegovina - Regional Olympiad, 1
Find all triplets of integers $(x,y,z)$ such that $$xy(x^2-y^2)+yz(y^2-z^2)+zx(z^2-x^2)=1$$
2022 VIASM Summer Challenge, Problem 2
Give $P(x) = {x^{2022}} + {a_{2021}}{x^{2021}} + ... + {a_1}x + 1$ is a polynomial with real coefficents.
a) Assume that $2021a_{2021}^2 - 4044{a_{2020}} < 0.$ Prove that: $P(x)$ can't have $2022$ real roots.
b) Assume that $a_1^2 + a_2^2 + ... + a_{2021}^2 \le \frac{4}{{2021}}.$ Prove that: $P(x)\ge 0$, for all $x\in \mathbb{R}.$
2008 AIME Problems, 7
Let $ r$, $ s$, and $ t$ be the three roots of the equation
\[ 8x^3\plus{}1001x\plus{}2008\equal{}0.\]Find $ (r\plus{}s)^3\plus{}(s\plus{}t)^3\plus{}(t\plus{}r)^3$.
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.
2019 IFYM, Sozopol, 4
The inscribed circle of an acute $\Delta ABC$ is tangent to $AB$ and $AC$ in $K$ and $L$ respectively. The altitude $AH$ intersects the angle bisectors of $\angle ABC$ and $\angle ACB$ in $P$ and $Q$ respectively. Prove that the middle point $M$ of $AH$ lies on the radical axis of the circumscribed circles of $\Delta KPB$ and $\Delta LQC$.
2024 Princeton University Math Competition, 8
Let $\triangle ABC$ be a triangle. Let points $D$ and $E$ be on segment $BC$ in the order $B, D, E, C$ such that $\angle BAD =$ $\angle DAE = \angle EAC.$ Suppose also that $BD = F_{2024}, DE = F_{2025}, EC = F_{2027},$ where $F_k$ is the $k$th Fibonacci number where $F_1 = F_2 = 1.$ To the nearest degree, $\angle BAC$ is $n^\circ.$ Find $n.$
Russian TST 2017, P2
Prove that every rational number is representable as $x^4+y^4-z^4-t^4$ with rational $x,y,z,t$.
2006 China Western Mathematical Olympiad, 2
Find the smallest positive real $k$ satisfying the following condition: for any given four DIFFERENT real numbers $a,b,c,d$, which are not less than $k$, there exists a permutation $(p,q,r,s)$ of $(a,b,c,d)$, such that the equation $(x^{2}+px+q)(x^{2}+rx+s)=0$ has four different real roots.
1987 AIME Problems, 7
Let $[r,s]$ denote the least common multiple of positive integers $r$ and $s$. Find the number of ordered triples $(a,b,c)$ of positive integers for which $[a,b] = 1000$, $[b,c] = 2000$, and $[c,a] = 2000$
2011 LMT, 14
Let $L,E,T,M,$ and $O$ be digits that satisfy $LEET+LMT=TOOL.$
Given that $O$ has the value of $0,$ digits may be repeated, and $L\neq0,$ what is the value of the $4$-digit integer $ELMO?$