Found problems: 85335
2002 National Olympiad First Round, 11
What is the coefficient of $x^5$ in the expansion of $(1 + x + x^2)^9$?
$
\textbf{a)}\ 1680
\qquad\textbf{b)}\ 882
\qquad\textbf{c)}\ 729
\qquad\textbf{d)}\ 450
\qquad\textbf{e)}\ 246
$
1996 French Mathematical Olympiad, Problem 3
(a) Let there be given a rectangular parallelepiped. Show that some four of its vertices determine a tetrahedron whose all faces are right triangles.
(b) Conversely, prove that every tetrahedron whose all faces are right triangles can be obtained by selecting four vertices of a rectangular parallelepiped.
(c) Now investigate such tetrahedra which also have at least two isosceles faces. Given the length $a$ of the shortest edge, compute the lengths of the other edges.
2017 Korea USCM, 5
Evaluate the following limit.
\[\lim_{n\to\infty} \sqrt{n} \int_0^\pi \sin^n x dx\]
1979 All Soviet Union Mathematical Olympiad, 272
Some numbers are written in the notebook. We can add to that list the arithmetic mean of some of them, if it doesn't equal to the number, already having been included in it. Let us start with two numbers, $0$ and $1$. Prove that it is possible to obtain :
a) $1/5$,
b) an arbitrary rational number between $0$ and $1$.
1993 Bundeswettbewerb Mathematik, 4
Given is a triangle $ABC$ with side lengths $a, b, c$ ($a = \overline{BC}$, $b = \overline{CA}$, $c = \overline{AB}$) and area $F$. The side $AB$ is extended beyond $A$ by a and beyond $B$ by $b$. Correspondingly, $BC$ is extended beyond $B$ and $C$ by $b$ and $c$, respectively. Eventually $CA$ is extended beyond $C$ and $A$ by $c$ and $a$, respectively. Connecting the outer endpoints of the extensions , a hexagon if formed with area $G$. Prove that $\frac{G}{F}>13$.
2011 Postal Coaching, 6
A positive integer is called [i]monotonic[/i] if when written in base $10$, the digits are weakly increasing. Thus $12226778$ is monotonic. Note that a positive integer cannot have first digit $0$. Prove that for every positive integer $n$, there is an $n$-digit monotonic number which is a perfect square.
1969 IMO Longlists, 5
$(BEL 5)$ Let $G$ be the centroid of the triangle $OAB.$
$(a)$ Prove that all conics passing through the points $O,A,B,G$ are hyperbolas.
$(b)$ Find the locus of the centers of these hyperbolas.
2018 Turkey EGMO TST, 1
Let $ABCD$ be a cyclic quadrilateral and $w$ be its circumcircle. For a given point $E$ inside $w$, $DE$ intersects $AB$ at $F$ inside $w$. Let $l$ be a line passes through $E$ and tangent to circle $AEF$. Let $G$ be any point on $l$ and inside the quadrilateral $ABCD$. Show that if $\angle GAD =\angle BAE$ and $\angle GCB + \angle GAB = \angle EAD + \angle AGD + \angle ABE$ then $BC$, $AD$ and $EG$ are concurrent.
1994 IMC, 1
a) Let $A$ be a $n\times n$, $n\geq 2$, symmetric, invertible matrix with real positive elements. Show that $z_n\leq n^2-2n$, where $z_n$ is the number of zero elements in $A^{-1}$.
b) How many zero elements are there in the inverse of the $n\times n$ matrix
$$A=\begin{pmatrix} 1&1&1&1&\ldots&1\\
1&2&2&2&\ldots&2\\
1&2&1&1&\ldots&1\\
1&2&1&2&\ldots&2\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\
1&2&1&2&\ldots&\ddots
\end{pmatrix}$$
2007 Mexico National Olympiad, 2
Given an equilateral $\triangle ABC$, find the locus of points $P$ such that $\angle APB=\angle BPC$.
2018 PUMaC Algebra A, 5
For $k \in \left \{ 0, 1, \ldots, 9 \right \},$ let $\epsilon_k \in \left \{-1, 1 \right \}$. If the minimum possible value of $\sum_{i = 1}^9 \sum_{j = 0}^{i -1} \epsilon_i \epsilon_j 2^{i + j}$ is $m$, find $|m|$.
2014 AMC 12/AHSME, 5
Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5:2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length o the square window?
[asy]
fill((0,0)--(25,0)--(25,25)--(0,25)--cycle,grey);
for(int i = 0; i < 4; ++i){
for(int j = 0; j < 2; ++j){
fill((6*i+2,11*j+3)--(6*i+5,11*j+3)--(6*i+5,11*j+11)--(6*i+2,11*j+11)--cycle,white);
}
}[/asy]
$\textbf{(A) }26\qquad\textbf{(B) }28\qquad\textbf{(C) }30\qquad\textbf{(D) }32\qquad\textbf{(E) }34$
2023 Chile Junior Math Olympiad, 3
Let $\vartriangle ABC$ be an equilateral triangle with side $1$. Four points are marked $P_1$, $P_2$, $P_3$, $P_4$ on side $AC$ and four points $Q_1$, $Q_2$, $Q_3$, $Q_4$ on side $AB$ (see figure) of such a way to generate $9$ triangles of equal area. Find the length of segment $AP_4$.
[img]https://cdn.artofproblemsolving.com/attachments/5/f/29243932262cb963b376244f4c981b1afe87c6.png[/img]
PS. Easier version of [url=https://artofproblemsolving.com/community/c6h3323141p30741525]2023 Chile NMO L2 P3[/url]
2014 Sharygin Geometry Olympiad, 8
Let $M$ be the midpoint of the chord $AB$ of a circle centered at $O$. Point $K$ is symmetric to $M$ with respect to $O$, and point $P$ is chosen arbitrarily on the circle. Let $Q$ be the intersection of the line perpendicular to $AB$ through $A$ and the line perpendicular to $PK$ through $P$. Let $H$ be the projection of $P$ onto $AB$. Prove that $QB$ bisects $PH$.
(Tran Quang Hung)
2016 Postal Coaching, 5
For even positive integer $n$ we put all numbers $1, 2, \cdots , n^2$ into the squares of an $n \times n$ chessboard (each number appears once and only once). Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that it is possible to have $\frac{S_1}{S_2}=\frac{39}{64}$.
1995 Korea National Olympiad, Problem 1
For any positive integer $m$,show that there exist integers $a,b$ satisfying
$\left | a \right |\leq m$, $ \left | b \right |\leq m$, $0< a+b\sqrt{2}\leq \frac{1+\sqrt{2}}{m+2}$
2019 European Mathematical Cup, 1
Every positive integer is marked with a number from the set $\{ 0,1,2\}$, according to the following rule:
$$\text{if a positive integer }k\text{ is marked with }j,\text{ then the integer }k+j\text{ is marked with }0.$$
Let $S$ denote the sum of marks of the first $2019$ positive integers. Determine the maximum possible value of $S$.
[i]Proposed by Ivan Novak[/i]
2023 CMIMC Integration Bee, 4
\[\int_0^\infty x e^{-\sqrt[3]{x}}\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2009 Sharygin Geometry Olympiad, 7
Given points $O, A_1, A_2, ..., A_n$ on the plane. For any two of these points the square of distance between them is natural number. Prove that there exist two vectors $\vec{x}$ and $\vec{y}$, such that for any point $A_i$, $\vec{OA_i }= k\vec{x}+l \vec{y}$, where $k$ and $l$ are some integer numbers.
(A.Glazyrin)
2019 BAMO, E/3
In triangle $\vartriangle ABC$, we have marked points $A_1$ on side $BC, B_1$ on side $AC$, and $C_1$ on side $AB$ so that $AA_1$ is an altitude, $BB_1$ is a median, and $CC_1$ is an angle bisector. It is known that $\vartriangle A_1B_1C_1$ is equilateral. Prove that $\vartriangle ABC$ is equilateral too.
(Note: A median connects a vertex of a triangle with the midpoint of the opposite side. Thus, for median $BB_1$ we know that $B_1$ is the midpoint of side $AC$ in $\vartriangle ABC$.)
2017 Dutch BxMO TST, 1
Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine
the minimum number of integers in a complete sequence of $n$ numbers.
May Olympiad L1 - geometry, 2008.4
Let $ABF$ be a right-angled triangle with $\angle AFB = 90$, a square $ABCD$ is externally to the triangle. If $FA = 6$, $FB = 8$ and $E$ is the circumcenter of the square $ABCD$, determine the value of $EF$
2003 IMO Shortlist, 2
Let $D_1$, $D_2$, ..., $D_n$ be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most $2003$ discs $D_i$. Prove that there exists a disc $D_k$ which intersects at most $7\cdot 2003 - 1 = 14020$ other discs $D_i$.
2022 3rd Memorial "Aleksandar Blazhevski-Cane", P6
For any integer $n\geq1$, we consider a set $P_{2n}$ of $2n$ points placed equidistantly on a circle. A [i]perfect matching[/i] on this point set is comprised of $n$ (straight-line) segments whose endpoints constitute $P_{2n}$. Let $\mathcal{M}_{n}$ denote the set of all non-crossing perfect matchings on $P_{2n}$. A perfect matching $M\in \mathcal{M}_{n}$ is said to be [i]centrally symmetric[/i], if it is invariant under point reflection at the circle center. Determine, as a function of $n$, the number of centrally symmetric perfect matchings within $\mathcal{M}_{n}$.
[i]Proposed by Mirko Petrusevski[/i]
2008 Harvard-MIT Mathematics Tournament, 3
Farmer John has $ 5$ cows, $ 4$ pigs, and $ 7$ horses. How many ways can he pair up the animals so that every pair consists of animals of different species? (Assume that all animals are distinguishable from each other.)