Found problems: 85335
1985 Bulgaria National Olympiad, Problem 3
A pyramid $MABCD$ with the top-vertex $M$ is circumscribed about a sphere with center $O$ so that $O$ lies on the altitude of the pyramid. Each of the planes $ACM,BDM,ABO$ divides the lateral surface of the pyramid into two parts of equal areas. The areas of the sections of the planes $ACM$ and $ABO$ inside the pyramid are in ratio $(\sqrt2+2):4$. Determine the angle $\delta$ between the planes $ACM$ and $ABO$, and the dihedral angle of the pyramid at the edge $AB$.
2013 Today's Calculation Of Integral, 872
Let $n$ be a positive integer.
(1) For a positive integer $k$ such that $1\leq k\leq n$, Show that :
\[\int_{\frac{k-1}{2n}\pi}^{\frac{k}{2n}\pi} \sin 2nt\cos t\ dt=(-1)^{k+1}\frac{2n}{4n^2-1}(\cos \frac{k}{2n}\pi +\cos \frac{k-1}{2n}\pi).\]
(2) Find the area $S_n$ of the part expressed by a parameterized curve $C_n: x=\sin t,\ y=\sin 2nt\ (0\leq t\leq \pi).$
If necessary, you may use ${\sum_{k=1}^{n-1} \cos \frac{k}{2n}\pi =\frac 12(\frac{1}{\tan \frac{\pi}{4n}}-1})\ (n\geq 2).$
(3) Find $\lim_{n\to\infty} S_n.$
2008 Greece Team Selection Test, 1
Find all possible values of $a\in \mathbb{R}$ and $n\in \mathbb{N^*}$ such that $f(x)=(x-1)^n+(x-2)^{2n+1}+(1-x^2)^{2n+1}+a$
is divisible by $\phi (x)=x^2-x+1$
2000 APMO, 4
Let $n,k$ be given positive integers with $n>k$. Prove that:
\[ \frac{1}{n+1} \cdot \frac{n^n}{k^k (n-k)^{n-k}} < \frac{n!}{k! (n-k)!} < \frac{n^n}{k^k(n-k)^{n-k}} \]
2012 Sharygin Geometry Olympiad, 10
In a convex quadrilateral all sidelengths and all angles are pairwise different.
a) Can the greatest angle be adjacent to the greatest side and at the same time the smallest angle be adjacent to the smallest side?
b) Can the greatest angle be non-adjacent to the smallest side and at the same time the smallest angle be non-adjacent to the greatest side?
Gheorghe Țițeica 2025, P3
Consider the plane vectors $\overrightarrow{OA_1},\overrightarrow{OA_2},\dots ,\overrightarrow{OA_n}$ with $n\geq 3$. Suppose that the inequality $$\big|\overrightarrow{OA_1}+\overrightarrow{OA_2}+\dots +\overrightarrow{OA_n}\big|\geq \big|\pm\overrightarrow{OA_1}\pm\overrightarrow{OA_2}\pm\dots \pm\overrightarrow{OA_n}\big|$$ takes place for all choiches of the $\pm$ signs. Show that there exists a line $\ell$ through $O$ such that all points $A_1,A_2,\dots ,A_n$ are all on one side of $\ell$.
[i]Cristi Săvescu[/i]
1996 North Macedonia National Olympiad, 5
Find the greatest $n$ for which there exist $n$ lines in space, passing through a single point, such that any two of them form the same angle.
2005 Junior Balkan Team Selection Tests - Romania, 10
Let $k,r \in \mathbb N$ and let $x\in (0,1)$ be a rational number given in decimal representation \[ x = 0.a_1a_2a_3a_4 \ldots . \] Show that if the decimals $a_k, a_{k+r}, a_{k+2r}, \ldots$ are canceled, the new number obtained is still rational.
[i]Dan Schwarz[/i]
2021 May Olympiad, 3
In a year that has $365$ days, what is the maximum number of "Tuesday the $13$th" there can be?
Note: The months of April, June, September and November have $30$ days each, February has $28$ and all others have $31$ days.
2017 Tournament Of Towns, 4
All the sides of the convex hexagon $ABCDEF$ are equal. In addition, $AD = BE = CF$.
Prove that a circle can be inscribed into this hexagon.
[i](Boyan Obukhov)[/i]
1995 All-Russian Olympiad, 1
A freight train departed from Moscow at $x$ hours and $y$ minutes and arrived at Saratov at $y$ hours and $z$ minutes. The length of its trip was $z$ hours and $x$ minutes. Find all possible values of $x$.
[i]S. Tokarev[/i]
PEN H Problems, 35
Find all cubic polynomials $x^3 +ax^2 +bx+c$ admitting the rational numbers $a$, $b$ and $c$ as roots.
2000 AMC 10, 22
One morning each member of Angela's family drank an $ 8$-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 7$
1999 Brazil National Olympiad, 5
There are $n$ football teams in [i]Tumbolia[/i]. A championship is to be organised in which each team plays against every other team exactly once. Ever match takes place on a sunday and each team plays at most one match each sunday. Find the least possible positive integer $m_n$ for which it is possible to set up a championship lasting $m_n$ sundays.
2012 IMO Shortlist, C1
Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x,y)$ by either $(y+1,x)$ or $(x-1,x)$. Prove that she can perform only finitely many such iterations.
[i]Proposed by Warut Suksompong, Thailand[/i]
2021 USAMTS Problems, 2
Sydney the squirrel is at $(0, 0)$ and is trying to get to $(2021, 2022)$. She can move only by reflecting her position over any line that can be formed by connecting two lattice points, provided that the reflection puts her on another lattice point. Is it possible for Sydney to reach $(2021, 2022)$?
2024 ELMO Shortlist, C1.5
Let $m, n \ge 2$ be distinct positive integers. In an infinite grid of unit squares, each square is filled with exactly one real number so that
[list]
[*]In each $m \times m$ square, the sum of the numbers in the $m^2$ cells is equal.
[*]In each $n \times n$ square, the sum of the numbers in the $n^2$ cells is equal.
[*]There exist two cells in the grid that do not contain the same number.
[/list]
Let $S$ be the set of numbers that appear in at least one square on the grid. Find, in terms of $m$ and $n$, the least possible value of $|S|$.
[i]Kiran Reddy[/i]
2014 Nordic, 3
Find all nonnegative integers $a, b, c$ such that
$$\sqrt{a} + \sqrt{b} + \sqrt{c} = \sqrt{2014}.$$
1998 Junior Balkan Team Selection Tests - Romania, 3
Let $ n $ be a natural number. Find all integer numbers that can be written as
$$ \frac{1}{a_1} +\frac{2}{a_2} +\cdots +\frac{n}{a_n} , $$
where $ a_1,a_2,...,a_n $ are natural numbers.
2014 India IMO Training Camp, 3
In how many ways rooks can be placed on a $8$ by $8$ chess board such that every row and every column has at least one rook?
(Any number of rooks are available,each square can have at most one rook and there is no relation of attacking between them)
2025 Israel TST, P3
Let \( n \) be a positive integer. A graph on \( 2n - 1 \) vertices is given such that the size of the largest clique in the graph is \( n \). Prove that there exists a vertex that is present in every clique of size \( n\)
PEN D Problems, 12
Suppose that $m>2$, and let $P$ be the product of the positive integers less than $m$ that are relatively prime to $m$. Show that $P \equiv -1 \pmod{m}$ if $m=4$, $p^n$, or $2p^{n}$, where $p$ is an odd prime, and $P \equiv 1 \pmod{m}$ otherwise.
2016 Romania National Olympiad, 3
[b]a)[/b] Let be two nonzero complex numbers $ a,b. $ Show that the area of the triangle formed by the representations of the affixes $ 0,a,b $ in the complex plane is $ \frac{1}{4}\left| \overline{a} b-a\overline{b} \right| . $
[b]b)[/b] Let be an equilateral triangle $ ABC, $ its circumcircle $ \mathcal{C} , $ its circumcenter $ O, $ and two distinct points $ P_1,P_2 $ in the interior of $ \mathcal{C} . $ Prove that we can form two triangles with sides $ P_1A,P_1B,P_1C, $ respectively, $ P_2A,P_2B,P_2C, $ whose areas are equal if and only if $ OP_1=OP_2. $
2019 Saudi Arabia JBMO TST, 1
Let $a, b$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove that
$$\frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{c}{a}+\frac{a}{c} \ge 2\sqrt2 \left( \sqrt{\frac{1-a}{a}}+\sqrt{\frac{1-b}{b}}+\sqrt{\frac{1-c}{c}}\right)$$
LMT Theme Rounds, 7
Let $R(x)$ be a function that takes a natural number as input and returns a rectangle. $R(1)$ is known to have integer side lengths. Let $p(x)$ be the perimeter of $R(x)$ and let $a(x)$ be the area of $R(x)$. Suppose that $p(x+5)=6 p(x)$ for all $x$ in the domain of $R$ and that $a(x+2)=12a(x)$ for all $x> 6$ in the domain of $R$. For $x \leq 6$, $a(x+1)=a(x)+2$. Suppose $p(16)=1296$, and let the side lengths of $R(11)$ be $a$ and $b$ with $a\leq b$. Find the ordered pair $(a,b)$.
[i]Proposed by Matthew Weiss