Found problems: 85335
2010 South East Mathematical Olympiad, 2
For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.
2015 Denmark MO - Mohr Contest, 3
Triangle $ABC$ is equilateral. The point $D$ lies on the extension of $AB$ beyond $B$, the point $E$ lies on the extension of $CB$ beyond $B$, and $|CD| = |DE|$. Prove that $|AD| = |BE|$.
[img]https://1.bp.blogspot.com/-QnAXFw3ijn0/XzR0YjqBQ3I/AAAAAAAAMU0/0TvhMQtBNjolYHtgXsQo2OPGJzEYSfCwACLcBGAsYHQ/s0/2015%2BMohr%2Bp3.png[/img]
2016 Peru IMO TST, 2
Determine how many $100$-positive integer sequences satisfy the two conditions following:
- At least one term of the sequence is equal to $4$ or $5$.
- Any two adjacent terms differ as a maximum in $2$.
2001 Paraguay Mathematical Olympiad, 3
Find a $10$-digit number, in which no digit is zero, that is divisible by the sum of their digits.
2014 Hanoi Open Mathematics Competitions, 15
Let $a_1,a_2,...,a_9 \ge - 1$ and $a^3_1+a^3_2+...+a^3_9= 0$.
Determine the maximal value of $M = a_1 + a_2 + ... + a_9$.
2021 Denmark MO - Mohr Contest, 1
Georg has a set of sticks. From these sticks he must create a closed figure with the property that each stick makes right angles with its neighbouring sticks. All the sticks must be used. If the sticks have the lengths $1, 1, 2, 2, 2, 3, 3$ and $4$, the figure might for example look like this: [img]https://cdn.artofproblemsolving.com/attachments/9/7/c16a3143a52ec6f442208c63b41f2df1ae735c.png[/img]
(a) Prove that he can create such a figure if the sticks have the lengths $1, 1, 1, 2, 2, 3, 4$ and $4$.
(b) Prove that it cannot be done if the sticks have the lengths $1, 2, 2, 3, 3, 3, 4, 4$ and $4$.
(c) Determine whether it is doable if the sticks have the lengths $1, 2, 2, 2, 3, 3, 3, 4, 4$ and $5$.
2017 SDMO (High School), 5
There are $n$ dots on the plane such that no three dots are collinear. Each dot is assigned a $0$ or a $1$. Each pair of dots is connected by a line segment. If the endpoints of a line segment are two dots with the same number, then the segment is assigned a $0$. Otherwise, the segment is assigned a $1$. Find all $n$ such that it is possible to assign $0$'s and $1$'s to the $n$ dots in a way that the corresponding line segments are assigned equally many $0$'s as $1$'s.
2012 Indonesia TST, 3
Let $P_1P_2\ldots P_n$ be an $n$-gon such that for all $i,j \in \{1,2,\ldots,n\}$ where $i \neq j$, there exists $k \neq i,j$ such that $\angle P_iP_kP_j = 60^\circ$. Prove that $n=3$.
2011 LMT, 17
Let $ABC$ be a triangle with $AB = 15$, $AC = 20$, and right angle at $A$. Let $D$ be the point on $\overline{BC}$ such that $\overline{AD}$ is perpendicular to $\overline{BC}$, and let $E$ be the midpoint of $\overline{AC}$. If $F$ is the point on $\overline{BC}$ such that $\overline{AD} \parallel \overline{EF}$, what is the area of quadrilateral $ADFE$?
2010 Sharygin Geometry Olympiad, 6
An arbitrary line passing through vertex $B$ of triangle $ABC$ meets side $AC$ at point $K$ and the circumcircle in point $M$. Find the locus of circumcenters of triangles $AMK$.
2024 Israel National Olympiad (Gillis), P3
A triangle is composed of circular cells arranged in $5784$ rows: the first row has one cell, the second has two cells, and so on (see the picture). The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called [b]diagonal[/b] if the two cells in it [i]aren't[/i] in the same row. What is the minimum possible amount of diagonal pairs in the division?
An example division into pairs is depicted in the image.
2016 Cono Sur Olympiad, 1
Let $\overline{abcd}$ be one of the 9999 numbers $0001, 0002, 0003, \ldots, 9998, 9999$. Let $\overline{abcd}$ be an [i]special[/i] number if $ab-cd$ and $ab+cd$ are perfect squares, $ab-cd$ divides $ab+cd$ and also $ab+cd$ divides $abcd$. For example 2016 is special. Find all the $\overline{abcd}$ special numbers.
[b]Note:[/b] If $\overline{abcd}=0206$, then $ab=02$ and $cd=06$.
2024 Simon Marais Mathematical Competition, A2
A positive integer $n$ is [i] tripariable [/i] if it is possible to partition the set $\{1, 2, \dots, n\}$ into disjoint pairs such that the sum of two elements in each pair is a power of $3$. For example $6$ is tripariable because $\{1, 2, \dots, n\}=\{1,2\}\cup\{3,6\}\cup\{4,5\}$ and $$1+2=3^1,\quad 3+6 = 3^2\quad\text{and}\quad4+5=3^2$$ are all powers of 3.
How many positive integers less than or equal to 2024 are tripariable?
2023 Stars of Mathematics, 4
Determine all positive integers $n{}$ for which there exist pairwise distinct integers $a_1,\ldots,a_n{}$ and $b_1,\ldots, b_n$ such that \[\prod_{i=1}^n(a_k^2+a_ia_k+b_i)=\prod_{i=1}^n(b_k^2+a_ib_k+b_i)=0, \quad \forall k=1,\ldots,n.\]
2008 Purple Comet Problems, 19
One side of a triangle has length $75$. Of the other two sides, the length of one is double the length of the other. What is the maximum possible area for this triangle
2007 Moldova Team Selection Test, 3
Consider a triangle $ABC$, with corresponding sides $a,b,c$, inradius $r$ and circumradius $R$. If $r_{A}, r_{B}, r_{C}$ are the radii of the respective excircles of the triangle, show that
\[a^{2}\left(\frac 2{r_{A}}-\frac{r}{r_{B}r_{C}}\right)+b^{2}\left(\frac 2{r_{B}}-\frac{r}{r_{A}r_{C}}\right)+c^{2}\left(\frac 2{r_{C}}-\frac{r}{r_{A}r_{B}}\right)=4(R+3r) \]
1994 Bulgaria National Olympiad, 5
Let $k$ be a positive integer and $r_n$ be the remainder when ${2 n} \choose {n}$ is divided by $k$.
Find all $k$ for which the sequence $(r_n)_{n=1}^{\infty}$ is eventually periodic.
2021 OMpD, 4
Determine the smallest positive integer $n$ with the following property: on a board $n \times n$, whose squares are painted in checkerboard pattern (that is, for any two squares with a common edge, one of them is black and the other is white), it is possible to place the numbers $1,2,3 , ... , n^2$, a number in each square, so if $B$ is the sum of the numbers written in the white squares and $P$ is the sum of the numbers written in the black squares, then $\frac {B}{P} = \frac{2021}{4321}$.
2017 Azerbaijan Junior National Olympiad, P2
For all $n>1$ let $f(n)$ be the sum of the smallest factor of $n$ that is not 1 and $n$ . The computer prints $f(2),f(3),f(4),...$ with order:$4,6,6,...$ ( Because $f(2)=2+2=4,f(3)=3+3=6,f(4)=4+2=6$ etc.). In this infinite sequence, how many times will be $ 2015$ and $ 2016$ written? (Explain your answer)
2013 IFYM, Sozopol, 1
Let $u_1=1,u_2=2,u_3=24,$ and
$u_{n+1}=\frac{6u_n^2 u_{n-2}-8u_nu_{n-1}^2}{u_{n-1}u_{n-2}}, n \geq 3.$
Prove that the elements of the sequence are natural numbers and that $n\mid u_n$ for all $n$.
2005 Austrian-Polish Competition, 10
Determine all pairs $(k,n)$ of non-negative integers such that the following inequality holds $\forall x,y>0$:
\[1+ \frac{y^n}{x^k} \geq \frac{(1+y)^n}{(1+x)^k}.\]
1979 IMO Longlists, 17
Find the real values of $p$ for which the equation
\[\sqrt{2p+ 1 - x^2} +\sqrt{3x + p + 4} = \sqrt{x^2 + 9x+ 3p + 9}\]
in $x$ has exactly two real distinct roots.($\sqrt t $ means the positive square root of $t$).
2010 Korea National Olympiad, 2
Let $ a, b, c $ be positive real numbers such that $ ab+bc+ca=1 $. Prove that
\[ \sqrt{ a^2 + b^2 + \frac{1}{c^2}} + \sqrt{ b^2 + c^2 + \frac{1}{a^2}} + \sqrt{ c^2 + a^2 + \frac{1}{b^2}} \ge \sqrt{33} \]
2024 Middle European Mathematical Olympiad, 4
A finite sequence $x_1,\dots,x_r$ of positive integers is a [i]palindrome[/i] if $x_i=x_{r+1-i}$ for all integers
$1 \le i \le r$.
Let $a_1,a_2,\dots$ be an infinite sequence of positive integers. For a positive integer $j \ge 2$, denote by
$a[j]$ the finite subsequence $a_1,a_2,\dots,a_{j-1}$. Suppose that there exists a strictly increasing infinite
sequence $b_1,b_2,\dots$ of positive integers such that for every positive integer $n$, the subsequence
$a[b_n]$ is a palindrome and $b_{n+2} \le b_{n+1}+b_n$. Prove that there exists a positive integer $T$ such
that $a_i=a_{i+T}$ for every positive integer $i$.
2019 Bundeswettbewerb Mathematik, 3
Let $ABCD$ be a square. Choose points $E$ on $BC$ and $F$ on $CD$ so that $\angle EAF=45^\circ$ and so that neither $E$ nor $F$ is a vertex of the square.
The lines $AE$ and $AF$ intersect the circumcircle of the square in the points $G$ and $H$ distinct from $A$, respectively.
Show that the lines $EF$ and $GH$ are parallel.