Found problems: 85335
2020 Mediterranean Mathematics Olympiad, 1
Determine all integers $m\ge2$ for which there exists an integer $n\ge1$ with
$\gcd(m,n)=d$ and $\gcd(m,4n+1)=1$.
[i]Proposed by Gerhard Woeginger, Austria[/i]
2010 ELMO Shortlist, 4
Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$.
[i]Evan O' Dorney.[/i]
2007 China Girls Math Olympiad, 6
For $ a,b,c\geq 0$ with $ a\plus{}b\plus{}c\equal{}1$, prove that
$ \sqrt{a\plus{}\frac{(b\minus{}c)^2}{4}}\plus{}\sqrt{b}\plus{}\sqrt{c}\leq \sqrt{3}$
2014 IMO Shortlist, A3
For a sequence $x_1,x_2,\ldots,x_n$ of real numbers, we define its $\textit{price}$ as \[\max_{1\le i\le n}|x_1+\cdots +x_i|.\] Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price $D$. Greedy George, on the other hand, chooses $x_1$ such that $|x_1 |$ is as small as possible; among the remaining numbers, he chooses $x_2$ such that $|x_1 + x_2 |$ is as small as possible, and so on. Thus, in the $i$-th step he chooses $x_i$ among the remaining numbers so as to minimise the value of $|x_1 + x_2 + \cdots x_i |$. In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price $G$.
Find the least possible constant $c$ such that for every positive integer $n$, for every collection of $n$ real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality $G\le cD$.
[i]Proposed by Georgia[/i]
1962 Putnam, B2
Let $S$ be the set of all subsets of the positive integers. Construct a function $f \colon \mathbb{R} \rightarrow S$ such that $f(a)$ is a proper subset of $f(b)$ whenever $a <b.$
2023 Mexican Girls' Contest, 5
Mia has $2$ green sticks of $\textbf{3cm}$ each one, $2$ blue sticks of $\textbf{4cm}$ each one and $2$ red sticks of $\textbf{5cm}$ each one. She wants to make a triangle using the $6$ sticks as it´s perimeter, all at once and without overlapping them. How many non-congruent triangles can make?
2007 Today's Calculation Of Integral, 178
Let $f(x)$ be a differentiable function such that $f'(x)+f(x)=4xe^{-x}\sin 2x,\ \ f(0)=0.$
Find $\lim_{n\to\infty}\sum_{k=1}^{n}f(k\pi).$
2024 China Team Selection Test, 19
$n$ is a positive integer. An equilateral triangle of side length $3n$ is split into $9n^2$ unit equilateral triangles, each colored one of red, yellow, blue, such that each color appears $3n^2$ times. We call a trapezoid formed by three unit equilateral triangles as a "standard trapezoid". If a "standard trapezoid" contains all three colors, we call it a "colorful trapezoid". Find the maximum possible number of "colorful trapezoids".
2005 MOP Homework, 5
Let $ABC$ be a triangle. Points $D$ and $E$ lie on sides $BC$ and $CA$, respectively, such that $BD=AE$. Segments $AD$ and $BE$ meet at $P$. The bisector of angle $BCA$ meet segments $AD$ and $BE$ at $Q$ and $R$, respectively. Prove that $\frac{PQ}{AD}=\frac{PR}{BE}$.
1996 Swedish Mathematical Competition, 2
In the country of Postonia, one wants to have only two values of stamps. These values should be integers greater than $1$ with the difference $2$, and should have the property that one can combine the stamps for any postage which is greater than or equal to the sum of these two values. What values can be chosen?
2018 Online Math Open Problems, 23
Consider all ordered pairs $(a, b)$ of positive integers such that $\frac{a^2 + b^2 + 2}{ab}$ is an integer and $a\le b$. We label all such pairs in increasing order by their distance from the origin. (It is guaranteed that no ties exist.) Thus $P_1 = (1, 1), P_2 = (1, 3)$, and so on. If $P_{2020} = (x, y),$ then compute the remainder when $x + y$ is divided by $2017$.
[i]Proposed by Ashwin Sah[/i]
2009 Baltic Way, 20
In the future city Baltic Way there are sixteen hospitals. Every night exactly four of them must be on duty for emergencies. Is it possible to arrange the schedule in such a way that after twenty nights every pair of hospitals have been on common duty exactly once?
1995 Romania Team Selection Test, 3
Let $f$ be an irreducible (in $Z[x]$) monic polynomial with integer coefficients and of odd degree greater than $1$. Suppose that the modules of the roots of $f$ are greater than $1$ and that $f(0)$ is a square-free number. Prove that
the polynomial $g(x) = f(x^3)$ is also irreducible
2005 Abels Math Contest (Norwegian MO), 4a
Show that for all positive real numbers $a, b$ and $c$, the inequality $(a+b)(a+c)\ge 2\sqrt{abc(a+b+c)}$ is true.
1988 Putnam, B6
Prove that there exist an infinite number of ordered pairs $(a,b)$ of integers such that for every positive integer $t$, the number $at+b$ is a triangular number if and only if $t$ is a triangular number. (The triangular numbers are the $t_n = n(n+1)/2$ with $n$ in $\{0,1,2,\dots\}$.)
II Soros Olympiad 1995 - 96 (Russia), 9.1
The exchange rates of the Dollar and the German mark during the week changed as follows:
$\begin{tabular}{|l|l|l|}
\hline
& Dollar & Mark \\ \hline
Monday & 4000 rub. & 2500 rub. \\ \hline
Tuesday & 4500 rub. & 2800 rub.\\ \hline
Wednesday & 5000 rub. & 2500 rub.\\ \hline
Thursday & 4500 rub. & 3000 rub.\\ \hline
Friday & 4000 rub. & 2500 rub.\\ \hline
Saturday & 4500 rub. & 3000 rub.\\ \hline
\end{tabular}$
What percentage was the maximum possible increase in capital this week by playing on changes in the exchange rates of these currencies? (The initial capital was in rubles. The final capital should also be in rubles. During the week, the available money can be distributed as desired into rubles, dollars and marks. The selling and purchasing rates are considered the same.)
1980 All Soviet Union Mathematical Olympiad, 291
The six-digit decimal number contains six different non-zero digits and is divisible by $37$. Prove that having transposed its digits you can obtain at least $23$ more numbers divisible by $37$
1997 IMC, 3
Show that $\sum^{\infty}_{n=1}\frac{(-1)^{n-1}\sin(\log n)}{n^\alpha}$ converges iff $\alpha>0$.
2014 IPhOO, 1
A capacitor is made with two square plates, each with side length $L$, of negligible thickness, and capacitance $C$. The two-plate capacitor is put in a microwave which increases the side length of each square plate by $ 1 \% $. By what percent does the voltage between the two plates in the capacitor change?
$ \textbf {(A) } \text {decreases by } 2\% \\ \textbf {(B) } \text {decreases by } 1\% \\ \textbf {(C) } \text {it does not change} \\ \textbf {(D) } \text {increases by } 1\% \\ \textbf {(E) } \text {increases by } 2\% $
[i]Problem proposed by Ahaan Rungta[/i]
2006 Canada National Olympiad, 5
The vertices of a right triangle $ABC$ inscribed in a circle divide the circumference into three arcs. The right angle is at $A$, so that the opposite arc $BC$ is a semicircle while arc $BC$ and arc $AC$ are supplementary. To each of three arcs, we draw a tangent such that its point of tangency is the mid point of that portion of the tangent intercepted by the extended lines $AB,AC$. More precisely, the point $D$ on arc $BC$ is the midpoint of the segment joining the points $D'$ and $D''$ where tangent at $D$ intersects the extended lines $AB,AC$. Similarly for $E$ on arc $AC$ and $F$ on arc $AB$. Prove that triangle $DEF$ is equilateral.
MathLinks Contest 4th, 7.1
Let $a, b, c, d$ be positive reals such that $abcd = 1$. Prove that
$$\frac{1}{a(b + 1)} +\frac{1}{b(c + 1)} +\frac{1}{c(d + 1)} +\frac{1}{d(a + 1)} \ge 2.$$
2016 Greece Junior Math Olympiad, 4
Find the number ot 6-tuples $(x_1, x_2,...,x_6)$, where $x_i=0,1 or 2$ and $x_1+x_2+...+x_6$ is even
2023 IFYM, Sozopol, 7
In the country of Drilandia, which has at least three cities, there are bidirectional roads connecting some pairs of cities such that any city can be reached from any other. Two cities are called [i]close[/i] if one can reach the other by using at most two intermediary cities. The mayor, Drilago, fortified the road system by building a direct road between each pair of close cities that were not already connected. Prove that after the expansion, there exists a journey that starts and ends at the same city, where each city except the first is visited exactly once, and the first city is visited twice (once at the beginning and once at the end).
2014 NIMO Problems, 5
Let $r$, $s$, $t$ be the roots of the polynomial $x^3+2x^2+x-7$. Then \[ \left(1+\frac{1}{(r+2)^2}\right)\left(1+\frac{1}{(s+2)^2}\right)\left(1+\frac{1}{(t+2)^2}\right)=\frac{m}{n} \] for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
[i]Proposed by Justin Stevens[/i]
2000 239 Open Mathematical Olympiad, 6
$n$ cockroaches are sitting on the plane at the vertices of the regular $ n $ -gon. They simultaneously begin to move at a speed of $ v $ on the sides of the polygon in the direction of the clockwise adjacent cockroach, then they continue moving in the initial direction with the initial speed. Vasya a entomologist moves on a straight line in the plane at a speed of $u$. After some time, it turned out that Vasya has crushed three cockroaches. Prove that $ v = u $.