Found problems: 85335
1995 All-Russian Olympiad Regional Round, 11.6
The sequence $ a_n$ satisfies $ a_{m\plus{}n}\plus{} a_{m\minus{}n}\equal{}\frac12(a_{2m}\plus{}a_{2n})$ for all $ m\geq n\geq 0$. If $ a_1\equal{}1$, find $ a_{1995}$.
2004 Harvard-MIT Mathematics Tournament, 3
Find \[ \lim_{x \to \infty} \left( \sqrt[3]{x^3 + x^2}-\sqrt[3]{x^3-x^2} \right). \]
2009 Danube Mathematical Competition, 4
Let be $ a,b,c $ positive integers.Prove that $ |a-b\sqrt{c}|<\frac{1}{2b} $ is true if and only if $ |a^{2}-b^{2}c|<\sqrt{c} $.
2015 District Olympiad, 1
Determine all natural numbers $ \overline{ab} $ with $ a<b $ which are equal with the sum of all the natural numbers between $ a $ and $ b, $ inclusively.
2013 NIMO Summer Contest, 1
What is the maximum possible score on this contest? Recall that on the NIMO 2013 Summer Contest, problems $1$, $2$, \dots, $15$ are worth $1$, $2$, \dots, $15$ points, respectively.
[i]Proposed by Evan Chen[/i]
2024 Germany Team Selection Test, 2
Let $a_1<a_2<a_3<\dots$ be positive integers such that $a_{k+1}$ divides $2(a_1+a_2+\dots+a_k)$ for every $k\geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_k$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_k$.
2014 Romania Team Selection Test, 4
Let $f$ be the function of the set of positive integers into itself, defi ned by $f(1) = 1$,
$f(2n) = f(n)$ and $f(2n + 1) = f(n) + f(n + 1)$. Show that, for any positive integer $n$, the
number of positive odd integers m such that $f(m) = n$ is equal to the number of positive
integers[color=#0000FF][b] less or equal to [/b][/color]$n$ and coprime to $n$.
[color=#FF0000][mod: the initial statement said less than $n$, which is wrong.][/color]
2013 Online Math Open Problems, 15
A permutation $a_1, a_2, ..., a_{13}$ of the numbers from $1$ to $13$ is given such that $a_i > 5$ for $i=1,2,3,4,5$. Determine the maximum possible value of \[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \][i]Evan Chen[/i]
1998 Swedish Mathematical Competition, 3
A cube side $5$ is made up of unit cubes. Two small cubes are [i]adjacent [/i] if they have a common face. Can we start at a cube adjacent to a corner cube and move through all the cubes just once? (The path must always move from a cube to an adjacent cube).
2014 Math Prize For Girls Problems, 8
A triangle has sides of length $\sqrt{13}$, $\sqrt{17}$, and $2 \sqrt{5}$. Compute the area of the triangle.
2022 Bulgaria JBMO TST, 3
For a positive integer $n$ let $t_n$ be the number of unordered triples of non-empty and pairwise disjoint subsets of a given set with $n$ elements. For example, $t_3 = 1$. Find a closed form formula for $t_n$ and determine the last digit of $t_{2022}$.
(I also give here that $t_4 = 10$, for a reader to check his/her understanding of the problem statement.)
2023 IMC, 7
Let $V$ be the set of all continuous functions $f\colon [0,1]\to \mathbb{R}$, differentiable on $(0,1)$, with the property that $f(0)=0$ and $f(1)=1$. Determine all $\alpha \in \mathbb{R}$ such that for every $f\in V$, there exists some $\xi \in (0,1)$ such that
\[f(\xi)+\alpha = f'(\xi)\]
2005 Purple Comet Problems, 6
$ABCDE$ is a regular pentagon. What is the degree measure of the acute angle at the intersection of line segments $AC$ and $BD$?
2008 Cono Sur Olympiad, 6
A palindrome is a number that is the same when its digits are reversed. Find all numbers that have at least one multiple that is a palindrome.
2014-2015 SDML (Middle School), 10
Suppose $a$ and $b$ are digits which are not both nine and not both zero. If the repeating decimal $.\overline{AB}$ is expressed as a fraction in lowest terms, how many possible denominators are there?
$\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$
2022 Junior Balkan Team Selection Tests - Romania, P1
Let $a\geq b\geq c\geq d$ be real numbers such that $(a-b)(b-c)(c-d)(d-a)=-3.$
[list=a]
[*]If $a+b+c+d=6,$ prove that $d<0,36.$
[*]If $a^2+b^2+c^2+d^2=14,$ prove that $(a+c)(b+d)\leq 8.$ When does equality hold?
[/list]
1987 Tournament Of Towns, (143) 4
On a chessboard a square is chosen . The sum of the squares of distances from its centre to the centre of all black squares is designated by $a$ and to the centre of all white squares by $b$. Prove that $a = b$.
(A. Andj ans, Riga)
1984 IMO Longlists, 47
Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.
2008 Romania National Olympiad, 3
Let $ a,b \in [0,1]$. Prove that \[ \frac 1{1\plus{}a\plus{}b} \leq 1 \minus{} \frac {a\plus{}b}2 \plus{} \frac {ab}3.\]
2008 International Zhautykov Olympiad, 3
Let $ a, b, c$ be positive integers for which $ abc \equal{} 1$. Prove that
$ \sum \frac{1}{b(a\plus{}b)} \ge \frac{3}{2}$.
1980 Kurschak Competition, 3
In a certain country there are two tennis clubs consisting of $1000 $ and $1001$ members respectively. All the members have different playing strength, and the descending order of palying strengths in each club is known. Find a procedure which determines, within $ 11$ games, who is in the $1001$st place among the $ 2001$ players in these clubs. It is assumed that a stronger player always beats a weaker one.
2021 LMT Fall, 8
Octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle where $A_1A_2=A_2A_3=A_3A_4=A_6A_7=13$ and $A_4A_5=A_5A_6=A_7A_8=A_8A_1=5. $ The sum of all possible areas of $A_1A_2A_3A_4A_5A_6A_7A_8$ can be expressed as $a+b\sqrt{c}$ where $\gcd{a,b}=1$ and $c$ is squarefree. Find $abc.$
[asy]
label("$A_1$",(5,0),E);
label("$A_2$",(2.92, -4.05),SE);
label("$A_3$",(-2.92,-4.05),SW);
label("$A_4$",(-5,0),W);
label("$A_5$",(-4.5,2.179),NW);
label("$A_6$",(-3,4), NW);
label("$A_7$",(3,4), NE);
label("$A_8$",(4.5,2.179),NE);
draw((5,0)--(2.9289,-4.05235));
draw((2.92898,-4.05325)--(-2.92,-4.05));
draw((-2.92,-4.05)--(-5,0));
draw((-5,0)--(-4.5, 2.179));
draw((-4.5, 2.179)--(-3,4));
draw((-3,4)--(3,4));
draw((3,4)--(4.5,2.179));
draw((4.5,2.179)--(5,0));
dot((0,0));
draw(circle((0,0),5));
[/asy]
2017 F = ma, 24
24) A ball of mass m moving at speed $v$ collides with a massless spring of spring constant $k$ mounted on a stationary box of mass $M$ in free space. No mechanical energy is lost in the collision. If the system does not rotate, what is the maximum compression $x$ of the spring?
A) $x = v\sqrt{\frac{mM}{(m + M)k}}$
B) $x = v\sqrt{\frac{m}{k}}$
C) $x = v\sqrt{\frac{M}{k}}$
D) $x = v\sqrt{\frac{m + M}{k}}$
E) $x = v\sqrt{\frac{(m + M)^3}{mMk}}$
2010 Lithuania National Olympiad, 4
Decimal digits $a,b,c$ satisfy
\[ 37\mid (a0a0\ldots a0b0c0c\ldots 0c)_{10} \]
where there are $1001$ a's and $1001$ c's. Prove that $b=a+c$.
2009 Junior Balkan MO, 1
Let $ ABCDE$ be a convex pentagon such that $ AB\plus{}CD\equal{}BC\plus{}DE$ and $ k$ a circle with center on side $ AE$ that touches the sides $ AB$, $ BC$, $ CD$ and $ DE$ at points $ P$, $ Q$, $ R$ and $ S$ (different from vertices of the pentagon) respectively. Prove that lines $ PS$ and $ AE$ are parallel.