Found problems: 85335
2019 Durer Math Competition Finals, 14
Let $S$ be the set of all positive integers less than $10,000$ whose last four digits in base $2$ are the same as its last four digits in base $5$. What remainder do we get if we divide the sum of all elements of $S$ by $10000$?
2023 Philippine MO, 5
Silverio is very happy for the 25th year of the PMO. In his jubilation, he ends up writing a finite sequence of As and Gs on a nearby blackboard. He then performs the following operation: if he finds at least one occurrence of the string "AG", he chooses one at random and replaces it with "GAAA". He performs this operation repeatedly until there is no more "AG" string on the blackboard. Show that for any initial sequence of As and Gs, Silverio will eventually be unable to continue doing the operation.
2013 Tournament of Towns, 4
Is it true that every integer is a sum of finite number of cubes of distinct integers?
1969 IMO, 3
For each of $k=1,2,3,4,5$ find necessary and sufficient conditions on $a>0$ such that there exists a tetrahedron with $k$ edges length $a$ and the remainder length $1$.
2015 Paraguayan Mathematical Olympiad, Problem 1
Alexa wrote the first $16$ numbers of a sequence:
\[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …\]
Then she continued following the same pattern, until she had $2015$ numbers in total.
What was the last number she wrote?
2003 Putnam, 6
For a set $S$ of nonnegative integers, let $r_S(n)$ denote the number of ordered pairs $(s_1, s_2)$ such that $s_1 \in S$, $s_2 \in S$, $s_1 \neq s_2$, and $s_1 + s_2 = n$. Is it possible to partition the nonnegative integers into two sets $A$ and $B$ in such a way that $r_A(n) = r_B(n)$ for all $n$?
2022 Germany Team Selection Test, 3
A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or
[*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter.
[i]Proposed by Aron Thomas[/i]
2017 239 Open Mathematical Olympiad, 2
Find all composite numbers $n$ such that for each decomposition of $n=xy$, $x+y$ be a power of $2$.
2000 Rioplatense Mathematical Olympiad, Level 3, 4
Let $a, b$ and $c$ be positive integers such that $a^2 + b^2 + 1 = c^2$ . Prove that $[a/2] + [c / 2]$ is even.
Note: $[x]$ is the integer part of $x$.
2023 Austrian MO Regional Competition, 4
Determine all pairs $(x, y)$ of positive integers such that for $d = gcd(x, y)$ the equation $$xyd = x + y + d^2$$
holds.
[i](Walther Janous)[/i]
2021 Poland - Second Round, 2
The point P lies on the side $CD$ of the parallelogram $ABCD$ with $\angle DBA = \angle CBP$. Point $O$ is the center of the circle passing through the points $D$ and $P$ and tangent to the straight line $AD$ at point $D$. Prove that $AO = OC$.
1993 Romania Team Selection Test, 2
Let $ABC$ be a triangle inscribed in the circle $\mathcal{C}(O,R)$ and circumscribed to the circle $\mathcal{C}(L,r)$. Denote $d=\dfrac{Rr}{R+r}$. Show that there exists a triangle $DEF$ such that for any interior point $M$ in $ABC$ there exists a point $X$ on the sides of $DEF$ such that $MX\le d$.
[i]Dan Brânzei[/i]
2020 DMO Stage 1, 3.
[b]Q .[/b]Prove that
$$\left(\sum_\text{cyc}(a-x)^4\right)\ +\ 2\left(\sum_\text{sym}x^3y\right)\ +\ 4\left(\sum_\text{cyc}x^2y^2\right)\ +\ 8xyza \geqslant \left(\sum_\text{cyc}(a-x)^2(a^2-x^2)\right)$$where $a=x+y+z$ and $x,y,z \in \mathbb{R}.$
[i]Proposed by srijonrick[/i]
2020 Latvia Baltic Way TST, 16
Given sequence $\{a_n\}$ satisfying:
$$ a_{n+1} = \frac{ lcm(a_n,a_{n-1})}{\gcd(a_n, a_{n-1})} $$
It is given that $a_{209} =209$ and $a_{361} = 361$. Find all possible values of $a_{2020}$.
2019 Junior Balkan Team Selection Tests - Romania, 4
The numbers from $1$ through $100$ are written in some order on a circle.
We call a pair of numbers on the circle [i]good [/i] if the two numbers are not neighbors on the circle and if at least one of the two arcs they determine on the circle only contains numbers smaller then both of them. What may be the total number of good pairs on the circle.
2011 Sharygin Geometry Olympiad, 4
Given the circle of radius $1$ and several its chords with the sum of lengths $1$. Prove that one can be inscribe a regular hexagon into that circle so that its sides don’t intersect those chords.
2019 Saudi Arabia JBMO TST, 1
Find the minimal positive integer $m$, so that there exist positive integers $n>k>1$, which satisfy
$11...1=11...1.m$, where the first number has $n$ digits $1$, and the second has $k$ digits $1$.
1975 AMC 12/AHSME, 27
If $p, q$ and $r$ are distinct roots of $x^3-x^2+x-2=0$, then $p^3+q^3+r^3$ equals
$ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{none of these} $
2009 Putnam, B2
A game involves jumping to the right on the real number line. If $ a$ and $ b$ are real numbers and $ b>a,$ the cost of jumping from $ a$ to $ b$ is $ b^3\minus{}ab^2.$ For what real numbers $ c$ can one travel from $ 0$ to $ 1$ in a finite number of jumps with total cost exactly $ c?$
2010 Iran Team Selection Test, 4
$S,T$ are two trees without vertices of degree 2. To each edge is associated a positive number which is called length of this edge. Distance between two arbitrary vertices $v,w$ in this graph is defined by sum of length of all edges in the path between $v$ and $w$. Let $f$ be a bijective function from leaves of $S$ to leaves of $T$, such that for each two leaves $u,v$ of $S$, distance of $u,v$ in $S$ is equal to distance of $f(u), f(v)$ in $T$. Prove that there is a bijective function $g$ from vertices of $S$ to vertices of $T$ such that for each two vertices $u,v$ of $S$, distance of $u,v$ in $S$ is equal to distance of $g(u)$ and $g(v)$ in $T$.
2020 Thailand TST, 6
Let $I$ be the incentre of acute-angled triangle $ABC$. Let the incircle meet $BC, CA$, and $AB$ at $D, E$, and $F,$ respectively. Let line $EF$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle DPA + \angle AQD =\angle QIP$.
(Slovakia)
2012 JBMO TST - Macedonia, 5
$ n\geq 4 $ points are given in a plane such that any 3 of them are not collinear. Prove that a triangle exist such that all the points are in its interior and there is exactly one point laying on each side.
1997 Moscow Mathematical Olympiad, 2
Prove that among the quadrilaterals with given lengths of the diagonals and the angle between them, the parallelogram has the least perimeter.
2024 Kyiv City MO Round 1, Problem 3
There are $2025$ people living on the island, each of whom is either a knight, i.e. always tells the truth, or a liar, which means they always lie. Some of the inhabitants of the island know each other, and everyone has at least one acquaintance, but no more than three. Each inhabitant of the island claims that there are exactly two liars among his acquaintances.
a) What is the smallest possible number of knights among the inhabitants of the island?
b) What is the largest possible number of knights among the inhabitants of the island?
[i]Proposed by Oleksii Masalitin[/i]
1949-56 Chisinau City MO, 48
Calculate $\sin^3 a + \cos^3 a$ if you know that $\sin a+ \cos a = m$.