Found problems: 85335
2019 Puerto Rico Team Selection Test, 3
Find the largest value that the expression can take $a^3b + b^3a$ where $a, b$ are non-negative real numbers, with $a + b = 3$.
2012 Online Math Open Problems, 27
Let $ABC$ be a triangle with circumcircle $\omega$. Let the bisector of $\angle ABC$ meet segment $AC$ at $D$ and circle $\omega$ at $M\ne B$. The circumcircle of $\triangle BDC$ meets line $AB$ at $E\ne B$, and $CE$ meets $\omega$ at $P\ne C$. The bisector of $\angle PMC$ meets segment $AC$ at $Q\ne C$. Given that $PQ = MC$, determine the degree measure of $\angle ABC$.
[i]Ray Li.[/i]
2002 Moldova National Olympiad, 3
There are $ 16$ persons in a company, each of which likes exactly $ 8$ other persons. Show that there exist two persons who like each other.
2018 MIG, 17
Two standard six sided dice labeled with the numbers $1$-$6$ are rolled, and the numbers that come up are multiplied. What is the probability that their product is a multiple of five?
$\textbf{(A) } \dfrac14\qquad\textbf{(B) } \dfrac5{18}\qquad\textbf{(C) } \dfrac{11}{36}\qquad\textbf{(D) } \dfrac13\qquad\textbf{(E) } \dfrac49$
1979 Kurschak Competition, 3
An $n \times n$ array of letters is such that no two rows are the same. Show that it must be possible to omit a column, so that the remaining table has no two rows the same.
2015 ISI Entrance Examination, 7
Let $\gamma_1, \gamma_2,\gamma_3 $ be three circles of unit radius which touch each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed by the common tangents be $\triangle XYZ$ . Find the length of each side of $\triangle XYZ$
2014 Contests, 1
Let $(x_{n}) \ n\geq 1$ be a sequence of real numbers with $x_{1}=1$ satisfying $2x_{n+1}=3x_{n}+\sqrt{5x_{n}^{2}-4}$
a) Prove that the sequence consists only of natural numbers.
b) Check if there are terms of the sequence divisible by $2011$.
2014 Greece Junior Math Olympiad, 4
We color the numbers $1, 2, 3,....,20$ with two colors white and black in such a way that both colors are used. Find the number of ways, we can perform this coloring if the product of white numbers and the product of black numbers have greatest common divisor equal to $1$.
2011 Baltic Way, 2
Let $f:\mathbb{Z}\to\mathbb{Z}$ be a function such that for all integers $x$ and $y$, the following holds:
\[f(f(x)-y)=f(y)-f(f(x)).\]
Show that $f$ is bounded.
1977 Vietnam National Olympiad, 3
Into how many regions do $n$ circles divide the plane, if each pair of circles intersects in two points and no point lies on three circles?
2019 Jozsef Wildt International Math Competition, W. 18
Let $\{c_k\}_{k\geq1}$ be a sequence with $0 \leq c_k \leq 1$, $c_1 \neq 0$, $\alpha > 1$. Let $C_n = c_1 + \cdots + c_n$. Prove $$\lim \limits_{n \to \infty}\frac{C_1^{\alpha}+\cdots+C_n^{\alpha}}{\left(C_1+\cdots +C_n\right)^{\alpha}}=0$$
2013 Tournament of Towns, 1
There are $100$ red, $100$ yellow and $100$ green sticks. One can construct a triangle using any three sticks all of different colours (one red, one yellow and one green). Prove that there is a colour such that one can construct a triangle using any three sticks of this colour.
2018 IFYM, Sozopol, 5
Point $X$ lies in a right-angled isosceles $\triangle ABC$ ($\angle ABC = 90^\circ$). Prove that
$AX+BX+\sqrt{2}CX \geq \sqrt{5}AB$
and find for which points $X$ the equality is met.
IV Soros Olympiad 1997 - 98 (Russia), 9.2
Solve the equation
$$2\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+(x+3)(x+5)}}}}=x$$
2022 ELMO Revenge, 1
Let $ABC$ and $DBC$ be triangles with incircles touching at a point $P$ on $BC.$ Points $A,D$ lie on the same side of $BC$ and $DB < AB < DC < AC.$ The bisector of $\angle BDC$ meets line $AP$ at $X,$ and the altitude from $A$ meets $DP$ at $Y.$ Point $Z$ lies on line $XY$ so $ZP \perp BC.$ Show the reflection of $A$ over $BC$ is on line $ZD.$
[i]Proposed by squareman (Evan Chang), USA[/i]
2001 Austria Beginners' Competition, 3
Find all real numbers $x$ such that $(x-1)^2(x-4)^2<(x-2)^2$.
2018 OMMock - Mexico National Olympiad Mock Exam, 4
For each positive integer $n$ let $s(n)$ denote the sum of the decimal digits of $n$. Find all pairs of positive integers $(a, b)$ with $a > b$ which simultaneously satisfy the following two conditions
$$a \mid b + s(a)$$
$$b \mid a + s(b)$$
[i]Proposed by Victor DomÃnguez[/i]
2014 Contests, 2
The points $P$ and $Q$ lie on the sides $BC$ and $CD$ of the parallelogram $ABCD$ so that $BP = QD$. Show that the intersection point between the lines $BQ$ and $DP$ lies on the line bisecting $\angle BAD$.
2015 India Regional MathematicaI Olympiad, 4
4. Suppose 36 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite.
2020 Tournament Of Towns, 4
For an infinite sequence $a_1, a_2,. . .$ denote as it's [i]first derivative[/i] is the sequence $a'_n= a_{n + 1} - a_n$ (where $n = 1, 2,..$.), and her $k$- th derivative as the first derivative of its $(k-1)$-th derivative ($k = 2, 3,...$). We call a sequence [i]good[/i] if it and all its derivatives consist of positive numbers.
Prove that if $a_1, a_2,. . .$ and $b_1, b_2,. . .$ are good sequences, then sequence $a_1\cdot b_1, a_2 \cdot b_2,..$ is also a good one.
R. Salimov
2007-2008 SDML (Middle School), 5
Maria and Joe are jogging towards each other on a long straight path. Joe is running at $10$ mph and Maria at $8$ mph. When they are $3$ miles apart, a fly begins to fly back and forth between them at a constant rate of $15$ mph, turning around instantaneously whenever it reachers one of the runners. How far, in miles, will the fly have traveled when Joe and Maria pass each other?
2016-2017 SDML (Middle School), 6
What is the probability that a random arrangement of the letters in the word 'ARROW' will have both R's next to each other?
$\text{(A) }\frac{1}{10}\qquad\text{(B) }\frac{2}{15}\qquad\text{(C) }\frac{1}{5}\qquad\text{(D) }\frac{3}{10}\qquad\text{(E) }\frac{2}{5}$
II Soros Olympiad 1995 - 96 (Russia), 11.4
Prove that the equation $x^6 - 100x+1 = 0$ has two roots, and both of these roots are positive.
a) Find the first non-zero digit in the decimal notation of the lesser root of this equation.
b) Find the first two non-zero digits in the decimal notation of the lesser root of this equation.
1978 Putnam, B3
The sequence $(Q_{n}(x))$ of polynomials is defined by
$$Q_{1}(x)=1+x ,\; Q_{2}(x)=1+2x,$$
and for $m \geq 1 $ by
$$Q_{2m+1}(x)= Q_{2m}(x) +(m+1)x Q_{2m-1}(x),$$
$$Q_{2m+2}(x)= Q_{2m+1}(x) +(m+1)x Q_{2m}(x).$$
Let $x_n$ be the largest real root of $Q_{n}(x).$ Prove that $(x_n )$ is an increasing sequence and that $\lim_{n\to \infty} x_n =0.$
2023 Turkey Olympic Revenge, 1
Find all $c\in \mathbb{R}$ such that there exists a function $f:\mathbb{R}\to \mathbb{R}$ satisfying $$(f(x)+1)(f(y)+1)=f(x+y)+f(xy+c)$$ for all $x,y\in \mathbb{R}$.
[i]Proposed by Kaan Bilge[/i]