Found problems: 85335
1984 Tournament Of Towns, (061) O2
Six altitudes are constructed from the three vertices of the base of a tetrahedron to the opposite sides of the three lateral faces. Prove that all three straight lines joining two base points of the altitudes in each lateral face are parallel to a certain plane.
(IF Sharygin, Moscow)
2005 Bosnia and Herzegovina Team Selection Test, 5
If for an arbitrary permutation $(a_1,a_2,...,a_n)$ of set ${1,2,...,n}$ holds $\frac{{a_k}^2}{a_{k+1}}\leq k+2$,
$k=1,2,...,n-1$, prove that $a_k=k$ for $k=1,2,...,n$
2014 Spain Mathematical Olympiad, 1
Is it possible to place the numbers $0,1,2,\dots,9$ on a circle so that the sum of any three consecutive numbers is a) 13, b) 14, c) 15?
2018 Romania National Olympiad, 2
Let $a, b, c, d$ be natural numbers such that $a + b + c + d = 2018$. Find the minimum value of the expression:
$$E = (a-b)^2 + 2(a-c)^2 + 3(a-d)^2+4(b-c)^2 + 5(b-d)^2 + 6(c-d)^2.$$
2008 Purple Comet Problems, 11
When Tim was Jim’s age, Kim was twice as old as Jim. When Kim was Tim’s age, Jim was 30. When Jim becomes Kim’s age, Tim will be 88. When Jim becomes Tim’s age, what will be the sum of the ages of Tim, Jim, and Kim?
2014 Math Hour Olympiad, 8-10.1
Sherlock and Mycroft are playing Battleship on a $4\times4$ grid. Mycroft hides a single $3\times1$ cruiser somewhere on the board. Sherlock can pick squares on the grid and fire upon them. What is the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser?
2010 Estonia Team Selection Test, 2
Let $n$ be a positive integer. Find the largest integer $N$ for which there exists a set of $n$ weights such that it is possible to determine the mass of all bodies with masses of $1, 2, ..., N$ using a balance scale .
(i.e. to determine whether a body with unknown mass has a mass $1, 2, ..., N$, and which namely).
2021 Azerbaijan IZhO TST, 4
Let $ABC$ be a triangle with incircle touching $BC, CA, AB$ at $D, E,
F,$ respectively. Let $O$ and $M$ be its circumcenter and midpoint of $BC.$ Suppose that circumcircles of $AEF$ and $ABC$ intersect at $X$ for the second time. Assume $Y \neq X$ is on the circumcircle of $ABC$ such that $OMXY$ is cyclic. Prove that circumcenter of $DXY$ lies on $BC.$
[i]Proposed by tenplusten.[/i]
MOAA Team Rounds, 2021.20
Compute the sum of all integers $x$ for which there exists an integer $y$ such that
\[x^3+xy+y^3=503.\]
[i]Proposed by Nathan Xiong[/i]
2021 JHMT HS, 7
A number line with the integers $1$ through $20,$ from left to right, is drawn. Ten coins are placed along this number line, with one coin at each odd number on the line. A legal move consists of moving one coin from its current position to a position of strictly greater value on the number line that is not already occupied by another coin. How many ways can we perform two legal moves in sequence, starting from the initial position of the coins (different two-move sequences that result in the same position are considered distinct)?
1989 Spain Mathematical Olympiad, 6
Prove that among any seven real numbers there exist two,$ a$ and $b$, such that $\sqrt3|a-b|\le |1+ab|$.
Give an example of six real numbers not having this property.
2001 Brazil Team Selection Test, Problem 1
given that p,q are two polynomials such that each one has at least one root and \[p(1+x+q(x)^2)=q(1+x+p(x)^2)\] then prove that p=q
2011 Purple Comet Problems, 28
Pictured below is part of a large circle with radius $30$. There is a chain of three circles with radius $3$, each internally tangent to the large circle and each tangent to its neighbors in the chain. There are two circles with radius $2$ each tangent to two of the radius $3$ circles. The distance between the centers of the two circles with radius $2$ can be written as $\textstyle\frac{a\sqrt b-c}d$, where $a,b,c,$ and $d$ are positive integers, $c$ and $d$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a+b+c+d$.
[asy]
size(200);
defaultpen(linewidth(0.5));
real r=aCos(79/81);
pair x=dir(270+r)*27,y=dir(270-r)*27;
draw(arc(origin,30,210,330));
draw(circle(x,3)^^circle(y,3)^^circle((0,-27),3));
path arcl=arc(y,5,0,180), arcc=arc((0,-27),5,0,180), arcr=arc(x,5,0,180);
pair centl=intersectionpoint(arcl,arcc), centr=intersectionpoint(arcc,arcr);
draw(circle(centl,2)^^circle(centr,2));
dot(x^^y^^(0,-27)^^centl^^centr,linewidth(2));
[/asy]
2015 Taiwan TST Round 2, 2
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively.
Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.
[i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]
2011 China Second Round Olympiad, 3
Given $n\ge 4$ real numbers $a_{n}>...>a_{1} > 0$. For $r > 0$, let $f_{n}(r)$ be the number of triples $(i,j,k)$ with $1\leq i<j<k\leq n$ such that $\frac{a_{j}-a_{i}}{a_{k}-a_{j}}=r$. Prove that ${f_{n}(r)}<\frac{n^{2}}{4}$.
2019 Jozsef Wildt International Math Competition, W. 22
Let $A$ and $B$ the series: $$A=\sum \limits_{n=1}^{\infty}\frac{C_{2n}^1}{C_{2n}^0+C_{2n}^1+\cdots +C_{2n}^{2n}},\ B=\sum \limits_{n=1}^{\infty}\frac{\Gamma \left(n+\frac{1}{2}\right) }{\Gamma \left(n+\frac{5}{2}\right)}$$Study if $\frac{A}{B}$ is irrational number.
Mid-Michigan MO, Grades 7-9, 2014
[b]p1.[/b] (a) Put the numbers $1$ to $6$ on the circle in such way that for any five consecutive numbers the sum of first three (clockwise) is larger than the sum of remaining two.
(b) Can you arrange these numbers so it works both clockwise and counterclockwise.
[b]p2.[/b] A girl has a box with $1000$ candies. Outside the box there is an infinite number of chocolates and muffins. A girl may replace:
$\bullet$ two candies in the box with one chocolate bar,
$\bullet$ two muffins in the box with one chocolate bar,
$\bullet$ two chocolate bars in the box with one candy and one muffin,
$\bullet$ one candy and one chocolate bar in the box with one muffin,
$\bullet$ one muffin and one chocolate bar in the box with one candy.
Is it possible that after some time it remains only one object in the box?
[b]p3.[/b] Find any integer solution of the puzzle: $WE+ST+RO+NG=128$ (different letters mean different digits between $1$ and $9$).
[b]p4.[/b] Two consecutive three‐digit positive integer numbers are written one after the other one. Show that the six‐digit number that is obtained is not divisible by $1001$.
[b]p5.[/b] There are $9$ straight lines drawn in the plane. Some of them are parallel some of them intersect each other. No three lines do intersect at one point. Is it possible to have exactly $17$ intersection points?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
PEN O Problems, 14
Let $p$ be a prime number, $p \ge 5$, and $k$ be a digit in the $p$-adic representation of positive integers. Find the maximal length of a non constant arithmetic progression whose terms do not contain the digit $k$ in their $p$-adic representation.
2011 Mongolia Team Selection Test, 1
Let $A=\{a^2+13b^2 \mid a,b \in\mathbb{Z}, b\neq0\}$. Prove that there
a) exist
b) exist infinitely many
$x,y$ integer pairs such that $x^{13}+y^{13} \in A$ and $x+y \notin A$.
(proposed by B. Bayarjargal)
2011 Mongolia Team Selection Test, 1
A group of the pupils in a class are called [i]dominant[/i] if any other pupil from the class has a friend in the group. If it is known that there exist at least 100 dominant groups, prove that there exists at least one more dominant group.
(proposed by B. Batbayasgalan, inspired by Komal problem)
2007 Belarusian National Olympiad, 7
Find solution in positive integers : $$n^5+n^4=7^m-1$$
2007 District Olympiad, 4
Let $A,B\in \mathcal{M}_n(\mathbb{R})$ such that $B^2=I_n$ and $A^2=AB+I_n$. Prove that:
\[\det A\le \left(\frac{1+\sqrt{5}}{2}\right)^n\]
2022 HMIC, 5
Let $\mathbb{F}_p$ be the set of integers modulo $p$. Call a function $f : \mathbb{F}_p^2 \to \mathbb{F}_p$ [i]quasiperiodic[/i] if there exist $a,b \in \mathbb{F}_p$, not both zero, so that $f(x + a, y + b) = f(x, y)$ for all $x,y \in \mathbb{F}_p$.
Find the number of functions $\mathbb{F}_p^2 \to \mathbb{F}_p$ that can be written as the sum of some number of quasiperiodic functions.
2022 Putnam, A3
Let $p$ be a prime number greater than 5. Let $f(p)$ denote the number of infinite sequences $a_1, a_2, a_3,\ldots$ such that $a_n \in \{1, 2,\ldots, p-1\}$ and $a_na_{n+2}\equiv1+a_{n+1}$ (mod $p$) for all $n\geq 1.$ Prove that $f(p)$ is congruent to 0 or 2 (mod 5).
Today's calculation of integrals, 883
Prove that for each positive integer $n$
\[\frac{4n^2+1}{4n^2-1}\int_0^{\pi} (e^{x}-e^{-x})\cos 2nx\ dx>\frac{e^{\pi}-e^{-\pi}-2}{4}\ln \frac{(2n+1)^2}{(2n-1)(n+3)}.\]