Found problems: 85335
2005 CentroAmerican, 2
Show that the equation $a^{2}b^{2}+b^{2}c^{2}+3b^{2}-c^{2}-a^{2}=2005$ has no integer solutions.
[i]Arnoldo Aguilar, El Salvador[/i]
2011 National Olympiad First Round, 26
The integers $0 \leq a < 2^{2008}$ and $0 \leq b < 8$ satisfy the equivalence $7(a+2^{2008}b) \equiv 1 \pmod{2^{2011}}$. Then $b$ is
$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None}$
2000 May Olympiad, 2
Given a parallelogram with area $1$ and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the $8$ lines formed we have a octagon inside of the parallelogram. Determine the area of this octagon
2019 Sharygin Geometry Olympiad, 20
Let $O$ be the circumcenter of triangle ABC, $H$ be its orthocenter, and $M$ be the midpoint of $AB$. The line $MH$ meets the line passing through $O$ and parallel to $AB$ at point $K$ lying on the circumcircle of $ABC$. Let $P$ be the projection of $K$ onto $AC$. Prove that $PH \parallel BC$.
1993 All-Russian Olympiad Regional Round, 10.7
Points $ M,N$ are taken on sides $ BC,CD$ respectively of parallelogram $ ABCD$. Let $ E\equal{}BD\cap AM, F\equal{}BD\cap AN$. Diagonal $ BD$ cuts triangle $ AMN$ into two parts. Prove that these two parts have equal area if and only if the point $ K$ given by $ EK\parallel{}AD, FK\parallel{}AB$ lies on segment $ MN$.
2021-IMOC, A9
For a given positive integer $n,$ find
$$\sum_{k=0}^{n} \left(\frac{\binom{n}{k} \cdot (-1)^k}{(n+1-k)^2} - \frac{(-1)^n}{(k+1)(n+1)}\right).$$
2016 Purple Comet Problems, 20
The 24 unshaded squares in the 5 × 5 grid below can be tiled with twelve 1 × 2 tiles. One such tiling is shown. Find the number of ways the grid can be tiled.
[center][img]https://snag.gy/KMoPrF.jpg[/img][/center]
2015 İberoAmerican, 1
The number $125$ can be written as a sum of some pairwise coprime integers larger than $1$. Determine the largest number of terms that the sum may have.
1985 Austrian-Polish Competition, 1
Show that if $a+b+c=0$ then $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})=9$.
2002 Croatia Team Selection Test, 1
In a certain language there are $n$ letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.
2013 Princeton University Math Competition, 1
Prove that \[ \frac{1}{a^2+2} + \frac{1}{b^2+2} + \frac{1}{c^2+2} \le \frac{1}{6ab+c^2} + \frac{1}{6bc+a^2} + \frac{1}{6ca+b^2} \] for all positive real numbers $a$, $b$ and $c$ satisfying $a^2+b^2+c^2=1$.
2002 Belarusian National Olympiad, 7
Several clocks lie on the table. It is known that at some moment the sum of distances between a point $X$ of the table and the ends of their minute hands is not equal to the sum of distances between $X$ and the ends of their hour hands.
Prove that there is a moment when the sum of distances between $X$ and the ends of their minute hands is greater than the sum of distances between $X$ and the ends of their hour hands.
(E. Barabanov, I. Voronovich)
2022 Lusophon Mathematical Olympiad, 5
Tow circumferences of radius $R_1$ and $R_2$ are tangent externally between each other. Besides that, they are both tangent to a semicircle with radius of 1, as shown in the figure. (Diagram is in the attachment)
a) If $A_1$ and $A_2$ are the tangency points of the two circumferences with the diameter of the semicircle, find the length of $\overline{A_1 A_2}$.
b) Prove that $R_{1}+R_{2}=2\sqrt{R_{1}R_{2}}(\sqrt{2}-\sqrt{R_{1}R_{2}})$.
2023 Iran MO (3rd Round), 2
find all $f : \mathbb{C} \to \mathbb{C}$ st:
$$f(f(x)+yf(y))=x+|y|^2$$
for all $x,y \in \mathbb{C}$
2020 Jozsef Wildt International Math Competition, W33
Let $p\in\mathbb N,f:[0,1]\to(0,\infty)$ be a continuous function and
$$a_n=\int^1_0x^p\sqrt[n]{f(x)}dx,n\in\mathbb N,n\ge2.$$
Demonstrate that:
a) $\lim_{n\to\infty}a_n=\frac1{p+1}$
b) $\lim_{n\to\infty}((p+1)a_n)^n=\exp\left((p+1)\int^1_0x^p\ln f(x)dx\right)$
[i]Proposed by Nicolae Papacu[/i]
2017 Harvard-MIT Mathematics Tournament, 4
Find all pairs $(a,b)$ of positive integers such that $a^{2017}+b$ is a multiple of $ab$.
2012 BAMO, 5
Find all nonzero polynomials $P(x)$ with integers coefficients that satisfy the following property: whenever $a$ and $b$ are relatively prime integers, then $P(a)$ and $P(b)$ are relatively prime as well. Prove that your answer is correct. (Two integers are [b]relatively prime[/b] if they have no common prime factors. For example, $-70$ and $99$ are relatively prime, while $-70$ and $15$ are not relatively prime.)
1973 All Soviet Union Mathematical Olympiad, 177
Given an angle with the vertex $O$ and a circle touching its sides in the points $A$ and $B$. A ray is drawn from the point $A$ parallel to $[OB)$. It intersects with the circumference in the point $C$. The segment $[OC]$ intersects the circumference in the point $E$. The straight lines $(AE)$ and $(OB)$ intersect in the point $K$. Prove that $|OK| = |KB|$.
2011 Chile National Olympiad, 1
Find all the solutions $(a, b, c)$ in the natural numbers, verifying $1\le a \le b \le c$, of the equation$$\frac34=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$
1979 AMC 12/AHSME, 1
[asy]
draw((-2,1)--(2,1)--(2,-1)--(-2,-1)--cycle);
draw((0,0)--(0,-1)--(-2,-1)--(-2,0)--cycle);
label("$F$",(0,0),E);
label("$A$",(-2,1),W);
label("$B$",(2,1),E);
label("$C$", (2,-1),E);
label("$D$",(-2,-1),WSW);
label("$E$",(-2,0),W);
label("$G$",(0,-1),S);
//Credit to TheMaskedMagician for the diagram
[/asy]
If rectangle $ABCD$ has area $72$ square meters and $E$ and $G$ are the midpoints of sides $AD$ and $CD$, respectively, then the area of rectangle $DEFG$ in square meters is
$\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$
1983 AMC 12/AHSME, 12
If $\log_7 \Big(\log_3 (\log_2 x) \Big) = 0$, then $x^{-1/2}$ equals
$\displaystyle \text{(A)} \ \frac{1}{3} \qquad \text{(B)} \ \frac{1}{2 \sqrt 3} \qquad \text{(C)} \ \frac{1}{3 \sqrt 3} \qquad \text{(D)} \ \frac{1}{\sqrt{42}} \qquad \text{(E)} \ \text{none of these}$
2024 IFYM, Sozopol, 6
Prove that for some positive integer \(N\), \(N\) points can be chosen on a circle such that there are at least \(1000N^2\) unordered quadruples \((A,B,C,D)\) of distinct selected points for which \(\displaystyle \frac{AC}{BC} = \frac{AD}{BD}\).
Croatia MO (HMO) - geometry, 2012.3
Let $ABCD$ be a cyclic quadrilateral such that $|AD| =|BD|$ and let $M$ be the intersection of its diagonals. Furthermore, let $N$ be the second intersection of the diagonal $AC$ with the circle passing through points $B, M$ and the center of the circle inscribed in triangle $BCM$. Prove that $AN \cdot NC = CD \cdot BN$
2000 Turkey MO (2nd round), 3
Find all continuous functions $f:[0,1]\to [0,1]$ for which there exists a positive integer $n$ such that $f^{n}(x)=x$ for $x \in [0,1]$ where $f^{0} (x)=x$ and $f^{k+1}=f(f^{k}(x))$ for every positive integer $k$.
1978 IMO Longlists, 48
Prove that it is possible to place $2n(2n + 1)$ parallelepipedic (rectangular) pieces of soap of dimensions $1 \times 2 \times (n + 1)$ in a cubic box with edge $2n + 1$ if and only if $n$ is even or $n = 1$.
[i]Remark[/i]. It is assumed that the edges of the pieces of soap are parallel to the edges of the box.