Found problems: 85335
1986 All Soviet Union Mathematical Olympiad, 432
Given $30$ equal cups with milk. An elf tries to make the amount of milk equal in all the cups. He takes a pair of cups and aligns the milk level in two cups. Can there be such an initial distribution of milk in the cups, that the elf will not be able to achieve his goal in a finite number of operations?
2009 Irish Math Olympiad, 2
For any positive integer $n$ define $$E(n)=n(n+1)(2n+1)(3n+1)\cdots (10n+1).$$
Find the greatest common divisor of $E(1),E(2),E(3),\dots ,E(2009).$
2011 Tokyo Instutute Of Technology Entrance Examination, 2
For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$.
(1) Find the minimum value of $f(x)$.
(2) Evaluate $\int_0^1 f(x)\ dx$.
[i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]
1994 Tournament Of Towns, (401) 3
Let $O$ be a point inside a convex polygon $A_1A_2... A_n$ such that $$\angle OA_1A_n \le \angle OA_1A_2, \angle OA_2A_1 \le \angle OA_2A_3, ..., \angle OA_{n-1}A_{n-2} \le \angle OA_{n-1}A_n, \angle OA_nA_{n-1} \le \angle OA_nA_1$$ and all of these angles are acute. Prove that $O$ is the centre of the circle inscribed in the polygon.
(V Proizvolov)
2021 Vietnam TST, 2
In a board of $2021 \times 2021$ grids, we pick $k$ unit squares such that every picked square shares vertice(s) with at most $1$ other picked square. Determine the maximum of $k$.
1991 IMO Shortlist, 7
$ ABCD$ is a terahedron: $ AD\plus{}BD\equal{}AC\plus{}BC,$ $ BD\plus{}CD\equal{}BA\plus{}CA,$ $ CD\plus{}AD\equal{}CB\plus{}AB,$ $ M,N,P$ are the mid points of $ BC,CA,AB.$ $ OA\equal{}OB\equal{}OC\equal{}OD.$ Prove that $ \angle MOP \equal{} \angle NOP \equal{}\angle NOM.$
2005 China Team Selection Test, 1
Triangle $ABC$ is inscribed in circle $\omega$. Circle $\gamma$ is tangent to $AB$ and $AC$ at points $P$ and $Q$ respectively. Also circle $\gamma$ is tangent to circle $\omega$ at point $S$. Let the intesection of $AS$ and $PQ$ be $T$. Prove that $\angle{BTP}=\angle{CTQ}$.
2023 Harvard-MIT Mathematics Tournament, 31
Let $$P=\prod_{i=0}^{2016} (i^3-i-1)^2.$$ The remainder when $P$ is divided by the prime $2017$ is not zero. Compute this remainder.
2017-2018 SDML (Middle School), 14
Amy made a list of every possible distinct five-digit positive integer that can be formed using each of the digits $1, 2, 3, 4,$ and $5$ exactly once in each integer. What is the sum of the integers on Amy's list?
$\mathrm{(A) \ } 3000000 \qquad \mathrm{(B) \ } 3600000 \qquad \mathrm {(C) \ } 3999960 \qquad \mathrm{(D) \ } 3999990 \qquad \mathrm{(E) \ } 5999940$
2004 Switzerland Team Selection Test, 7
The real numbers $a,b,c,d$ satisfy the equations:
$$\begin{cases} a =\sqrt{45-\sqrt{21-a}} \\ b =\sqrt{45+\sqrt{21-b}}\\ c =\sqrt{45-\sqrt{21+c}}\ \\ d=\sqrt{45+\sqrt{21+d}} \end {cases}$$
Prove that $abcd = 2004$.
1990 Nordic, 4
It is possible to perform three operations $f, g$, and $h$ for positive integers: $f(n) = 10n, g(n) = 10n + 4$, and $h(2n) = n$; in other words, one may write $0$ or $4$ in the end of the number and one may divide an even number by $2$. Prove: every positive integer can be constructed starting from $4$ and performing a finite number of the operations $f,
g,$ and $h$ in some order.
2014 Taiwan TST Round 1, 2
A triangle has side lengths $a$, $b$, $c$, and the altitudes have lengths $h_a$, $h_b$, $h_c$. Prove that
\[ \left( \frac{a}{h_a} \right)^2
+ \left( \frac{b}{h_b} \right)^2
+ \left( \frac{c}{h_c} \right)^2 \ge 4. \]
Kyiv City MO Juniors 2003+ geometry, 2020.9.41
The points $A, B, C, D$ are selected on the circle as followed so that $AB = BC = CD$. Bisectors of $\angle ABD$ and $\angle ACD$ intersect at point $E$. Find $\angle ABC$, if it is known that $AE \parallel CD$.
2024 ELMO Shortlist, C3
Let $n$ and $k$ be positive integers and $G$ be a complete graph on $n$ vertices. Each edge of $G$ is colored one of $k$ colors such that every triangle consists of either three edges of the same color or three edges of three different colors. Furthermore, there exist two different-colored edges. Prove that $n\le(k-1)^2$.
[i]Linus Tang[/i]
2021 Taiwan TST Round 1, 4
Let $n$ be a positive integer. For each $4n$-tuple of nonnegative real numbers $a_1,\ldots,a_{2n}$, $b_1,\ldots,b_{2n}$ that satisfy $\sum_{i=1}^{2n}a_i=\sum_{j=1}^{2n}b_j=n$, define the sets
\[A:=\left\{\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:i\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\},\]
\[B:=\left\{\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:j\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\}.\]
Let $m$ be the minimum element of $A\cup B$. Determine the maximum value of $m$ among those derived from all such $4n$-tuples $a_1,\ldots,a_{2n},b_1,\ldots,b_{2n}$.
[I]Proposed by usjl.[/i]
2024 Poland - Second Round, 4
Let $n$ be a positive integer. A regular hexagon $ABCDEF$ with side length $n$ is partitioned into $6n^2$ equilateral triangles with side length $1$. The hexagon is covered by $3n^2$ rhombuses with internal angles $60^{\circ}$ and $120^{\circ}$ such that each rhombus covers exactly two triangles and every triangle is covered by exactly one rhombus. Show that the diagonal $AD$ divides in half exactly $n$ rhombuses.
2017 Assam Mathematics Olympiad, 1
1)$k, l, m\in\mathbb{N}$
$2^{k+l} +2^{l+m}+2^{m+k}\le 2^{k+l+m+1} +1$
[color=#00f]Moved to HSO. ~ oVlad[/color]
2023 USEMO, 5
Let $n \ge 2$ be an integer. A cube of size $n \times n \times n$ is dissected into $n^3$ unit cubes. A nonzero real number is written at the center of each unit cube so that the sum of the $n^2$ numbers in each slab of size $1 \times n \times n$, $n \times 1 \times n$, or $n \times n \times 1$ equals zero. (There are a total of $3n$ such slabs, forming three groups of $n$ slabs each such that slabs in the same group are parallel and slabs in different groups are perpendicular.)
Could it happen that some plane in three-dimensional space separates the positive and the negative written numbers? (The plane should not pass through any of the numbers.)
[i]Nikolai Beluhov[/i]
2007 AIME Problems, 2
Find the number of ordered triple $(a,b,c)$ where $a$, $b$, and $c$ are positive integers, $a$ is a factor of $b$, $a$ is a factor of $c$, and $a+b+c=100$.
2009 IMC, 1
Suppose that $f,g:\mathbb{R}\to \mathbb{R}$ satisfying
\[ f(r)\le g(r)\quad \forall r\in \mathbb{Q} \]
Does this imply $f(x)\le g(x)\quad \forall x\in \mathbb{R}$ if
[list]
(a)$f$ and $g$ are non-decreasing ?
(b)$f$ and $g$ are continuous?[/list]
2008 China Team Selection Test, 1
Given a rectangle $ ABCD,$ let $ AB \equal{} b, AD \equal{} a ( a\geq b),$ three points $ X,Y,Z$ are put inside or on the boundary of the rectangle, arbitrarily. Find the maximum of the minimum of the distances between any two points among the three points. (Denote it by $ a,b$)
2018 AMC 12/AHSME, 9
What is \[ \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) ? \]
$
\textbf{(A) }100,100 \qquad
\textbf{(B) }500,500\qquad
\textbf{(C) }505,000 \qquad
\textbf{(D) }1,001,000 \qquad
\textbf{(E) }1,010,000 \qquad
$
2024 Serbia Team Selection Test, 6
In the plane, there is a figure in the form of an $L$-tromino, which is composed of $3$ unit squares, which we will denote by $\Phi_0$. On every move, we choose an arbitrary straight line in the plane and using it we construct a new figure. The $\Phi_n$, obtained in the $n$-th move, is obtained as the union of the figure $\Phi_{n-1}$ and its axial reflection with respect to the chosen line. Also, for the move to be valid, it is necessary that the surface of the newly obtained piece to be twice as large as the previous one. Is it possible to cover the whole plane in that process?
2024 Brazil EGMO TST, 3
Consider 90 distinct positive integers. Show that there exist two of them whose least common multiple is greater than 2024.
2021 Science ON grade VI, 3
Consider positive integers $a<b$ and the set $C\subset\{a,a+1,a+2,\dots ,b-2,b-1,b\}$. Suppose $C$ has more than $\frac{b-a+1}{2}$ elements. Prove that there are two elements $x,y\in C$ that satisfy $x+y=a+b$.
[i] (From "Radu Păun" contest, Radu Miculescu)[/i]