Found problems: 85335
2016 Saint Petersburg Mathematical Olympiad, 2
Given the positive numbers $x_1, x_2,..., x_n$, such that $x_i \le 2x_j$ with $1 \le i < j \le n$.
Prove that there are positive numbers $y_1\le y_2\le...\le y_n$, such that $x_k \le y_k \le 2x_k$ for all $k=1,2,..., n$
1968 All Soviet Union Mathematical Olympiad, 096
The circumference with the radius $100$ cm is drawn on the cross-lined paper with the side of the squares $1$ cm. It neither comes through the vertices of the squares, nor touches the lines. How many squares can it pass through?
2006 All-Russian Olympiad Regional Round, 8.3
Four drivers took part in the round-robin racing. Their cars started simultaneously from one point and moved at constant speeds. It is known that after the start of the race, for any three cars there was a moment when they met. Prove that after the start of the race there will be a moment when all 4 cars meet. (We consider races to be infinitely long in time.)
2021 Austrian MO Beginners' Competition, 4
Let $p$ be a prime number and let $m$ and $n$ be positive integers with $p^2 + m^2 = n^2$.
Prove that $m> p$.
(Karl Czakler)
1966 All Russian Mathematical Olympiad, 082
The distance from $A$ to $B$ is $d$ kilometres. A plane flying with the constant speed in the constant direction along and over the line $(AB)$ is being watched from those points. Observers have reported that the angle to the plane from the point $A$ has changed by $\alpha$ degrees and from $B$ --- by $\beta$ degrees within one second. What can be the minimal speed of the plane?
2018 Flanders Math Olympiad, 1
In the triangle $\vartriangle ABC$ we have $| AB |^3 = | AC |^3 + | BC |^3$. Prove that $\angle C> 60^o$ .
2007 Hanoi Open Mathematics Competitions, 12
Calculate the sum $\frac{1}{2.7.12} + \frac{1}{7.12.17} + ... + \frac{1}{1997.2002.2007}$.
2018 PUMaC Team Round, 14
Find the sum of the positive integer solutions to the equation $\left\lfloor\sqrt[3]{x}\right\rfloor+\left\lfloor\sqrt[4]{x}\right\rfloor=4.$
2021 Kosovo National Mathematical Olympiad, 3
Prove that for any natural numbers $a,b,c$ and $d$ there exist infinetly natural numbers $n$ such that $a^n+b^n+c^n+d^n$ is composite.
2017 Austria Beginners' Competition, 1
The nonnegative real numbers $a$ and $b$ satisfy $a + b = 1$. Prove that:
$$\frac{1}{2} \leq \frac{a^3+b^3}{a^2+b^2} \leq 1$$
When do we have equality in the right inequality and when in the left inequality?
[i]Proposed by Walther Janous [/i]
2002 Moldova National Olympiad, 1
Find all real solutions of the equation: $ [x]\plus{}[x\plus{}\dfrac{1}{2}]\plus{}[x\plus{}\dfrac{2}{3}]\equal{}2002$
MBMT Geometry Rounds, 2017
[hide=R stands for Ramanujan, P stands for Pascal]they had two problem sets under those two names[/hide]
[b]R1.[/b] What is the distance between the points $(6, 0)$ and $(-2, 0)$?
[b]R2 / P1.[/b] Angle $X$ has a degree measure of $35$ degrees. What is the supplement of the complement of angle $X$?
[i]The complement of an angle is $90$ degrees minus the angle measure. The supplement of an angle is $180$ degrees minus the angle measure.
[/i]
[b]R3.[/b] A cube has a volume of $729$. What is the side length of the cube?
[b]R4 / P2.[/b] A car that always travels in a straight line starts at the origin and goes towards the point $(8, 12)$. The car stops halfway on its path, turns around, and returns back towards the origin. The car again stops halfway on its return. What are the car’s final coordinates?
[b]R5.[/b] A full, cylindrical soup can has a height of $16$ and a circular base of radius $3$. All the soup in the can is used to fill a hemispherical bowl to its brim. What is the radius of the bowl?
[b]R6.[/b] In square $ABCD$, the numerical value of the length of the diagonal is three times the numerical value of the area of the square. What is the side length of the square?
[b]R7.[/b] Consider triangle $ABC$ with $AB = 3$, $BC = 4$, and $AC = 5$. The altitude from $B$ to $AC$ intersects $AC$ at $H$. Compute $BH$.
[b]R8.[/b] Mary shoots $5$ darts at a square with side length $2$. Let $x$ be equal to the shortest distance between any pair of her darts. What is the maximum possible value of $x$?
[b]P3.[/b] Let $ABC$ be an isosceles triangle such that $AB = BC$ and all of its angles have integer degree measures. Two lines, $\ell_1$ and $\ell_2$, trisect $\angle ABC$. $\ell_1$ and $\ell_2$ intersect $AC$ at points $D$ and $E$ respectively, such that $D$ is between $A$ and $E$. What is the smallest possible integer degree measure of $\angle BDC$?
[b]P4.[/b] In rectangle $ABCD$, $AB = 9$ and $BC = 8$. $W$, $X$, $Y$ , and $Z$ are on sides $AB$, $BC$, $CD$, and $DA$, respectively, such that $AW = 2WB$, $CX = 3BX$, $CY = 2DY$ , and $AZ = DZ$. If $WY$ and $XZ$ intersect at $O$, find the area of $OWBX$.
[b]P5.[/b] Consider a regular $n$-gon with vertices $A_1A_2...A_n$. Find the smallest value of $n$ so that there exist positive integers $i, j, k \le n$ with $\angle A_iA_jA_k = \frac{34^o}{5}$.
[b]P6.[/b] In right triangle $ABC$ with $\angle A = 90^o$ and $AB < AC$, $D$ is the foot of the altitude from $A$ to $BC$, and $M$ is the midpoint of $BC$. Given that $AM = 13$ and $AD = 5$, what is $\frac{AB}{AC}$ ?
[b]P7.[/b] An ant is on the circumference of the base of a cone with radius $2$ and slant height $6$. It crawls to the vertex of the cone $X$ in an infinite series of steps. In each step, if the ant is at a point $P$, it crawls along the shortest path on the exterior of the cone to a point $Q$ on the opposite side of the cone such that $2QX = PX$. What is the total distance that the ant travels along the exterior of the cone?
[b]P8.[/b] There is an infinite checkerboard with each square having side length $2$. If a circle with radius $1$ is dropped randomly on the checkerboard, what is the probability that the circle lies inside of exactly $3$ squares?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2015 IMO Shortlist, N7
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.
[i]Proposed by James Rickards, Canada[/i]
1960 Putnam, A5
Find all polynomials $f(x)$ with real coefficients having the property $f(g(x))=g(f(x))$ for every polynomial $g(x)$ with real coefficients.
2013 F = Ma, 7
A light car and a heavy truck have the same momentum. The truck weighs ten times as much as the car. How do their kinetic energies compare?
$\textbf{(A)}$ The truck's kinetic energy is larger by a factor of $100$
$\textbf{(B)}$ They truck's kinetic energy is larger by a factor of $10$
$\textbf{(C)}$ They have the same kinetic energy
$\textbf{(D)}$ The car's kinetic energy is larger by a factor of $10$
$\textbf{(E)}$ The car's kinetic energy is larger by a factor of $100$
2018 Oral Moscow Geometry Olympiad, 4
Given a triangle $ABC$ ($AB> AC$) and a circle circumscribed around it. Construct with a compass and a ruler the midpoint of the arc $BC$ (not containing vertex $A$), with no more than two lines (straight or circles).
2006 Harvard-MIT Mathematics Tournament, 8
Compute $\displaystyle\int_0^{\pi/3}x\tan^2(x)dx$.
1958 AMC 12/AHSME, 36
The sides of a triangle are $ 30$, $ 70$, and $ 80$ units. If an altitude is dropped upon the side of length $ 80$, the larger segment cut off on this side is:
$ \textbf{(A)}\ 62\qquad
\textbf{(B)}\ 63\qquad
\textbf{(C)}\ 64\qquad
\textbf{(D)}\ 65\qquad
\textbf{(E)}\ 66$
2005 Germany Team Selection Test, 1
Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals.
Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded?
[i]Proposed by Mihai Bălună, Romania[/i]
2009 AMC 12/AHSME, 2
Paula the painter had just enough paint for $ 30$ identically sized rooms. Unfortunately, on the way to work, three cans of paint fell of her truck, so she had only enough paint for $ 25$ rooms. How many cans of paint did she use for the $ 25$ rooms?
$ \textbf{(A)}\ 10 \qquad
\textbf{(B)}\ 12 \qquad
\textbf{(C)}\ 15 \qquad
\textbf{(D)}\ 18 \qquad
\textbf{(E)}\ 25$
1997 Poland - Second Round, 1
For the real number $a$ find the number of solutions $(x, y, z)$ of a system of the equations:
$\left\{\begin{array}{lll} x+y^2+z^2=a \\ x^2+y+z^2=a \\ x^2+y^2+z=a\end{array}\right.$
1956 AMC 12/AHSME, 6
In a group of cows and chickens, the number of legs was $ 14$ more than twice the number of heads. The number of cows was:
$ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 14$
2007 CentroAmerican, 1
In a remote island, a language in which every word can be written using only the letters $a$, $b$, $c$, $d$, $e$, $f$, $g$ is spoken. Let's say two words are [i]synonymous[/i] if we can transform one into the other according to the following rules:
i) Change a letter by another two in the following way: \[a \rightarrow bc,\ b \rightarrow cd,\ c \rightarrow de,\ d \rightarrow ef,\ e \rightarrow fg,\ f\rightarrow ga,\ g\rightarrow ab\]
ii) If a letter is between other two equal letters, these can be removed. For example, $dfd \rightarrow f$.
Show that all words in this language are synonymous.
2003 Federal Math Competition of S&M, Problem 3
Let $ABCD$ be a rectangle. Determine the set of all points $P$ from the region between the parallel lines $AB$ and $CD$ such that $\angle APB=\angle CPD$.
2017 Vietnamese Southern Summer School contest, Problem 2
Find all functions $f:\mathbb{R}\mapsto \mathbb{R}$ satisfy:
$$f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)$$
for all real numbers $x,y$.