Found problems: 85335
1974 Miklós Schweitzer, 1
Let $ \mathcal{F}$ be a family of subsets of a ground set $ X$ such that $ \cup_{F \in \mathcal{F}}F=X$, and
(a) if $ A,B \in \mathcal{F}$, then $ A \cup B \subseteq C$ for some $ C \in \mathcal{F};$
(b) if $ A_n \in \mathcal{F} \;(n=0,1,...)\ , B \in \mathcal{F},$ and $ A_0 \subset A_1 \subset...,$ then, for some $ k \geq 0, \;A_n \cap B=A_k \cap B$ for all $ n \geq k$.
Show that there exist pairwise disjoint sets ${ X_{\gamma} \;( \gamma \in \Gamma}\ )$, with $ X= \cup \{ X_{\gamma} : \;\gamma \in \Gamma \ \},$ such that every $ X_{\gamma}$ is contained in some member of $ \mathcal{F}$, and every element of $ \mathcal{F}$ is contained in the union of finitely many $ X_{\gamma}$'s.
[i]A. Hajnal[/i]
2018 PUMaC Number Theory A, 7
Find the smallest positive integer $G$ such that there exist distinct positive integers $a, b, c$ with the following properties:
$\: \bullet \: \gcd(a, b, c) = G$.
$\: \bullet \: \text{lcm}(a, b) = \text{lcm}(a, c) = \text{lcm}(b, c)$.
$\: \bullet \: \frac{1}{a} + \frac{1}{b}, \frac{1}{a} + \frac{1}{c},$ and $\frac{1}{b} + \frac{1}{c}$ are reciprocals of integers.
$\: \bullet \: \gcd(a, b) + \gcd(a, c) + \gcd(b, c) = 16G$.
1991 Arnold's Trivium, 67
What is the dimension of the space of solutions continuous on $x^2+y^2\ge1$ of the problem
\[\Delta u=0\text{ for }x^2+y^2>1\]
\[\partial u/\partial n=0\text{ for }x^2+y^2=1\]
2002 National High School Mathematics League, 3
Function $f(x)=\frac{x}{1-2^x}-\frac{x}{2}$ is
$\text{(A)}$ an even function, not an odd function.
$\text{(B)}$ an odd function, not an even function.
$\text{(C)}$ an even function, also an odd function.
$\text{(D)}$ neither an even function, nor an odd function.
2005 Bulgaria Team Selection Test, 1
Let $ABC$ be an acute triangle. Find the locus of the points $M$, in the interior of $\bigtriangleup ABC$, such that $AB-FG= \frac{MF.AG+MG.BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to the lines $BC$ and $AC$, respectively.
2014 Postal Coaching, 1
Suppose $p,q,r$ are three distinct primes such that $rp^3+p^2+p=2rq^2+q^2+q$. Find all possible values of $pqr$.
2014 HMNT, 2
Let $ABC$ be a triangle with $\angle B = 90^o$. Given that there exists a point $D$ on $AC$ such that $AD = DC$ and $BD = BC$, compute the value of the ratio $\frac{AB}{BC}$ .
2021 OMpD, 1
Let $ABCDEF$ be a regular hexagon with sides $1m$ and $O$ as its center. Suppose that $OPQRST$ is a regular hexagon, so that segments $OP$ and $AB$ intersect at $X$ and segments $OT$ and $CD$ intersect at $Y$, as shown in the figure below. Determine the area of the pentagon $OXBCY$.
2021 IMO Shortlist, C5
Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right.
Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors.
[i]Proposed by Gurgen Asatryan, Armenia[/i]
2023 Harvard-MIT Mathematics Tournament, 10
Let $\zeta= e^{2\pi i/99}$ and $\omega e^{2\pi i/101}$. The polynomial $$x^{9999} + a_{9998}x^{9998} + ...+ a_1x + a_0$$ has roots $\zeta^m + \omega^n$ for all pairs of integers $(m, n)$ with $0 \le m < 99$ and $0 \le n < 101$. Compute $a_{9799} + a_{9800} + ...+ a_{9998}$.
2020 LMT Spring, 23
Let $\triangle ABC$ be a triangle such that $AB=AC=40$ and $BC=79.$ Let $X$ and $Y$ be the points on segments $AB$ and $AC$ such that $AX=5, AY=25.$ Given that $P$ is the intersection of lines $XY$ and $BC,$ compute $PX\cdot PY-PB\cdot PC.$
2009 AMC 10, 15
The figures $ F_1$, $ F_2$, $ F_3$, and $ F_4$ shown are the first in a sequence of figures. For $ n\ge3$, $ F_n$ is constructed from $ F_{n \minus{} 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $ F_{n \minus{} 1}$ had on each side of its outside square. For example, figure $ F_3$ has $ 13$ diamonds. How many diamonds are there in figure $ F_{20}$?
[asy]unitsize(3mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
path d=(1/2,0)--(0,sqrt(3)/2)--(-1/2,0)--(0,-sqrt(3)/2)--cycle;
marker m=marker(scale(5)*d,Fill);
path f1=(0,0);
path f2=(0,0)--(-1,1)--(1,1)--(1,-1)--(-1,-1);
path[] g2=(-1,1)--(-1,-1)--(0,0)^^(1,-1)--(0,0)--(1,1);
path f3=f2--(-2,-2)--(-2,0)--(-2,2)--(0,2)--(2,2)--(2,0)--(2,-2)--(0,-2);
path[] g3=g2^^(-2,-2)--(0,-2)^^(2,-2)--(1,-1)^^(1,1)--(2,2)^^(-1,1)--(-2,2);
path[] f4=f3^^(-3,-3)--(-3,-1)--(-3,1)--(-3,3)--(-1,3)--(1,3)--(3,3)--
(3,1)--(3,-1)--(3,-3)--(1,-3)--(-1,-3);
path[] g4=g3^^(-2,-2)--(-3,-3)--(-1,-3)^^(3,-3)--(2,-2)^^(2,2)--(3,3)^^
(-2,2)--(-3,3);
draw(f1,m);
draw(shift(5,0)*f2,m);
draw(shift(5,0)*g2);
draw(shift(12,0)*f3,m);
draw(shift(12,0)*g3);
draw(shift(21,0)*f4,m);
draw(shift(21,0)*g4);
label("$F_1$",(0,-4));
label("$F_2$",(5,-4));
label("$F_3$",(12,-4));
label("$F_4$",(21,-4));[/asy]$ \textbf{(A)}\ 401 \qquad \textbf{(B)}\ 485 \qquad \textbf{(C)}\ 585 \qquad \textbf{(D)}\ 626 \qquad \textbf{(E)}\ 761$
2001 Turkey MO (2nd round), 1
Find all ordered triples of positive integers $(x,y,z)$ such that
\[3^{x}+11^{y}=z^{2}\]
1962 AMC 12/AHSME, 33
The set of $ x$-values satisfying the inequality $ 2 \leq |x\minus{}1| \leq 5$ is:
$ \textbf{(A)}\ \minus{}4 \leq x \leq \minus{}1 \text{ or } 3 \leq x \leq 6 \qquad
\textbf{(B)}\ 3 \leq x \leq 6 \text{ or } \minus{}6 \leq x \leq \minus{}3 \qquad
\textbf{(C)}\ x \leq \minus{}1 \text{ or } x \geq 3 \qquad
\textbf{(D)}\ \minus{}1 \leq x \leq 3 \qquad
\textbf{(E)}\ \minus{}4 \leq x \leq 6$
2022 Israel TST, 3
A class has 30 students. To celebrate 'Tu BiShvat' each student chose some dried fruits out of $n$ different kinds. Say two students are friends if they both chose from the same type of fruit. Find the minimal $n$ so that it is possible that each student has exactly \(6\) friends.
Russian TST 2022, P1
A convex 51-gon is given. For each of its vertices and each diagonal that does not contain this vertex, we mark in red a point symmetrical to the vertex relative to the middle of the diagonal. Prove that strictly inside the polygon there are no more than 20400 red dots.
[i]Proposed by P. Kozhevnikov[/i]
2007 Hanoi Open Mathematics Competitions, 6
Let $P(x) = x^3 + ax^2 + bx + 1$ and $|P(x)| \leq 1$ for all x such that $|x| \leq 1$.
Prove that $|a| + |b| \leq 5$.
2017-IMOC, C1
On a blackboard , the 2016 numbers $\frac{1}{2016} , \frac{2}{2016} ,... \frac{2016}{2016}$ are written.
One can perfurm the following operation : Choose any numbers in the blackboard, say $a$ and$ b$ and replace them by $2ab-a-b+1$.
After doing 2015 operation , there will only be one number $t$ Onthe blackboard .
Find all possible values of $ t$.
2022 Brazil Team Selection Test, 3
Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right.
Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors.
[i]Proposed by Gurgen Asatryan, Armenia[/i]
1955 Miklós Schweitzer, 10
[b]10.[/b] Show that if a convex polyhedron has vertices of regular distribution and congruent faces, then it is regular. (A system of points is said to be of regular distribution if every point of the system can be transformed into any other point by congruent transformations mapping the system onto itself.) [b](G. 11)[/b]
2017 MMATHS, 4
In a triangle $ABC$, let $A_0$ be the point where the interior angle bisector of angle $A$ meets with side $BC$. Similarly define $B_0$ and $C_0$. Prove that $\angle B_0A_0C_0 = 90^o$ if and only if $\angle BAC = 120^o$.
1984 IMO Longlists, 22
In a permutation $(x_1, x_2, \dots , x_n)$ of the set $1, 2, \dots , n$ we call a pair $(x_i, x_j )$ discordant if $i < j$ and $x_i > x_j$. Let $d(n, k)$ be the number of such permutations with exactly $k$ discordant pairs. Find $d(n, 2)$ and $d(n, 3).$
2007 Tuymaada Olympiad, 2
Two polynomials $ f(x)=a_{100}x^{100}+a_{99}x^{99}+\dots+a_{1}x+a_{0}$ and $ g(x)=b_{100}x^{100}+b_{99}x^{99}+\dots+b_{1}x+b_{0}$ of degree $ 100$ differ from each other by a permutation of coefficients. It is known that $ a_{i}\ne b_{i}$ for $ i=0, 1, 2, \dots, 100$. Is it possible that $ f(x)\geq g(x)$ for all real $ x$?
2023 Germany Team Selection Test, 1
In a triangle $\triangle ABC$ with orthocenter $H$, let $BH$ and $CH$ intersect $AC$ and $AB$ at $E$ and $F$, respectively. If the tangent line to the circumcircle of $\triangle ABC$ passing through $A$ intersects $BC$ at $P$, $M$ is the midpoint of $AH$, and $EF$ intersects $BC$ at $G$, then prove that $PM$ is parallel to $GH$.
[i]Proposed by Sreejato Bhattacharya[/i]
2023 Taiwan TST Round 3, 5
Let $N$ be a positive integer. Kingdom Wierdo has $N$ castles, with at most one road between each pair of cities. There are at most four guards on each road. To cost down, the King of Wierdos makes the following policy:
(1) For any three castles, if there are roads between any two of them, then any of these roads cannot have four guards.
(2) For any four castles, if there are roads between any two of them, then for any one castle among them, the roads from it toward the other three castles cannot all have three guards.
Prove that, under this policy, the total number of guards on roads in Kingdom Wierdo is smaller than or equal to $N^2$.
[i]Remark[/i]: Proving that the number of guards does not exceed $cN^2$ for some $c > 1$ independent of $N$ will be scored based on the value of $c$.
[i]Proposed by usjl[/i]