Found problems: 85335
2023 Flanders Math Olympiad, 3
The vertices of a regular $4$-gon, $6$-gon and $12$-goncan be brought together in one point to form a complete angle of $360^o$ (see figure). [center][img]https://cdn.artofproblemsolving.com/attachments/b/1/e9245179b7e0f5acb98b226bdc6db87fd72ad5.png[/img] [/center]
Determine all triples $a, b, c \in N$ with $a < b < c$ for which the angles of a regular $a$-gon, $b$-gon and $c$-gon together also form $360^o$ .
2023 District Olympiad, P2
Let $(G,\cdot)$ be a grup with neutral element $e{}$, and let $H{}$ and $K$ be proper subgroups of $G$, satisfying $H\cap K=\{e\}$. It is known that $(G\setminus(H\cup K))\cup\{e\}$ is closed under the operation of $G$. Prove that $x^2=e$ for all the elements $x{}$ of $G{}$.
2002 Turkey MO (2nd round), 2
Let $ABC$ be a triangle, and points $D,E$ are on $BA,CA$ respectively such that $DB=BC=CE$. Let $O,I$ be the circumcenter, incenter of $\triangle ABC$. Prove that the circumradius of $\triangle ADE$ is equal to $OI$.
2013 Purple Comet Problems, 21
Evaluate $(2-\sec^2{1^\circ})(2-\sec^2{2^\circ})(2-\sec^2{3^\circ})\cdots(2-\sec^2{89^\circ}).$
2021 AMC 10 Fall, 13
A square with side length $3$ is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length $2$ has two vertices on the other square and the other two on sides of the triangle, as shown. What is the area of the triangle?
[asy]
//diagram by kante314
draw((0,0)--(8,0)--(4,8)--cycle, linewidth(1.5));
draw((2,0)--(2,4)--(6,4)--(6,0)--cycle, linewidth(1.5));
draw((3,4)--(3,6)--(5,6)--(5,4)--cycle, linewidth(1.5));
[/asy]
$(\textbf{A})\: 19\frac14\qquad(\textbf{B}) \: 20\frac14\qquad(\textbf{C}) \: 21 \frac34\qquad(\textbf{D}) \: 22\frac12\qquad(\textbf{E}) \: 23\frac34$
Swiss NMO - geometry, 2014.1
The points $A, B, C$ and $D$ lie in this order on the circle $k$. Let $t$ be the tangent at $k$ through $C$ and $s$ the reflection of $AB$ at $AC$. Let $G$ be the intersection of the straight line $AC$ and $BD$ and $H$ the intersection of the straight lines $s$ and $CD$. Show that $GH$ is parallel to $t$.
2024 Taiwan TST Round 2, G
Let $ABC$ be a triangle with $O$ as its circumcenter. A circle $\Gamma$ tangents $OB, OC$ at $B, C$, respectively. Let $D$ be a point on $\Gamma$ other than $B$ with $CB=CD$, $E$ be the second intersection of $DO$ and $\Gamma$, and $F$ be the second intersection of $EA$ and $\Gamma$. Let $X$ be a point on the line $AC$ so that $XB\perp BD$. Show that one half of $\angle ADF$ is equal to one of $\angle BDX$ and $\angle BXD$.
[i]Proposed by usjl[/i]
2005 Tournament of Towns, 3
Among 6 coins one is counterfeit (its weight differs from that real one and neither weights is known). Using scales that show the total weight of coins placed on the cup, find the counterfeit coin in 3 weighings.
[i](4 points)[/i]
Revenge EL(S)MO 2024, 2
Prove that for any convex quadrilateral there exist an inellipse and circumellipse which are homothetic.
Proposed by [i]Benny Wang + Oron Wang[/i]
2021 USMCA, 27
You are participating in a virtual stock market, with many different stocks. For a stock $S$, there is a list of prices where the $i$th number is the price of the stock on day $i$. On each day $i$, you are given the stock's current price (in dollars), and you can either buy a share of stock $S$, sell your share of stock $S$, or do nothing, but you may only take one of these actions per day, and you may not have more than one share of stock $S$ at a time. Each stock is independent, so for example on the first day, you may buy a share of $S$ and a share of $T$, and on the second day you may sell your share of $T$.
At USMCA Trading LLC, you are given $2021!$ different stocks, where each stock's list of prices corresponds to a unique permutation of the first $2021$ positive integers, to trade for $2021$ days. You start out with $M$ dollars, and at the end of $2021$ days, you end up with $N$ dollars. Assume $M$ is large enough so that you can never run out of money during the $2021$ days. What is the maximum possible value of $N - M$?
2008 Iran MO (3rd Round), 3
Prove that for each $ n$:
\[ \sum_{k\equal{}1}^n\binom{n\plus{}k\minus{}1}{2k\minus{}1}\equal{}F_{2n}\]
2005 Federal Math Competition of S&M, Problem 4
Inside a circle $k$ of radius $R$ some round spots are made. The area of each spot is $1$. Every radius of circle $k$, as well as every circle concentric with $k$, meets in no more than one spot. Prove that the total area of all the spots is less than
$$\pi\sqrt R+\frac12R\sqrt R.$$
Maryland University HSMC part II, 1999
[b]p1.[/b] Twelve tables are set up in a row for a Millenium party. You want to put $2000$ cupcakes on the tables so that the numbers of cupcakes on adjacent tables always differ by one (for example, if the $5$th table has $20$ cupcakes, then the $4$th table has either $19$ or $21$ cupcakes, and the $6$th table has either $19$ or $21$ cupcakes).
a) Find a way to do this.
b) Suppose a Y2K bug eats one of the cupcakes, so you have only $1999$ cupcakes. Show that it is impossible to arrange the cupcakes on the tables according to the above conditions.
[b]p2.[/b] Let $P$ and $Q$ lie on the hypotenuse $AB$ of the right triangle $CAB$ so that $|AP|=|PQ|=|QB|=|AB|/3$. Suppose that $|CP|^2+|CQ|^2=5$. Prove that $|AB|$ has the same value for all such triangles, and find that value. Note: $|XY|$ denotes the length of the segment $XY$.
[b]p3.[/b] Let $P$ be a polynomial with integer coefficients and let $a, b, c$ be integers. Suppose $P(a)=b$, $P(b)=c$, and $P(c)=a$. Prove that $a=b=c$.
[b]p4.[/b] A lattice point is a point $(x,y)$ in the plane for which both $x$ and $y$ are integers. Each lattice point is painted with one of $1999$ available colors. Prove that there is a rectangle (of nonzero height and width) whose corners are lattice points of the same color.
[b]p5.[/b] A $1999$-by-$1999$ chocolate bar has vertical and horizontal grooves which divide it into $1999^2$ one-by-one squares. Caesar and Brutus are playing the following game with the chocolate bar: A move consists of a player picking up one chocolate rectangle; breaking it along a groove into two smaller rectangles; and then either putting both rectangles down or eating one piece and putting the other piece down. The players move alternately. The one who cannot make a move at his turn (because there are only one-by-one squares left) loses. Caesar starts. Which player has a winning strategy? Describe a winning strategy for that player.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2024 Mexican Girls' Contest, 6
On a \(4 \times 4\) board, each cell is colored either black or white such that each row and each column have an even number of black cells. How many ways can the board be colored?
2015 ASDAN Math Tournament, 2
Jonah recently harvested a large number of lychees and wants to split them into groups. Unfortunately, for all $n$ where $3\leq n\leq8$, when the lychees are distributed evenly into $n$ groups, $n-1$ lychees remain. What is the smallest possible number of lychees that Jonah could have?
1991 IMO, 1
Suppose $ \,G\,$ is a connected graph with $ \,k\,$ edges. Prove that it is possible to label the edges $ 1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.
[b]Note: Graph-Definition[/b]. A [b]graph[/b] consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices $ \,u,v\,$ belongs to at most one edge. The graph $ G$ is connected if for each pair of distinct vertices $ \,x,y\,$ there is some sequence of vertices $ \,x \equal{} v_{0},v_{1},v_{2},\cdots ,v_{m} \equal{} y\,$ such that each pair $ \,v_{i},v_{i \plus{} 1}\;(0\leq i < m)\,$ is joined by an edge of $ \,G$.
2024 Ukraine National Mathematical Olympiad, Problem 8
There are $2024$ cities in a country, some pairs of which are connected by bidirectional flights. For any distinct cities $A, B, C, X, Y, Z$, it is possible to fly directly from some of the cities $A, B, C$ to some of the cities $X, Y, Z$. Prove that it is possible to plan a route $T_1\to T_2 \to \ldots \to T_{2022}$ that passes through $2022$ distinct cities.
[i]Proposed by Lior Shayn[/i]
MOAA Team Rounds, 2021.10
For how many nonempty subsets $S \subseteq \{1, 2, \ldots , 10\}$ is the sum of all elements in $S$ even?
[i]Proposed by Andrew Wen[/i]
2019 Saudi Arabia JBMO TST, 4
Find all positive integers $k>1$, such that there exist positive integer $n$, such that the number $A=17^{18n}+4.17^{2n}+7.19^{5n}$ is product of $k$ consecutive positive integers.
2005 Turkey MO (2nd round), 5
If $a,b,c$ are the sides of a triangle and $r$ the inradius of the triangle, prove that
\[\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\le \frac{1}{4r^2} \]
2016 IFYM, Sozopol, 3
Let $x\leq y\leq z$ be real numbers such that $x+y+z=12$, $x^2+y^2+z^2=54$. Prove that:
a) $x\leq 3$ and $z\geq 5$
b) $xy$, $yz$, $zx\in [9,25]$
2023 Switzerland - Final Round, 4
Determine the smallest possible value of the expression $$\frac{ab+1}{a+b}+\frac{bc+1}{b+c}+\frac{ca+1}{c+a}$$ where $a,b,c \in \mathbb{R}$ satisfy $a+b+c = -1$ and $abc \leqslant -3$
1988 IMO Shortlist, 6
In a given tedrahedron $ ABCD$ let $ K$ and $ L$ be the centres of edges $ AB$ and $ CD$ respectively. Prove that every plane that contains the line $ KL$ divides the tedrahedron into two parts of equal volume.
2024 AIME, 5
Let ABCDEF be an equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are the extensions of AB, CD and EF has side lengths 200, 240 and 300 respectively. Find the side length of the hexagon.
2016 Indonesia MO, 8
Determine with proof, the number of permutations $a_1,a_2,a_3,...,a_{2016}$ of $1,2,3,...,2016$ such that the value of $|a_i-i|$ is fixed for all $i=1,2,3,...,2016$, and its value is an integer multiple of $3$.