Found problems: 85335
2010 AMC 12/AHSME, 23
Monic quadratic polynomials $ P(x)$ and $ Q(x)$ have the property that $ P(Q(x))$ has zeroes at $ x\equal{}\minus{}23,\minus{}21,\minus{}17, \text{and} \minus{}15$, and $ Q(P(x))$ has zeroes at $ x\equal{}\minus{}59, \minus{}57, \minus{}51, \text{and} \minus{}49$. What is the sum of the minimum values of $ P(x)$ and $ Q(x)$?
$ \textbf{(A)}\ \text{\minus{}100} \qquad \textbf{(B)}\ \text{\minus{}82} \qquad \textbf{(C)}\ \text{\minus{}73} \qquad \textbf{(D)}\ \text{\minus{}64} \qquad \textbf{(E)}\ 0$
2022 District Olympiad, P1
Let $e$ be the identity of monoid $(M,\cdot)$ and $a\in M$ an invertible element. Prove that
[list=a]
[*]The set $M_a:=\{x\in M:ax^2a=e\}$ is nonempty;
[*]If $b\in M_a$ is invertible, then $b^{-1}\in M_a$ if and only if $a^4=e$;
[*]If $(M_a,\cdot)$ is a monoid, then $x^2=e$ for all $x\in M_a.$
[/list]
[i]Mathematical Gazette[/i]
2015 Paraguayan Mathematical Olympiad, Problem 2
Consider all sums that add up to $2015$. In each sum, the addends are consecutive positive integers, and all sums have less than $10$ addends. How many such sums are there?
2006 Singapore Team Selection Test, 1
In the plane containing a triangle $ABC$, points $A'$, $B'$ and $C'$ distinct from the vertices of $ABC$ lie on the lines $BC$, $AC$ and $AB$ respectively such that $AA'$, $BB'$ and $CC'$ are concurrent at $G$ and $AG/GA' = BG/GB' = CG/GC'$.
Prove that $G$ is the centroid of $ABC$.
2010 Dutch IMO TST, 5
The polynomial $A(x) = x^2 + ax + b$ with integer coefficients has the following property:
for each prime $p$ there is an integer $k$ such that $A(k)$ and $A(k + 1)$ are both divisible by $p$.
Proof that there is an integer $m$ such that $A(m) = A(m + 1) = 0$.
2010 Contests, 1
Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$.
[i]Carl Lian and Brian Hamrick.[/i]
2023 Auckland Mathematical Olympiad, 4
Which digit must be substituted instead of the star so that the following large number $$\underbrace{66...66}_{2023} \star \underbrace{55...55}_{2023}$$ is divisible by $7$?
2023 Canadian Mathematical Olympiad Qualification, 7
(a) Let $u$, $v$, and $w$ be the real solutions to the equation $x^3 - 7x + 7 = 0$. Show that there exists a quadratic polynomial $f$ with rational coefficients such that $u = f(v)$, $v = f(w)$, and $w = f(u)$.
(b) Let $u$, $v$, and $w$ be the real solutions to the equation $x^3 -7x+4 = 0$. Show that there does not exist a quadratic polynomial $f $with rational coefficients such that $u = f(v)$, $v = f(w)$, and $w = f(u)$.
2015 Dutch BxMO/EGMO TST, 3
Let $n \ge 2$ be a positive integer. Each square of an $n\times n$ board is coloured red or blue. We put dominoes on the board, each covering two squares of the board. A domino is called [i]even [/i] if it lies on two red or two blue squares and [i]colourful [/i] if it lies on a red and a blue square. Find the largest positive integer $k$ having the following property: regardless of how the red/blue-colouring of the board is done, it is always possible to put $k$ non-overlapping dominoes on the board that are either all [i]even [/i] or all [i]colourful[/i].
2013 International Zhautykov Olympiad, 2
Given convex hexagon $ABCDEF$ with $AB \parallel DE$, $BC \parallel EF$, and $CD \parallel FA$ . The distance between the lines $AB$ and $DE$ is equal to the distance between the lines $BC$ and $EF$ and to the distance between the lines $CD$ and $FA$. Prove that the sum $AD+BE+CF$ does not exceed the perimeter of hexagon $ABCDEF$.
2016 Hong Kong TST, 1
During a school year 44 competitions were held. Exactly 7 students won in each of the competition. For any two competitions, there exists exactly 1 student who won both competitions. Is it true that there exists a student who won all the competitions?
Geometry Mathley 2011-12, 10.3
Let $ABC$ be a triangle inscribed in a circle $(O)$. d is the tangent at $A$ of $(O), P$ is an arbitrary point in the plane. $D,E, F$ are the projections of $P$ on $BC,CA,AB$. Let $DE,DF$ intersect the line $d$ at $M,N$ respectively. The circumcircle of triangle $DEF$ meets $CA,AB$ at $K,L$ distinct from $E, F$. Prove that $KN$ meets $LM$ at a point on the circumcircle of triangle $DEF$.
Trần Quang Hùng
1990 AIME Problems, 12
A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form
\[ a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6}, \]
where $a$, $b$, $c$, and $d$ are positive integers. Find $a + b + c + d$.
2018 PUMaC Live Round, 2.2
Let $ABC$ be a triangle with side lengths $13,14,15$. The points on the interior of $ABC$ with distance at least $1$ from each side are shaded. The area of the shaded region can be written in simplest form as $\tfrac{m}{n}$. Find $m+n$.
2005 Bundeswettbewerb Mathematik, 1
Two players $A$ and $B$ have one stone each on a $100 \times 100$ chessboard. They move their stones one after the other, and a move means moving one's stone to a neighbouring field (horizontally or vertically, not diagonally). At the beginning of the game, the stone of $A$ lies in the lower left corner, and the one of $B$ in the lower right corner. Player $A$ starts.
Prove: Player $A$ is, independently from that what $B$ does, able to reach, after finitely many steps, the field $B$'s stone is lying on at that moment.
1985 ITAMO, 4
A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly 1/1985.
[asy]
size(200);
pair A=(0,1), B=(1,1), C=(1,0), D=origin;
draw(A--B--C--D--A--(1,1/6));
draw(C--(0,5/6)^^B--(1/6,0)^^D--(5/6,1));
pair point=( 0.5 , 0.5 );
//label("$A$", A, dir(point--A));
//label("$B$", B, dir(point--B));
//label("$C$", C, dir(point--C));
//label("$D$", D, dir(point--D));
label("$1/n$", (11/12,1), N, fontsize(9));[/asy]
2010 Kosovo National Mathematical Olympiad, 1
Solve the equation
$|x+1|-|x-1|=2$.
2011 China Second Round Olympiad, 4
If ${\cos^5 x}-{\sin^5 x}<7({\sin^3 x}-{\cos ^3 x}) $ (for $x\in [ 0,2\pi) $), then find the range of $x$.
1980 Poland - Second Round, 1
Students $ A $ and $ B $ play according to the following rules: student $ A $ selects a vector $ \overrightarrow{a_1} $ of length 1 in the plane, then student $ B $ gives the number $ s_1 $, equal to $ 1 $ or $ - $1; then the student $ A $ chooses a vector $ \overrightarrow{a_1} $ of length $ 1 $, and in turn the student $ B $ gives a number $ s_2 $ equal to $ 1 $ or $ -1 $ etc. $ B $ wins if for a certain $ n $ vector $ \sum_{j=1}^n \varepsilon_j \overrightarrow{a_j} $ has a length greater than the number $ R $ determined before the start of the game. Prove that student $B$ can achieve a win in no more than $R^2 + 1$ steps regardless of partner $A$'s actions.
1987 Iran MO (2nd round), 1
Calculate the product:
\[A=\sin 1^\circ \times \sin 2^\circ \times \sin 3^\circ \times \cdots \times \sin 89^\circ\]
2010 Junior Balkan MO, 4
A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares.
Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.
2017 China Team Selection Test, 4
Show that there exists a degree $58$ monic polynomial
$$P(x) = x^{58} + a_1x^{57} + \cdots + a_{58}$$
such that $P(x)$ has exactly $29$ positive real roots and $29$ negative real roots and that $\log_{2017} |a_i|$ is a positive integer for all $1 \leq i \leq 58$.
2013 Iran Team Selection Test, 8
Find all Arithmetic progressions $a_{1},a_{2},...$ of natural numbers for which there exists natural number $N>1$ such that for every $k\in \mathbb{N}$:
$a_{1}a_{2}...a_{k}\mid a_{N+1}a_{N+2}...a_{N+k}$
1999 IMO Shortlist, 5
Find all the functions $f: \mathbb{R} \to\mathbb{R}$ such that
\[f(x-f(y))=f(f(y))+xf(y)+f(x)-1\]
for all $x,y \in \mathbb{R} $.
1996 All-Russian Olympiad, 5
At the vertices of a cube are written eight pairwise distinct natural numbers, and on each of its edges is written the greatest common divisor of the numbers at the endpoints of the edge. Can the sum of the numbers written at the vertices be the same as the sum of the numbers written at the edges?
[i]A. Shapovalov[/i]