Found problems: 85335
1984 IMO Longlists, 30
Decide whether it is possible to color the $1984$ natural numbers $1, 2, 3, \cdots, 1984$ using $15$ colors so that no geometric sequence of length $3$ of the same color exists.
1979 AMC 12/AHSME, 10
If $P_1P_2P_3P_4P_5P_6$ is a regular hexagon whose apothem (distance from the center to midpoint of a side) is $2$, and $Q_i$ is the midpoint of side $P_iP_{i+1}$ for $i=1,2,3,4$, then the area of quadrilateral $Q_1Q_2Q_3Q_4$ is
$\textbf{(A) }6\qquad\textbf{(B) }2\sqrt{6}\qquad\textbf{(C) }\frac{8\sqrt{3}}{3}\qquad\textbf{(D) }3\sqrt{3}\qquad\textbf{(E) }4\sqrt{3}$
2022 Novosibirsk Oral Olympiad in Geometry, 1
Cut a square with three straight lines into three triangles and four quadrilaterals.
2013 Today's Calculation Of Integral, 877
Let $f(x)=\lim_{n\to\infty} \frac{\sin^{n+2}x+\cos^{n+2}x}{\sin^n x+\cos^n x}$ for $0\leq x\leq \frac{\pi}2.$
Evaluate $\int_0^{\frac{\pi}2} f(x)\ dx.$
Kvant 2022, M2684
Let $a_1,a_2,\ldots,a_n$ ($n\geq 2$) be nonnegative real numbers whose sum is $\frac{n}{2}$. For every $i=1,\ldots,n$ define
$$b_i=a_i+a_ia_{i+1}+a_ia_{i+1}a_{i+2}+\cdots+ a_ia_{i+1}\cdots a_{i+n-2}+2a_ia_{i+1}\cdots a_{i+n-1}$$
where $a_{j+n}=a_j$ for every $j$. Prove that $b_i\geq 1$ holds for at least one index $i$.
1996 German National Olympiad, 5
Given two non-intersecting chords $AB$ and $CD$ of a circle $k$ and a length $a <CD$. Determine a point $X$ on $k$ with the following property: If lines $XA$ and $XB$ intersect $CD$ at points $P$ and $Q$ respectively, then $PQ = a$. Show how to construct all such points $X$ and prove that the obtained points indeed have the desired property.
2020 LIMIT Category 1, 13
On the side $AC$ of an acute triangle $\triangle ABC$, a point $D$ is taken such that $2AD=CD=2, BD\perp AC$. A circle of radius $2$ passes through $A$ and $D$ and is tangent to the circumcircle of $\triangle BDC$. Find $[\text{Area}(\triangle ABC)]$ where $[.]$ is the greatest integer function.
2006 MOP Homework, 3
Prove that the following inequality holds with the exception of finitely many positive integers $n$:
$\sum^{n}_{i=1}\sum^{n}_{j=1}gcd(i,j)>4n^2$.
2018 Romania National Olympiad, 3
Let $f:[a,b] \to \mathbb{R}$ be an integrable function and $(a_n) \subset \mathbb{R}$ such that $a_n \to 0.$
$\textbf{a) }$ If $A= \{m \cdot a_n \mid m,n \in \mathbb{N}^* \},$ prove that every open interval of strictly positive real numbers contains elements from $A.$
$\textbf{b) }$ If, for any $n \in \mathbb{N}^*$ and for any $x,y \in [a,b]$ with $|x-y|=a_n,$ the inequality $\left| \int_x^yf(t)dt \right| \leq |x-y|$ is true, prove that $$\left| \int_x^y f(t)dt \right| \leq |x-y|, \: \forall x,y \in [a,b]$$
[i]Nicolae Bourbacut[/i]
2023 Malaysian IMO Training Camp, 5
Let $ABCD$ be a cyclic quadrilateral, with circumcircle $\omega$ and circumcenter $O$. Let $AB$ intersect $CD$ at $E$, $AD$ intersect $BC$ at $F$, and $AC$ intersect $BD$ at $G$.
The points $A_1, B_1, C_1, D_1$ are chosen on rays $GA$, $GB$, $GC$, $GD$ such that:
$\bullet$ $\displaystyle \frac{GA_1}{GA} = \frac{GB_1}{GB} = \frac{GC_1}{GC} = \frac{GD_1}{GD}$
$\bullet$ The points $A_1, B_1, C_1, D_1, O$ lie on a circle.
Let $A_1B_1$ intersect $C_1D_1$ at $K$, and $A_1D_1$ intersect $B_1C_1$ at $L$. Prove that the image of the circle $(A_1B_1C_1D_1)$ under inversion about $\omega$ is a line passing through the midpoints of $KE$ and $LF$.
[i]Proposed by Anzo Teh Zhao Yang & Ivan Chan Kai Chin[/i]
2014 Kyiv Mathematical Festival, 3a
a) There are 8 teams in a Quidditch tournament. Each team plays every other team once without draws. Prove that there exist teams $A,B,C,D$ such that pairs of teams $A,B$ and $C,D$ won the same number of games in total.
b) There are 25 teams in a Quidditch tournament. Each team plays every other team once without draws. Prove that there exist teams $A,B,C,D,E,F$ such that pairs of teams $A,B,$ $~$ $C,D$ and $E,F$ won the same number of games in total.
2016 Macedonia National Olympiad, Problem 3
Solve the equation in the set of natural numbers $xyz+yzt+xzt+xyt = xyzt + 3$
2002 Greece Junior Math Olympiad, 4
Prove that $1\cdot2\cdot3\cdots 2002<\left(\frac{2003}{2}\right)^{2002}.$
2012 CHMMC Fall, 4
A lattice point $(x, y, z) \in Z^3$ can be seen from the origin if the line from the origin does not contain any other lattice point $(x', y', z')$ with $$(x')^2 + (y')^2 + (z')^2 < x^2 + y^2 + z^2.$$ Let $p$ be the probability that a randomly selected point on the cubic lattice $Z^3$ can be seen from the origin. Given that
$$\frac{1}{p}= \sum^{\infty}_{n=i} \frac{k}{n^s}$$
for some integers $ i, k$, and $s$, find $i, k$ and $s$.
2017 ASDAN Math Tournament, 8
Compute
$$\int_0^1\frac{2xe^x-1}{2x^2e^x+2}dx.$$
2021 Indonesia MO, 8
On a $100 \times 100$ chessboard, the plan is to place several $1 \times 3$ boards and $3 \times 1$ board, so that
[list]
[*] Each tile of the initial chessboard is covered by at most one small board.
[*] The boards cover the entire chessboard tile, except for one tile.
[*] The sides of the board are placed parallel to the chessboard.
[/list]
Suppose that to carry out the instructions above, it takes $H$ number of $1 \times 3$ boards and $V$ number of $3 \times 1$ boards. Determine all possible pairs of $(H,V)$.
[i]Proposed by Muhammad Afifurrahman, Indonesia[/i]
2022 Malaysia IMONST 2, 1
Given a polygon $ABCDEFGHIJ$.
How many diagonals does the polygon have?
2009 Korea Junior Math Olympiad, 2
In an acute triangle $\triangle ABC$, let $A',B',C'$ be the reflection of $A,B,C$ with respect to $BC,CA,AB$. Let $D = B'C \cap BC'$, $E = CA' \cap C'A$, $F = A'B \cap AB'$. Prove that $AD,BE,CF$ are concurrent
2021 CCA Math Bonanza, I4
Given that nonzero real numbers $x$ and $y$ satisfy $x+\frac{1}{y}=3$ and $y+\frac{1}{x}=4$, what is $xy+\frac{1}{xy}$?
[i]2021 CCA Math Bonanza Individual Round #4[/i]
2009 Tournament Of Towns, 5
A country has two capitals and several towns. Some of them are connected by roads. Some of the roads are toll roads where a fee is charged for driving along them. It is known that any route from the south capital to the north capital contains at least ten toll roads. Prove that all toll roads can be distributed among ten companies so that anybody driving from the south capital to the north capital must pay each of these companies.
[i](5 points)[/i]
2023 Harvard-MIT Mathematics Tournament, 6
Suppose $a_1, a_2, ... , a_{100}$ are positive real numbers such that $$a_k =\frac{ka_{k-1}}{a_{k-1} - (k - 1)}$$ for $k = 2, 3, ... , 100$. Given that $a_{20} = a_{23}$, compute $a_{100}$.
2002 Manhattan Mathematical Olympiad, 3
Prove that for any polygon with all equal angles and for any interior point $A$, the sum of distances from $A$ to the sides of the polygon does not depend on the position of $A$.
2007 Hanoi Open Mathematics Competitions, 13
Let be given triangle $ABC$. Find all points $M$ such that area of $\vartriangle MAB$= area of $\vartriangle MAC$
2011 Morocco National Olympiad, 1
Let $a$ and $b$ be two positive real numbers such that $a+b=ab$.
Prove that $\frac{a}{b^{2}+4}+\frac{b}{a^{2}+4}\geq \frac{1}{2}$.
2022 APMO, 3
Find all positive integers $k<202$ for which there exist a positive integers $n$ such that
$$\bigg {\{}\frac{n}{202}\bigg {\}}+\bigg {\{}\frac{2n}{202}\bigg {\}}+\cdots +\bigg {\{}\frac{kn}{202}\bigg {\}}=\frac{k}{2}$$