Found problems: 85335
2012 Online Math Open Problems, 36
Let $s_n$ be the number of solutions to $a_1 + a_2 + a_3 +a _4 + b_1 + b_2 = n$, where $a_1,a_2,a_3$ and $a_4$ are elements of the set $\{2, 3, 5, 7\}$ and $b_1$ and $b_2$ are elements of the set $\{ 1, 2, 3, 4\}$. Find the number of $n$ for which $s_n$ is odd.
[i]Author: Alex Zhu[/i]
[hide="Clarification"]$s_n$ is the number of [i]ordered[/i] solutions $(a_1, a_2, a_3, a_4, b_1, b_2)$ to the equation, where each $a_i$ lies in $\{2, 3, 5, 7\}$ and each $b_i$ lies in $\{1, 2, 3, 4\}$. [/hide]
2021 BMT, 23
Shivani has a single square with vertices labeled $ABCD$. She is able to perform the following transformations:
$\bullet$ She does nothing to the square.
$\bullet$ She rotates the square by $90$, $180$, or $270$ degrees.
$\bullet$ She reflects the square over one of its four lines of symmetry.
For the first three timesteps, Shivani only performs reflections or does nothing. Then for the next three timesteps, she only performs rotations or does nothing. She ends up back in the square’s original configuration. Compute the number of distinct ways she could have achieved this.
1906 Eotvos Mathematical Competition, 2
Let $K, L,M,N$ designate the centers of the squares erected on the four sides (outside) of a rhombus. Prove that the polygon $KLMN$ is a square.
2021 Belarusian National Olympiad, 9.8
Given a positive integer $n$. An inversion of a permutation is the amount of pairs $(i,j)$ such that $i<j$ and the $i$-th number is smaller than $j$-th number in the permutation.
Prove that for every positive integer $k \leq n$ there exist exactly $\frac{n!}{k}$ permutations in which the inversion is divisible by $k$.
2010 Brazil Team Selection Test, 1
For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied:
[list][*] $a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$,
[*] If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$[/list]
Determine $N(n)$ for all $n\geq 2$.
[i]Proposed by Dan Schwarz, Romania[/i]
2003 Iran MO (2nd round), 1
We call the positive integer $n$ a $3-$[i]stratum[/i] number if we can divide the set of its positive divisors into $3$ subsets such that the sum of each subset is equal to the others.
$a)$ Find a $3-$stratum number.
$b)$ Prove that there are infinitely many $3-$stratum numbers.
2012 Kosovo National Mathematical Olympiad, 1
Find the value of
$(1+2)(1+2^2)(1+2^4)(1+2^8)...(1+2^{2048})$.
2010 CHMMC Fall, 1
Susan plays a game in which she rolls two fair standard six-sided dice with sides labeled one
through six. She wins if the number on one of the dice is three times the number on the other
die. If Susan plays this game three times, compute the probability that she wins at least once.
1999 Kazakhstan National Olympiad, 1
Prove that for any real numbers $ a_1, a_2, \dots, a_ {100} $ there exists a real number $ b $ such that all numbers $ a_i + b $ ($ 1 \leq i \leq 100 $) are irrational.
2008 Tuymaada Olympiad, 4
Point $ I_1$ is the reflection of incentre $ I$ of triangle $ ABC$ across the side $ BC$. The circumcircle of $ BCI_1$ intersects the line $ II_1$ again at point $ P$. It is known that $ P$ lies outside the incircle of the triangle $ ABC$. Two tangents drawn from $ P$ to the latter circle touch it at points $ X$ and $ Y$. Prove that the line $ XY$ contains a medial line of the triangle $ ABC$.
[i]Author: L. Emelyanov[/i]
2021 CHMMC Winter (2021-22), 1
Let $ABC$ be a right triangle with hypotenuse $\overline{AC}$ and circumcenter $O$. Point $E$ lies on $\overline{AB}$ such that $AE = 9$, $EB = 3$, point $F$ lies on $\overline{BC}$ such that $BF = 6$, $FC = 2$. Now suppose $W, X, Y$, and $Z$ are the midpoints of $\overline{EB}$, $\overline{BF}$, $\overline{FO}$, and $\overline{OE}$, respectively. Compute the area of quadrilateral $W XY Z$.
1975 Bulgaria National Olympiad, Problem 4
In the plane are given a circle $k$ with radii $R$ and the points $A_1,A_2,\ldots,A_n$, lying on $k$ or outside $k$. Prove that there exist infinitely many points $X$ from the given circumference for which
$$\sum_{i=1}^n A_iX^2\ge2nR^2.$$
Does there exist a pair of points on different sides of some diameter, $X$ and $Y$ from $k$, such that
$$\sum_{i=1}^n A_iX^2\ge2nR^2\text{ and }\sum_{i=1}^n A_iY^2\ge2nR^2?$$
[i]H. Lesov[/i]
2010 National Olympiad First Round, 21
A right circular cone and a right cylinder with same height $20$ does not have same circular base but the circles are coplanar and their centers are same. If the cone and the cylinder are at the same side of the plane and their base radii are $20$ and $10$, respectively, what is the ratio of the volume of the part of the cone inside the cylinder over the volume of the part of the cone outside the cylinder?
$ \textbf{(A)}\ 3
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ \frac{5}{3}
\qquad\textbf{(D)}\ \frac{4}{3}
\qquad\textbf{(E)}\ 1
$
2013 Tournament of Towns, 5
A point in the plane is called a node if both its coordinates are integers. Consider a triangle with vertices at nodes containing exactly two nodes inside. Prove that the straight line connecting these nodes either passes through a vertex or is parallel to a side of the triangle.
2005 IMC, 6
6. If $ p,q$ are rationals, $r=p+\sqrt{7}q$, then prove there exists a matrix
$\left(\begin{array}{cc}a&b\\c&d\end{array}\right) \in M_{2}(Z)- ( \pm I_{2})$
for which $\frac{ar+b}{cr+d}=r$ and $det(A)=1$
Ukrainian TYM Qualifying - geometry, I.5
The heights of a triangular pyramid intersect at one point. Prove that all flat angles at any vertex of the surface are either acute, or right, or obtuse.
2010 Iran MO (3rd Round), 3
prove that for each natural number $n$ there exist a polynomial with degree $2n+1$ with coefficients in $\mathbb{Q}[x]$ such that it has exactly $2$ complex zeros and it's irreducible in $\mathbb{Q}[x]$.(20 points)
1998 Polish MO Finals, 1
Define the sequence $a_1, a_2, a_3, ...$ by $a_1 = 1$, $a_n = a_{n-1} + a_{[n/2]}$. Does the sequence contain infinitely many multiples of $7$?
1990 China Team Selection Test, 3
Prove that for every integer power of 2, there exists a multiple of it with all digits (in decimal expression) not zero.
2025 Caucasus Mathematical Olympiad, 1
For given positive integers $a$ and $b$, let us consider the equation$$a + \gcd(b, x) = b + \gcd(a, x).$$
[list=a]
[*]For $a = 20$ and $b = 25$, find the least positive integer $x$ satisfying this equation.
[*]Prove that for any positive integers $a$ and $b$, there exist infinitely many positive integers $x$ satisfying this equation.
[/list]
[i](Here, $\gcd(m, n)$ denotes the greatest common divisor of positive integers $m$ and $n$.)[/i]
2015 Gulf Math Olympiad, 2
a) Let $UVW$ , $U'V'W'$ be two triangles such that $ VW = V'W' , UV = U'V' , \angle WUV = \angle W'U'V'.$
Prove that the angles $\angle VWU , \angle V'W'U'$ are equal or supplementary.
b) $ABC$ is a triangle where $\angle A$ is [b]obtuse[/b]. take a point $P$ inside the triangle , and extend $AP,BP,CP$ to meet the sides $BC,CA,AB$ in $K,L,M$ respectively. Suppose that $PL = PM .$
1) If $AP$ bisects $\angle A$ , then prove that $AB = AC$ .
2) Find the angles of the triangle $ABC$ if you know that $AK,BL,CM$ are angle bisectors of the triangle $ABC$ and that $2AK = BL$.
PEN S Problems, 36
For every natural number $n$, denote $Q(n)$ the sum of the digits in the decimal representation of $n$. Prove that there are infinitely many natural numbers $k$ with $Q(3^{k})>Q(3^{k+1})$.
VMEO III 2006 Shortlist, N14
For any natural number $n = \overline{a_i...a_2a_1}$, consider the number $$T(n) =10 \sum_{i \,\, even} a_i+\sum_{i \,\, odd} a_i.$$ Let's find the smallest positive integer $A$ such that is sum of the natural numbers $n_1,n_2,...,n_{148}$ as well as of $m_1,m_2,...,m_{149}$ and matches the pattern:
$A=n_1+n_2+...+n_{148}=m_1+m_2+...+m_{149}$
$T(n_1)=T(n_2)=...=T(n_{148})$
$T(m_1)=T(m_2)=...=T(m_{148})$
2006 District Olympiad, 2
Let $n,p \geq 2$ be two integers and $A$ an $n\times n$ matrix with real elements such that $A^{p+1} = A$.
a) Prove that $\textrm{rank} \left( A \right) + \textrm{rank} \left( I_n - A^p \right) = n$.
b) Prove that if $p$ is prime then \[ \textrm{rank} \left( I_n - A \right) = \textrm{rank} \left( I_n - A^2 \right) = \ldots = \textrm{rank} \left( I_n - A^{p-1} \right) . \]
2023 SAFEST Olympiad, 3
A binoku is a $9 \times 9$ grid that is divided into nine $3 \times 3$ subgrids with the following properties:
- each cell contains either a $0$ or a $1$,
- each row contains at least one $0$ and at least one $1$,
- each column contains at least one $0$ and at least one $1$, and
- each of the nine subgrids contains at least one $0$ and at least one $1$.
An incomplete binoku is obtained from a binoku by removing the numbers from some of the cells. What is the largest number of empty cells that an incomplete binoku can contain if it can be completed into a binoku in a unique way?
[i]Proposed by Stijn Cambie, South Korea[/i]