Found problems: 85335
2024 Baltic Way, 7
A $45 \times 45$ grid has had the central unit square removed. For which positive integers $n$ is it possible to cut the remaining area into $1 \times n$ and $n\times 1$ rectangles?
2010 India IMO Training Camp, 3
For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:
(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$;
(b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$.
Determine $N(n)$ for all $n\ge 2$.
2014 ASDAN Math Tournament, 6
In triangle $ABC$, we have that $AB=AC$, $BC=16$, and that the area of $\triangle ABC$ is $120$. Compute the length of $AB$.
1986 All Soviet Union Mathematical Olympiad, 437
Prove that the sum of all numbers representable as $\frac{1}{mn}$, where $m,n$ -- natural numbers, $1 \le m < n \le1986$, is not an integer.
2017 China Team Selection Test, 1
Let $n$ be a positive integer. Let $D_n$ be the set of all divisors of $n$ and let $f(n)$ denote the smallest natural $m$ such that the elements of $D_n$ are pairwise distinct in mod $m$. Show that there exists a natural $N$ such that for all $n \geq N$, one has $f(n) \leq n^{0.01}$.
2001 Bulgaria National Olympiad, 3
Let $p$ be a prime number congruent to $3$ modulo $4$, and consider the equation $(p+2)x^{2}-(p+1)y^{2}+px+(p+2)y=1$.
Prove that this equation has infinitely many solutions in positive integers, and show that if $(x,y) = (x_{0}, y_{0})$ is a solution of the equation in positive integers, then $p | x_{0}$.
2001 Bundeswettbewerb Mathematik, 4
A square $ R$ of sidelength $ 250$ lies inside a square $ Q$ of sidelength $ 500$. Prove that: One can always find two points $ A$ and $ B$ on the perimeter of $ Q$ such that the segment $ AB$ has no common point with the square $ R$, and the length of this segment $ AB$ is greater than $ 521$.
2000 All-Russian Olympiad Regional Round, 8.6
The electric train traveled from platform A to platform B in $X$ minutes ($0< X<60$). Find $X$ if it is known that as at the moment departure from A, and at the time of arrival at B, the angle between hourly and the minute hand was equal to $X$ degrees.
2013 JBMO Shortlist, 4
A rectangle in xy Cartesian System is called latticed if all it's vertices have integer coordinates.
a) Find a latticed rectangle of area $2013$, whose sides are not parallel to the axes.
b) Show that if a latticed rectangle has area $2011$, then their sides are parallel to the axes.
2013 Paraguay Mathematical Olympiad, 5
Let $ABC$ be an obtuse triangle, with $AB$ being the largest side.
Draw the angle bisector of $\measuredangle BAC$. Then, draw the perpendiculars to this angle bisector from vertices $B$ and $C$, and call their feet $P$ and $Q$, respectively.
$D$ is the point in the line $BC$ such that $AD \perp AP$.
Prove that the lines $AD$, $BQ$ and $PC$ are concurrent.
1972 Bulgaria National Olympiad, Problem 4
Find maximal possible number of points lying on or inside a circle with radius $R$ in such a way that the distance between every two points is greater than $R\sqrt2$.
[i]H. Lesov[/i]
2013 AMC 10, 9
In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on $20\%$ of her three-point shots and $30\%$ of her two-point shots. Shenille attempted $30$ shots. How many points did she score?
$ \textbf{(A)}\ 12\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 36 $
2023 Math Prize for Girls Problems, 9
The ring shown below is made out of 18 congruent regular hexagons. How many ways are there to tile the ring using tiles that consist of two hexagons, each congruent to any one of the 18 in the design, joined edge-to-edge? (The central hexagon, in black, is not to be covered with a tile and the ring cannot be rotated or reflected.)
2020 Polish Junior MO First Round, 1.
Determine all natural numbers $n$, such that it's possible to insert one digit at the right side of $n$ to obtain $13n$.
2018 PUMaC Algebra B, 4
If $a_1, a_2, \ldots$ is a sequence of real numbers such that for all $n$,
$$\sum_{k = 1}^n a_k \left( \frac{k}{n} \right)^2 = 1,$$
find the smallest $n$ such that $a_n < \frac{1}{2018}$.
2021 South East Mathematical Olympiad, 5
To commemorate the $43rd$ anniversary of the restoration of mathematics competitions, a mathematics enthusiast
arranges the first $2021$ integers $1,2,\dots,2021$ into a sequence $\{a_n\}$ in a certain order, so that the sum of any consecutive $43$ items in the sequence is a multiple of $43$.
(1) If the sequence of numbers is connected end to end into a circle, prove that the sum of any consecutive $43$ items on the circle is also a multiple of $43$;
(2) Determine the number of sequences $\{a_n\}$ that meets the conditions of the question.
MOAA Team Rounds, 2021.17
Compute the remainder when $10^{2021}$ is divided by $10101$.
[i]Proposed by Nathan Xiong[/i]
2014 Belarus Team Selection Test, 1
Given triangle $ABC$ with $\angle A = a$. Let $AL$ be the bisector of the triangle $ABC$. Let the incircle of $\vartriangle ABC$ touch the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $X$ be the intersection point of the lines $AQ$ and $LP$. Prove that the lines $BX$ and $AL$ are perpendicular.
(V. Karamzin)
2020 Sharygin Geometry Olympiad, 6
Circles $\omega_1$ and $\omega_2$ meet at point $P,Q$. Let $O$ be the common point of external tangents of $\omega_1$ and $\omega_2$. A line passing through $O$ meets $\omega_1$ and $\omega_2$ at points $A,B$ located on the same side with respect to line segment $PQ$.The line $PA$ meets $\omega_2$ for the second time at $C$ and the line $QB$ meets $\omega_1$ for the second time at $D$. Prove that $O-C-D$ are collinear.
2022 AMC 12/AHSME, 18
Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive integer $n$ such that performing the sequence of transformations transformations $T_1, T_2, T_3, \dots, T_n$ returns the point $(1,0)$ back to itself?
$\textbf{(A) } 359 \qquad \textbf{(B) } 360\qquad \textbf{(C) } 719 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 721$
1991 Arnold's Trivium, 71
Calculate the charge density on the conducting boundary $x^2 + y^2 + z^2 = 1$ of a cavity in which a charge $q = 1$ is placed at distance $r$ from the centre.
1990 AMC 12/AHSME, 30
If $R_n=\frac{1}{2}(a^n+b^n)$ where $a=3+2\sqrt{2}$, $b=3-2\sqrt{2}$, and $n=0,1,2, ...,$ then $R_{12345}$ is an integer. Its units digit is
$ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 9 $
2018 Puerto Rico Team Selection Test, 2
Let $ABC$ be an acute triangle and let $P,Q$ be points on $BC$ such that $\angle QAC =\angle ABC$ and $\angle PAB = \angle ACB$. We extend $AP$ to $M$ so that $ P$ is the midpoint of $AM$ and we extend $AQ$ to $N$ so that $Q$ is the midpoint of $AN$. If T is the intersection point of $BM$ and $CN$, show that quadrilateral $ABTC$ is cyclic.
2010 China Girls Math Olympiad, 4
Let $x_1,x_2,\cdots,x_n$ be real numbers with $x_1^2+x_2^2+\cdots+x_n^2=1$. Prove that
\[\sum_{k=1}^{n}\left(1-\dfrac{k}{{\displaystyle \sum_{i=1}^{n} ix_i^2}}\right)^2 \cdot \dfrac{x_k^2}{k} \leq \left(\dfrac{n-1}{n+1}\right)^2 \sum_{k=1}^{n} \dfrac{x_k^2}{k}\]
Determine when does the equality hold?
2023 Caucasus Mathematical Olympiad, 4
Let $n>k>1$ be positive integers and let $G$ be a graph with $n$ vertices such that among any $k$ vertices, there is a vertex connected to the rest $k-1$ vertices. Find the minimal possible number of edges of $G$.
Proposed by V. Dolnikov