Found problems: 85335
2018 IMO, 6
A convex quadrilateral $ABCD$ satisfies $AB\cdot CD = BC\cdot DA$. Point $X$ lies inside $ABCD$ so that \[\angle{XAB} = \angle{XCD}\quad\,\,\text{and}\quad\,\,\angle{XBC} = \angle{XDA}.\] Prove that $\angle{BXA} + \angle{DXC} = 180^\circ$.
[i]Proposed by Tomasz Ciesla, Poland[/i]
2021 Belarusian National Olympiad, 9.5
Prove that for some positive integer $n$ there exist positive integers $a$,$b$ and $c$ such that $a^2-n=xy$, $b^2-n=yz$ and $c^2-n=xz$ where $x,y$ and $z$ - some pairwise different positive integers.
2010 Dutch IMO TST, 4
Let $ABCD$ be a cyclic quadrilateral satisfying $\angle ABD = \angle DBC$. Let $E$ be the intersection of the diagonals $AC$ and $BD$. Let $M$ be the midpoint of $AE$, and $N$ be the midpoint of $DC$. Show that $MBCN$ is a cyclic quadrilateral.
2010 Contests, 4
Let $P(x)=ax^3+bx^2+cx+d$ be a polynomial with real coefficients such that \[\min\{d,b+d\}> \max\{|{c}|,|{a+c}|\}\]
Prove that $P(x)$ do not have a real root in $[-1,1]$.
1990 China Team Selection Test, 2
Finitely many polygons are placed in the plane. If for any two polygons of them, there exists a line through origin $O$ that cuts them both, then these polygons are called "properly placed". Find the least $m \in \mathbb{N}$, such that for any group of properly placed polygons, $m$ lines can drawn through $O$ and every polygon is cut by at least one of these $m$ lines.
2018 Harvard-MIT Mathematics Tournament, 1
Four standard six-sided dice are rolled. Find the probability that, for each pair of dice, the product of the two numbers rolled on those dice is a multiple of 4.
2018 IFYM, Sozopol, 1
$A = \{a_1, a_2, . . . , a_k\}$ is a set of positive integers for which the sum of some (we can have only one number too) different numbers from the set is equal to a different number i.e. there $2^k - 1$ different sums of different numbers from $A$. Prove that the following inequality holds:
$\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}<2$
VII Soros Olympiad 2000 - 01, 9.4
The distance between cities $A$ and $B$ is $30$ km. A bus departed from $A$, which makes a stop every $5$ km for $2$ minutes. The bus moves between stops at a speed of $80$ km / h. Simultaneously with the departure of the bus from $A$, a cyclist leaves $B$ to meet it, traveling at a speed of $27$ km / h. How far from $A$ will the cyclist meet the bus?
2009 Stars Of Mathematics, 4
Determine all non-constant polynomials $ f\in \mathbb{Z}[X]$ with the property that there exists $ k\in\mathbb{N}^*$ such that for any prime number $ p$, $ f(p)$ has at most $ k$ distinct prime divisors.
2012 Rioplatense Mathematical Olympiad, Level 3, 5
Let $a \ge 2$ and $n \ge 3$ be integers . Prove that one of the numbers $a^n+ 1 , a^{n + 1}+ 1 , ... , a^{2 n-2}+ 1$ does not share any odd divisor greater than $1$ with any of the other numbers.
2020 Purple Comet Problems, 16
Find the maximum possible value of $$\left( \frac{a^3}{b^2c}+\frac{b^3}{c^2a}+\frac{c^3}{a^2b} \right)^2$$ where $a, b$, and $c$ are nonzero real numbers satisfying $$a \sqrt[3]{\frac{a}{b}}+b\sqrt[3]{\frac{b}{c}}+c\sqrt[3]{\frac{c}{a}}=0$$
2010 VJIMC, Problem 2
If $A,B\in M_2(C)$ such that $AB-BA=B^2$ then prove that
\[AB=BA\]
2020 Jozsef Wildt International Math Competition, W14
Let $\{F_n\}_{n\ge1}$ be the Fibonacci sequence defined by $F_1=F_2=1$ and for all $n\ge3$, $F_n=F_{n-1}+F_{n-2}$. Prove that among the first $10000000000000002$ terms of the sequence there is one term that ends up with $8$ zeroes.
[i]Proposed by José Luis Díaz-Barrero[/i]
1987 China Team Selection Test, 1
Given a convex figure in the Cartesian plane that is symmetric with respect of both axis, we construct a rectangle $A$ inside it with maximum area (over all posible rectangles). Then we enlarge it with center in the center of the rectangle and ratio lamda such that is covers the convex figure. Find the smallest lamda such that it works for all convex figures.
1965 German National Olympiad, 5
Determine all triples of nonzero decimal digits $(x,y,z)$ for which the equality $\sqrt{ \underbrace{xxx...x}_{2n}- \underbrace{yy...y}_{n}}= \underbrace{zzz...z}_{n}$ holds for at least two different natural numbers $n$.
2008 Cuba MO, 8
Let $ABC$ an acute-angle triangle. Let $R$ be a rectangle with vertices in the edges of $ABC$. Let $O$ be the center of $R$.
a) Find the locus of all the points $O$.
b) Decide if there is a point that is the center of three of these rectangles.
2016 CMIMC, 5
Let $\mathcal{P}$ be a parallelepiped with side lengths $x$, $y$, and $z$. Suppose that the four space diagonals of $\mathcal{P}$ have lengths $15$, $17$, $21$, and $23$. Compute $x^2+y^2+z^2$.
1971 All Soviet Union Mathematical Olympiad, 158
A switch has two inputs $1, 2$ and two outputs $1, 2$. It either connects $1$ to $1$ and $2$ to $2$, or $1$ to $2$ and $2$ to 1. If you have three inputs $1, 2, 3$ and three outputs $1, 2, 3$, then you can use three switches, the first across $1$ and $2$, then the second across $2$ and $3$, and finally the third across $1$ and $2$. It is easy to check that this allows the output to be any permutation of the inputs and that at least three switches are required to achieve this. What is the minimum number of switches required for $4$ inputs, so that by suitably setting the switches the output can be any permutation of the inputs?
2024 Indonesia MO, 5
Each integer is colored with exactly one of the following colors: red, blue, or orange, and all three colors are used in the coloring. The coloring also satisfies the following properties:
1. The sum of a red number and an orange number results in a blue-colored number,
2. The sum of an orange and blue number results in an orange-colored number;
3. The sum of a blue number and a red number results in a red-colored number.
(a) Prove that $0$ and $1$ must have distinct colors.
(b) Determine all possible colorings of the integers which also satisfy the properties stated above.
2009 India IMO Training Camp, 3
Let $ a,b$ be two distinct odd natural numbers.Define a Sequence $ { < a_n > }_{n\ge 0}$ like following:
$ a_1 \equal{} a \\
a_2 \equal{} b \\
a_n \equal{} \text{largest odd divisor of }(a_{n \minus{} 1} \plus{} a_{n \minus{} 2})$.
Prove that there exists a natural number $ N$ such that $ a_n \equal{} gcd(a,b) \forall n\ge N$.
2013 EGMO, 3
Let $n$ be a positive integer.
(a) Prove that there exists a set $S$ of $6n$ pairwise different positive integers, such that the least common multiple of any two elements of $S$ is no larger than $32n^2$.
(b) Prove that every set $T$ of $6n$ pairwise different positive integers contains two elements the least common multiple of which is larger than $9n^2$.
III Soros Olympiad 1996 - 97 (Russia), 9.3
Draw the set of projections of a square given on a plane onto all possible lines passing through a given point $O$ of the plane lying outside the square.
1978 Polish MO Finals, 5
For a given real number $a$, define the sequence $(a_n)$ by $a_1 = a$ and
$$a_{n+1} =\begin{cases}
\dfrac12 \left(a_n -\dfrac{1}{a_n}\right) \,\,\, if \,\,\, a_n \ne 0, \\
0 \,\,\, if \,\,\, a_n = 0 \end{cases}$$
Prove that the sequence $(a_n)$ contains infinitely many nonpositive terms.
2011 China Team Selection Test, 1
Let $H$ be the orthocenter of an acute trangle $ABC$ with circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) of $\Gamma$, and let $M$ be a point on the arc $CA$ (not containing $B$) of $\Gamma$ such that $H$ lies on the segment $PM$. Let $K$ be another point on $\Gamma$ such that $KM$ is parallel to the Simson line of $P$ with respect to triangle $ABC$. Let $Q$ be another point on $\Gamma$ such that $PQ \parallel BC$. Segments $BC$ and $KQ$ intersect at a point $J$. Prove that $\triangle KJM$ is an isosceles triangle.
2002 Turkey MO (2nd round), 2
Two circles are externally tangent to each other at a point $A$ and internally tangent to a third circle $\Gamma$ at points $B$ and $C.$ Let $D$ be the midpoint of the secant of $\Gamma$ which is tangent to the smaller circles at $A.$ Show that $A$ is the incenter of the triangle $BCD$ if the centers of the circles are not collinear.