Found problems: 85335
2022 Mexico National Olympiad, 5
Let $n>1$ be a positive integer and $d_1<d_2<\dots<d_m$ be its $m$ positive divisors, including $1$ and $n$. Lalo writes the following $2m$ numbers on a board:
\[d_1,d_2\dots, d_m,d_1+d_2,d_2+d_3,\dots,d_{m-1}+d_m,N \]
where $N$ is a positive integer. Afterwards, Lalo erases any number that is repeated (for example, if a number appears twice, he erases one of them). Finally, Lalo realizes that the numbers left on the board are exactly all the divisors of $N$. Find all possible values that $n$ can take.
1956 AMC 12/AHSME, 41
The equation $ 3y^2 \plus{} y \plus{} 4 \equal{} 2(6x^2 \plus{} y \plus{} 2)$ where $ y \equal{} 2x$ is satisfied by:
$ \textbf{(A)}\ \text{no value of }x \qquad\textbf{(B)}\ \text{all values of }x \qquad\textbf{(C)}\ x \equal{} 0\text{ only}$
$ \textbf{(D)}\ \text{all integral values of }x\text{ only} \qquad\textbf{(E)}\ \text{all rational values of }x\text{ only}$
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 $k\in \mathbb{N}$:
$a_{1}a_{2}...a_{k}\mid a_{N+1}a_{N+2}...a_{N+k}$