Found problems: 85335
2018 ELMO Shortlist, 2
Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$ holds for all integers $n>N.$
[i]Proposed by Carl Schildkraut[/i]
2023 Durer Math Competition Finals, 16
For the Dürer final results announcement, four loudspeakers are used to provide sound in the hall. However, there are only two sockets in the wall from which the power comes. To solve the problem, Ádám got two extension cords and two power strips. One plug can be plugged into an extension cord, and two plugs can be plugged into a power strip. Gábor, in his haste before the announcement of the results, quickly plugs the $8$ plugs into the $8$ holes. Every possible way of plugging has the same probability, and it is also possible for Gábor to plug something into itself. What is the probability that all $4$ speakers will have sound at the results announcement? For the solution, give the sum of the numerator and the denominator in the simplified form of the probability. A speaker sounds when it is plugged directly or indirectly into the wall.
2017 Brazil Team Selection Test, 2
Let $\tau(n)$ be the number of positive divisors of $n$. Let $\tau_1(n)$ be the number of positive divisors of $n$ which have remainders $1$ when divided by $3$. Find all positive integral values of the fraction $\frac{\tau(10n)}{\tau_1(10n)}$.
2001 Croatia National Olympiad, Problem 4
Let $S$ be a set of $100$ positive integers less than $200$. Prove that there exists a nonempty subset $T$ of $S$ the product of whose elements is a perfect square.
2011 Mediterranean Mathematics Olympiad, 2
Let $A$ be a finite set of positive reals, let $B = \{x/y\mid x,y\in A\}$ and let $C = \{xy\mid x,y\in A\}$.
Show that $|A|\cdot|B|\le|C|^2$.
[i](Proposed by Gerhard Woeginger, Austria)[/i]
2023 Germany Team Selection Test, 3
Let $f(x)$ be a monic polynomial of degree $2023$ with positive integer coefficients.
Show that for any sufficiently large integer $N$ and any prime number $p>2023N$, the product
\[f(1)f(2)\dots f(N)\]
is at most $\binom{2023}{2}$ times divisible by $p$.
[i]Proposed by Ashwin Sah[/i]
2014 International Zhautykov Olympiad, 3
Four segments divide a convex quadrilateral into nine quadrilaterals. The points of intersections of these segments lie on the diagonals of the quadrilateral (see figure). It is known that the quadrilaterals 1, 2, 3, 4 admit inscribed circles. Prove that the quadrilateral 5 also has an inscribed circle.
[asy]
pair A,B,C,D,E,F,G,H,I,J,K,L;
A=(-4.0,4.0);B=(-1.06,4.34);C=(-0.02,4.46);D=(4.14,4.93);E=(3.81,0.85);F=(3.7,-0.42);
G=(3.49,-3.05);H=(1.37,-2.88);I=(-1.46,-2.65);J=(-2.91,-2.52);K=(-3.14,-1.03);L=(-3.61,1.64);
draw(A--D);draw(D--G);draw(G--J);draw(J--A);
draw(A--G);draw(D--J);
draw(B--I);draw(C--H);draw(E--L);draw(F--K);
pair R,S,T,U,V;
R=(-2.52,2.56);S=(1.91,2.58);T=(-0.63,-0.11);U=(-2.37,-1.94);V=(2.38,-2.06);
label("1",R,N);label("2",S,N);label("3",T,N);label("4",U,N);label("5",V,N);
[/asy]
[i]Proposed by Nairi M. Sedrakyan, Armenia[/i]
2024 IFYM, Sozopol, 6
Let $P(x)$ be a polynomial in one variable with integer coefficients. Prove that the number of pairs $(m,n)$ of positive integers such that $2^n + P(n) = m!$, is finite.
2025 Thailand Mathematical Olympiad, 2
A school sent students to compete in an academic olympiad in $11$ differents subjects, each consist of $5$ students. Given that for any $2$ different subjects, there exists a student compete in both subjects. Prove that there exists a student who compete in at least $4$ different subjects.
1970 AMC 12/AHSME, 18
$\sqrt{3+2\sqrt{2}}-\sqrt{3-2\sqrt{2}}$ is equal to
$\textbf{(A) }2\qquad\textbf{(B) }2\sqrt{3}\qquad\textbf{(C) }4\sqrt{2}\qquad\textbf{(D) }\sqrt{6}\qquad \textbf{(E) }2\sqrt{2}$
2005 VJIMC, Problem 4
Let $(x_n)_{n\ge2}$ be a sequence of real numbers such that $x_2>0$ and $x_{n+1}=-1+\sqrt[n]{1+nx_n}$ for $n\ge2$. Find
(a) $\lim_{n\to\infty}x_n$,
(b) $\lim_{n\to\infty}nx_n$.
2025 Portugal MO, 4
Let $ABCD$ be a square with $2cm$ side length and with center $T$. A rhombus $ARTE$ is drawn where point $E$ lies on line $DC$. What is the area of $ARTE$?
2009 Peru IMO TST, 2
300 bureaucrats are split into three comissions of 100 people. Each two bureaucrats are either familiar to each other or non familiar to each other. Prove that there exists two bureaucrats from two distinct commissions such that the third commission contains either 17 bureaucrats familiar to both of them, or 17 bureaucrats familiar to none of them.
_________________________________________
This problem is taken from Russian Olympiad 2007-2008 district round 9.8
$ Tipe$
2024 LMT Fall, 4
A rhombus has vertices at $(0,0)$, $(6, 8)$, $(16, 8)$, and $(10, 0)$. A line with slope $m$ passes through the point $(3, 1)$ and splits the rhombus into $2$ regions of equal area. Find $m$.
2021-IMOC, C11
In an $m \times n$ grid, each square is either filled or not filled. For each square, its [i]value[/i] is defined as $0$ if it is filled and is defined as the number of neighbouring filled cells if it is not filled. Here, two squares are neighbouring if they share a common vertex or side. Let $f(m,n)$ be the largest total value of squares in the grid. Determine the minimal real constant $C$ such that $$\frac{f(m,n)}{mn} \le C$$holds for any positive integers $m,n$
[i]CSJL[/i]
2019 Math Prize for Girls Olympiad, 1
Let $A_1$, $A_2$, $\ldots\,$, $A_n$ be finite sets. Prove that
\[
\Bigl| \bigcup_{1 \le i \le n} A_i \Bigr|
\ge \frac{1}{2} \sum_{1 \le i \le n} \left| A_i \right|
- \frac{1}{6} \sum_{1 \le i < j \le n} \left| A_i \cap A_j \right| \, .
\]
Recall that if $S$ is a finite set, then its cardinality $|S|$ is the number of elements of $S$.
2020 Purple Comet Problems, 6
A given in finite geometric series with first term $a \ne 0$ and common ratio $2r$ sums to a value that is $6$ times the sum of an infi nite geometric series with first term $2a$ and common ratio $r$. Then $r = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2023-IMOC, G4
Given triangle $ABC$. $D$ is a point on $BC$. $AC$ meets $(ABD)$ again at $E$,and $AB$ meets $(ACD)$ again at $F$. $M$ is the midpoint of $EF$. $BC$ meets $(DEF)$ again at $P$. Prove that $\angle BAP = \angle MAC$.
2016 Saudi Arabia IMO TST, 1
Let $ABC$ be a triangle whose incircle $(I)$ touches $BC, CA, AB$ at $D, E, F$, respectively. The line passing through $A$ and parallel to $BC$ cuts $DE, DF$ at $M, N$, respectively. The circumcircle of triangle $DMN$ cuts $(I)$ again at $L$.
a) Let $K$ be the intersection of $N E$ and $M F$. Prove that $K$ is the orthocenter of the triangle $DMN$.
b) Prove that $A, K, L$ are collinear.
2025 CMIMC Combo/CS, 5
Consider a $12$-card deck containing all four suits of $2, 3,$ and $4.$ A [i]double[/i] is defined as two cards directly next to each other in the deck, with the same value. Suppose we scan the deck left to right, and whenever we encounter a double, we remove all the cards up to that point (including the double). Let $N$ denote the number of times we have to remove cards. What is the expected value of $N$?
2016 MMPC, 2
Let $s_1,s_2,s_3,s_4,...$ be a sequence (infinite list) of $1$s and $0$s. For example $1,0,1,0,1,0,...$, that is, $s_n=1$ if $n$ is odd and $s_n=0$ if $n$ is even, is such a sequence. Prove that it is possible to delete infinitely many terms in $s_1,s_2,s_3,s_4,...$ so that the resulting sequence is the original sequence. For the given example, one can delete $s_3,s_4,s_7,s_8,s_{11},s_{12},...$
2014 EGMO, 1
Determine all real constants $t$ such that whenever $a$, $b$ and $c$ are the lengths of sides of a triangle, then so are $a^2+bct$, $b^2+cat$, $c^2+abt$.
2018 Saint Petersburg Mathematical Olympiad, 3
Point $T$ lies on the bisector of $\angle B$ of acuteangled $\triangle ABC$. Circle $S$ with diameter $BT$ intersects $AB$ and $BC$ at points $P$ and $Q$. Circle, that goes through point $A$ and tangent to $S$ at $P$ intersects line $AC$ at $X$. Circle, that goes through point $C$ and tangent to $S$ at $Q$ intersects line $AC$ at $Y$. Prove, that $TX=TY$
2011 Sharygin Geometry Olympiad, 5
A line passing through vertex $A$ of regular triangle $ABC$ doesn’t intersect segment $BC$. Points $M$ and $N$ lie on this line, and $AM = AN = AB$ (point $B$ lies inside angle $MAC$). Prove that the quadrilateral formed by lines $AB, AC, BN, CM$ is cyclic.
2015 Postal Coaching, Problem 1
$O$ is the centre of the circumcircle of triangle $ABC$, and $M$ is its orthocentre. Point $A$ is reflected in the perpendicular bisector of the side $BC$,$ B$ is reflected in the perpendicular bisector of the side $CA$, and finally $C$ is reflected in the perpendicular bisector of the side $AB$. The images are denoted by $A_1, B_1, C_1$ respectively. Let $K$ be the centre of the inscribed circle of triangle $A_1B_1C_1$. Prove that $O$ bisects the line segment $MK$.