Found problems: 85335
2011 Mongolia Team Selection Test, 3
Let $m$ and $n$ be positive integers such that $m>n$ and $m \equiv n \pmod{2}$. If $(m^2-n^2+1) \mid n^2-1$, then prove that $m^2-n^2+1$ is a perfect square.
(proposed by G. Batzaya, folklore)
2016 Iran Team Selection Test, 1
A real function has been assigned to every cell of an $n \times n$ table. Prove that a function can be assigned to each row and each column of this table such that the function assigned to each cell is equivalent to the combination of functions assigned to the row and the column containing it.
2010 Germany Team Selection Test, 1
Consider 2009 cards which are lying in sequence on a table. Initially, all cards have their top face white and bottom face black. The cards are enumerated from 1 to 2009. Two players, Amir and Ercole, make alternating moves, with Amir starting. Each move consists of a player choosing a card with the number $k$ such that $k < 1969$ whose top face is white, and then this player turns all cards at positions $k,k+1,\ldots,k+40.$ The last player who can make a legal move wins.
(a) Does the game necessarily end?
(b) Does there exist a winning strategy for the starting player?
[i]Also compare shortlist 2009, combinatorics problem C1.[/i]
2016 NZMOC Camp Selection Problems, 5
Find all polynomials $P(x)$ with real coefficients such that the polynomial $$Q(x) = (x + 1)P(x-1) -(x-1)P(x)$$ is constant.
2008 iTest Tournament of Champions, 5
While running from an unrealistically rendered zombie, Willy Smithers runs into a vacant lot in the shape of a square, $100$ meters on a side. Call the four corners of the lot corners $1$, $2$, $3$, and $4$, in clockwise order. For $k = 1, 2, 3, 4$, let $d_k$ be the distance between Willy and corner $k$. Let
(a) $d_1<d_2<d_4<d_3$,
(b) $d_2$ is the arithmetic mean of $d_1$ and $d_3$, and
(c) $d_4$ is the geometric mean of $d_2$ and $d_3$.
If $d_1^2$ can be written in the form $\dfrac{a-b\sqrt c}d$, where $a,b,c,$ and $d$ are positive integers, $c$ is square-free, and the greatest common divisor of $a$, $b$, and $d$ is $1,$ find the remainder when $a+b+c+d$ is divided by $2008$.
2011 Croatia Team Selection Test, 1
We define a sequence $a_n$ so that $a_0=1$ and
\[a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + d & \textrm{ otherwise. } \end{cases} \]
for all postive integers $n$.
Find all positive integers $d$ such that there is some positive integer $i$ for which $a_i=1$.
2005 All-Russian Olympiad Regional Round, 11.5
Prove that for any polynomial $P$ with integer coefficients and any natural number $k$ there exists a natural number $n$ such that $P(1) + P(2) + ...+ P(n)$ is divisible by $k$.
2017 Polish Junior Math Olympiad First Round, 7.
Let $a$ and $b$ be positive integers such that the prime number $a+b+1$ divides the integer $4ab-1$. Prove that $a=b$.
1955 AMC 12/AHSME, 46
The graphs of $ 2x\plus{}3y\minus{}6\equal{}0$, $ 4x\minus{}3y\minus{}6\equal{}0$, $ x\equal{}2$, and $ y\equal{}\frac{2}{3}$ intersect in:
$ \textbf{(A)}\ \text{6 points} \qquad
\textbf{(B)}\ \text{1 point} \qquad
\textbf{(C)}\ \text{2 points} \qquad
\textbf{(D)}\ \text{no points} \\
\textbf{(E)}\ \text{an unlimited number of points}$
1992 China Team Selection Test, 1
A triangle $ABC$ is given in the plane with $AB = \sqrt{7},$ $BC = \sqrt{13}$ and $CA = \sqrt{19},$ circles are drawn with centers at $A,B$ and $C$ and radii $\frac{1}{3},$ $\frac{2}{3}$ and $1,$ respectively. Prove that there are points $A',B',C'$ on these three circles respectively such that triangle $ABC$ is congruent to triangle $A'B'C'.$
2018 Miklós Schweitzer, 1
Let $S\subset \mathbb{R}$ be a closed set and $f:\mathbb{R}^{2n}\to \mathbb{R}$ be a continuous function. Define a graph $G$ as follows: Let $x$ be a vertex of $G$ iff $x\in \mathbb{R}^{n}$ and $f(x,x)\not\in S$, then connect the vertices $x$ and $y$ by an edge in $G$ iff $f(x,y)\in S$ or $f(y,x)\in S$. Show that the chromatic number of $G$ is countable.
2020 China Team Selection Test, 4
Show that the following equation has finitely many solutions $(t,A,x,y,z)$ in positive integers
$$\sqrt{t(1-A^{-2})(1-x^{-2})(1-y^{-2})(1-z^{-2})}=(1+x^{-1})(1+y^{-1})(1+z^{-1})$$
LMT Team Rounds 2010-20, 2020.S19
Let $ABC$ be a triangle such that such that $AB=14, BC=13$, and $AC=15$. Let $X$ be a point inside triangle $ABC$. Compute the minimum possible value of $(\sqrt{2}AX+BX+CX)^2$.
2016 Harvard-MIT Mathematics Tournament, 5
Nine pairwise noncongruent circles are drawn in the plane such that any two circles intersect twice. For each pair of circles, we draw the line through these two points, for a total of $\binom 92 = 36$ lines. Assume that all $36$ lines drawn are distinct. What is the maximum possible number of points which lie on at least two of the drawn lines?
1996 All-Russian Olympiad, 6
Three sergeants and several solders serve in a platoon. The sergeants take turns on duty. The commander has given the following orders:
(a) Each day, at least one task must be issued to a soldier.
(b) No soldier may have more than two task or receive more than one tasks in a single day.
(c) The lists of soldiers receiving tasks for two different days must not be the same.
(d) The first sergeant violating any of these orders will be jailed.
Can at least one of the sergeants, without conspiring with the others, give tasks according to these rules and avoid being jailed?
[i]M. Kulikov[/i]
1947 Moscow Mathematical Olympiad, 131
Calculate (without calculators, tables, etc.) with accuracy to $0.00001$ the product $\left(1-\frac{1}{10}\right)\left(1-\frac{1}{10^2}\right)...\left(1-\frac{1}{10^{99}}\right)$
2010 Harvard-MIT Mathematics Tournament, 8
Let $f(n)=\displaystyle\sum_{k=2}^\infty \dfrac{1}{k^n\cdot k!}.$ Calculate $\displaystyle\sum_{n=2}^\infty f(n)$.
1988 National High School Mathematics League, 5
In rectangular coordinate system, define that if and only if both $x$-axis and $y$-axis of a point are integers, we call it itegral point. $I$ is set of all lines, $M$ is set of lines that pass exactly one intengral point, $N$ is set of lines that pass no itengral point, $P$ is set of lines that pass infinitely many itengral points. Then, how many conclusions are right?
(1)$M\cup N\cup P=I$.
(2)$N\neq\varnothing$.
(3)$M\neq\varnothing$.
(4)$P\neq\varnothing$.
$\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}4\qquad$
1957 AMC 12/AHSME, 44
In triangle $ ABC$, $ AC \equal{} CD$ and $ \angle CAB \minus{} \angle ABC \equal{} 30^\circ$. Then $ \angle BAD$ is:
[asy]defaultpen(linewidth(.8pt));
unitsize(2.5cm);
pair A = origin;
pair B = (2,0);
pair C = (0.5,0.75);
pair D = midpoint(C--B);
draw(A--B--C--cycle);
draw(A--D);
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
label("$D$",D,NE);[/asy]$ \textbf{(A)}\ 30^\circ\qquad \textbf{(B)}\ 20^\circ\qquad \textbf{(C)}\ 22\frac {1}{2}^\circ\qquad \textbf{(D)}\ 10^\circ\qquad \textbf{(E)}\ 15^\circ$
2015 IMO Shortlist, A4
Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]for all real numbers $x$ and $y$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2007 Thailand Mathematical Olympiad, 13
Let $S = \{1, 2,..., 8\}$. How many ways are there to select two disjoint subsets of $S$?
2015 Peru Cono Sur TST, P4
In a small city there are $n$ bus routes, with $n > 1$, and each route has exactly $4$ stops. If any two routes have exactly one common stop, and each pair of stops belongs to exactly one route, find all possible values of $n$.
2010 Sharygin Geometry Olympiad, 2
Two points $A$ and $B$ are given. Find the locus of points $C$ such that triangle $ABC$ can be covered by a circle with radius $1$.
(Arseny Akopyan)
2019 Peru EGMO TST, 5
Define the sequence sequence $a_0,a_1, a_2,....,a_{2018}, a_{2019}$ of real numbers as follows:
$\bullet$ $a_0 = 1$.
$\bullet$ $a_{n + 1} = a_n - \frac{a_n^2}{2019}$ for $n = 0, 1, ...,2018$.
Prove that $a_{2019} < \frac12 <a_{2018}$.
1979 All Soviet Union Mathematical Olympiad, 283
Given $n$ points (in sequence)$ A_1, A_2, ... , A_n$ on a line. All the segments $A_1A_2$, $A_2A_3$,$ ...$, $A_{n-1}A_n$ are shorter than $1$. We need to mark $(k-1)$ points so that the difference of every two segments, with the ends in the marked points, is shorter than $1$. Prove that it is possible
a) for $k=3$,
b) for every $k$ less than $(n-1)$.