Found problems: 85335
2022 Yasinsky Geometry Olympiad, 1
In the triangle $ABC$, the median $AM$ is extended to the intersection with the circumscribed circle at point $D$. It is known that $AB = 2AM$ and $AD = 4AM$. Find the angles of the triangle $ABC$.
(Gryhoriy Filippovskyi)
2015 Auckland Mathematical Olympiad, 1
The teacher wrote on the blackboard quadratic polynomial $x^2 + 10x + 20$. Then in turn each student in the class either increased or decreased by $1$ either the coefficient of $x$ or the constant term. At the end the quadratic polynomial became $x^2+20x+10$. Is it true that at certain moment a quadratic polynomial with integer roots was on the board?
2010 Princeton University Math Competition, 1
Let the operation $\bigstar$ be defined by $x\bigstar y=y^x-xy$. Calculate $(3\bigstar4)-(4\bigstar3)$.
2010 Sharygin Geometry Olympiad, 7
Each of two regular polygons $P$ and $Q$ was divided by a line into two parts. One part of $P$ was attached to one part of $Q$ along the dividing line so that the resulting polygon was regular and not congruent to $P$ or $Q$. How many sides can it have?
1970 IMO Longlists, 35
Find for every value of $n$ a set of numbers $p$ for which the following statement is true: Any convex $n$-gon can be divided into $p$ isosceles triangles.
2011 Bangladesh Mathematical Olympiad, HS
[size=130][b]Higher Secondary: 2011[/b]
[/size]
Time: 4 Hours
[b]Problem 1:[/b]
Prove that for any non-negative integer $n$ the numbers $1, 2, 3, ..., 4n$ can be divided in tow mutually exclusive classes with equal number of members so that the sum of numbers of each class is equal.
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=709
[b]Problem 2:[/b]
In the first round of a chess tournament, each player plays against every other player exactly once. A player gets $3, 1$ or $-1$ points respectively for winning, drawing or losing a match. After the end of the first round, it is found that the sum of the scores of all the players is $90$. How many players were there in the tournament?
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=708
[b]Problem 3:[/b]
$E$ is the midpoint of side $BC$ of rectangle $ABCD$. $A$ point $X$ is chosen on $BE$. $DX$ meets extended $AB$ at $P$. Find the position of $X$ so that the sum of the areas of $\triangle BPX$ and $\triangle DXC$ is maximum with proof.
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=683
[b]Problem 4:[/b]
Which one is larger 2011! or, $(1006)^{2011}$? Justify your answer.
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=707
[b]Problem 5:[/b]
In a scalene triangle $ABC$ with $\angle A = 90^{\circ}$, the tangent line at $A$ to its circumcircle meets line $BC$ at $M$ and the incircle touches $AC$ at $S$ and $AB$ at $R$. The lines $RS$ and $BC$ intersect at $N$ while the lines $AM$ and $SR$ intersect at $U$. Prove that the triangle $UMN$ is isosceles.
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=706
[b]Problem 6:[/b]
$p$ is a prime and sum of the numbers from $1$ to $p$ is divisible by all primes less or equal to $p$. Find the value of $p$ with proof.
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=693
[b]Problem 7:[/b]
Consider a group of $n > 1$ people. Any two people of this group are related by mutual friendship or mutual enmity. Any friend of a friend and any enemy of an enemy is a friend. If $A$ and $B$ are friends/enemies then we count it as $1$ [b]friendship/enmity[/b]. It is observed that the number of friendships and number of enmities are equal in the group. Find all possible values of $n$.
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=694
[b]Problem 8:[/b]
$ABC$ is a right angled triangle with $\angle A = 90^{\circ}$ and $D$ be the midpoint of $BC$. A point $F$ is chosen on $AB$. $CA$ and $DF$ meet at $G$ and $GB \parallel AD$. $CF$ and $AD$ meet at $O$ and $AF = FO$. $GO$ meets $BC$ at $R$. Find the sides of $ABC$ if the area of $GDR$ is $\dfrac{2}{\sqrt{15}}$
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=704
[b]Problem 9:[/b]
The repeat of a natural number is obtained by writing it twice in a row (for example, the repeat of $123$ is $123123$). Find a positive integer (if any) whose repeat is a perfect square.
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=703
[b]Problem 10:[/b]
Consider a square grid with $n$ rows and $n$ columns, where $n$ is odd (similar to a chessboard). Among the $n^2$ squares of the grid, $p$ are black and the others are white. The number of black squares is maximized while their arrangement is such that horizontally, vertically or diagonally neighboring black squares are separated by at least one white square between them. Show that there are infinitely many triplets of integers $(p, q, n)$ so that the number of white squares is $q^2$.
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=702
The problems of the Junior categories are available in [url=http://matholympiad.org.bd/forum/]BdMO Online forum[/url]:
http://matholympiad.org.bd/forum/viewtopic.php?f=25&t=678
2022 Centroamerican and Caribbean Math Olympiad, 3
Let $ABC$ an acutangle triangle with orthocenter $H$ and circumcenter $O$. Let $D$ the intersection of $AO$ and $BH$. Let $P$ be the point on $AB$ such that $PH=PD$. Prove that the points $B, D, O$ and $P$ lie on a circle.
2025 District Olympiad, P2
Solve in $\mathbb{R}$ the equation $$\frac{1}{x}+\frac{1}{\lfloor x\rfloor} + \frac{1}{\{x\}} = 0.$$
[i]Mathematical Gazette[/i]
2009 China National Olympiad, 1
Given an acute triangle $ PBC$ with $ PB\neq PC.$ Points $ A,D$ lie on $ PB,PC,$ respectively. $ AC$ intersects $ BD$ at point $ O.$ Let $ E,F$ be the feet of perpendiculars from $ O$ to $ AB,CD,$ respectively. Denote by $ M,N$ the midpoints of $ BC,AD.$
$ (1)$: If four points $ A,B,C,D$ lie on one circle, then $ EM\cdot FN \equal{} EN\cdot FM.$
$ (2)$: Determine whether the converse of $ (1)$ is true or not, justify your answer.
1988 AMC 12/AHSME, 10
In an experiment, a scientific constant $C$ is determined to be $2.43865$ with an error of at most $\pm 0.00312$. The experimenter wishes to announce a value for $C$ in which every digit is significant. That is, whatever $C$ is, the announced value must be the correct result when C is rounded to that number of digits. The most accurate value the experimenter can announce for $C$ is
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 2.4\qquad\textbf{(C)}\ 2.43\qquad\textbf{(D)}\ 2.44\qquad\textbf{(E)}\ 2.439 $
2016 IMC, 2
Let $k$ and $n$ be positive integers. A sequence $\left( A_1, \dots , A_k \right)$ of $n\times n$ real matrices is [i]preferred[/i] by Ivan the Confessor if $A_i^2\neq 0$ for $1\le i\le k$, but $A_iA_j=0$ for $1\le i$, $j\le k$ with $i\neq j$. Show that $k\le n$ in all preferred sequences, and give an example of a preferred sequence with $k=n$ for each $n$.
(Proposed by Fedor Petrov, St. Petersburg State University)
2016 Mathematical Talent Reward Programme, MCQ: P 12
Let $f(x)=(x-1)(x-2)(x-3)$. Consider $g(x)=min\{f(x),f'(x)\}$. Then the number of points of discontinuity are
[list=1]
[*] 0
[*] 1
[*] 2
[*] More than 2
[/list]
2015 Online Math Open Problems, 15
Let $a$, $b$, $c$, and $d$ be positive real numbers such that
\[a^2 + b^2 - c^2 - d^2 = 0 \quad \text{and} \quad a^2 - b^2 - c^2 + d^2 = \frac{56}{53}(bc + ad).\]
Let $M$ be the maximum possible value of $\tfrac{ab+cd}{bc+ad}$. If $M$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $100m + n$.
[i]Proposed by Robin Park[/i]
1954 Moscow Mathematical Olympiad, 273
Given a piece of graph paper with a letter assigned to each vertex of every square such that on every segment connecting two vertices that have the same letter and are on the same line of the mesh, there is at least one vertex with another letter. What is the least number of distinct letters needed to plot such a picture, along the sides of the cells?
Russian TST 2019, P1
Suppose that $A$, $B$, $C$, and $D$ are distinct points, no three of which lie on a line, in the Euclidean plane. Show that if the squares of the lengths of the line segments $AB$, $AC$, $AD$, $BC$, $BD$, and $CD$ are rational numbers, then the quotient
\[\frac{\mathrm{area}(\triangle ABC)}{\mathrm{area}(\triangle ABD)}\]
is a rational number.
2019 Iran Team Selection Test, 5
A sub-graph of a complete graph with $n$ vertices is chosen such that the number of its edges is a multiple of $3$ and degree of each vertex is an even number. Prove that we can assign a weight to each triangle of the graph such that for each edge of the chosen sub-graph, the sum of the weight of the triangles that contain that edge equals one, and for each edge that is not in the sub-graph, this sum equals zero.
[i]Proposed by Morteza Saghafian[/i]
2022-IMOC, C1
Given a positive integer $k$, a pigeon and a seagull play a game on an $n\times n$ board. The pigeon goes first, and they take turns doing the operations. The pigeon will choose $m$ grids and lay an egg in each grid he chooses. The seagull will choose a $k\times k$ grids and eat all the eggs inside them. If at any point every grid in the $n\times n $ board has an egg in it, then the pigeon wins. Else, the seagull wins. For every integer $n\geq k$, find all $m$ such that the pigeon wins.
[i]Proposed by amano_hina[/i]
2018 Hanoi Open Mathematics Competitions, 2
In triangle $ABC,\angle BAC = 60^o, AB = 3a$ and $AC = 4a, (a > 0)$. Let $M$ be point on the segment $AB$ such that $AM =\frac13 AB, N$ be point on the side $AC$ such that $AN =\frac12AC$. Let $I$ be midpoint of $MN$. Determine the length of $BI$.
A. $\frac{a\sqrt2}{19}$ B. $\frac{2a}{\sqrt{19}}$ C. $\frac{19a\sqrt{19}}{2}$ D. $\frac{19a}{\sqrt2}$ E. $\frac{a\sqrt{19}}{2}$
2021 JHMT HS, 5
For real numbers $x,$ let $T_x$ be the triangle with vertices $(5, 5^3),$ $(8, 8^3),$ and $(x, x^3)$ in $\mathbb{R}^2.$ Over all $x$ in the interval $[5, 8],$ the area of the triangle $T_x$ is maximized at $x = \sqrt{n},$ for some positive integer $n.$ Compute $n.$
1988 IMO Longlists, 91
A regular 14-gon with side $a$ is inscribed in a circle of radius one. Prove \[ \frac{2-a}{2 \cdot a} > \sqrt{3 \cdot \cos \left( \frac{\pi}{7} \right)}. \]
2011 Saudi Arabia Pre-TST, 2.4
Let $ABCD$ be a rectangle of center $O$, such that $\angle DAC = 60^o$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$ and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.
2001 All-Russian Olympiad Regional Round, 8.7
Is it possible to paint the cells of a $5\times 5$ board in $4$ colors so that the cells standing at the intersection of any two rows and any two columns were painted in at least $ 3$ colors?
2020 USOJMO, 3
An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A [i]beam[/i] is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:
[list=]
[*]The two $1 \times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \cdot {2020}^2$ possible positions for a beam.)
[*]No two beams have intersecting interiors.
[*]The interiors of each of the four $1 \times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam.
[/list]
What is the smallest positive number of beams that can be placed to satisfy these conditions?
[i]Proposed by Alex Zhai[/i]
2011 AMC 12/AHSME, 18
A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
$ \textbf{(A)}\ 5\sqrt{2}-7 \qquad
\textbf{(B)}\ 7-4\sqrt{3} \qquad
\textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad
\textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad
\textbf{(E)}\ \frac{\sqrt{3}}{9} $
2010 LMT, 6
Al travels for $20$ miles per hour rolling down a hill in his chair for two hours, then four miles per hour climbing a hill for six hours. What is his average speed, in miles per hour?