Found problems: 85335
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 $3$.
1987 All Soviet Union Mathematical Olympiad, 458
The convex $n$-gon ($n\ge 5$) is cut along all its diagonals. Prove that there are at least a pair of parts with the different areas.
1999 VJIMC, Problem 3
Let $A_1,\ldots,A_n$ be points of an ellipsoid with center $O$ in $\mathbb R^n$ such that $OA_i$, for $i=1,\ldots,n$, are mutually orthogonal. Prove that the distance of the point $O$ from the hyperplane $A_1A_2\ldots A_n$ does not depend on the choice of the points $A_1,\ldots,A_n$.
2010 Brazil Team Selection Test, 3
Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram.
Prove that $GR=GS$.
[i]Proposed by Hossein Karke Abadi, Iran[/i]
2016 Harvard-MIT Mathematics Tournament, 31
For a positive integer $n$, denote by $\tau(n)$ the number of positive integer divisors of $n$, and denote by $\phi(n)$ the number of positive integers that are less than or equal to $n$ and relatively prime to $n$. Call a positive integer $n$ $\emph{good}$ if $\varphi (n) + 4\tau (n) = n$. For example, the number $44$ is good because $\varphi (44) + 4\tau (44)= 44$.
Find the sum of all good positive integers $n$.
2020 Candian MO, 5#
If A,B are invertible and the set {A<sup>k</sup> - B<sup>k</sup> | k is a natural number} is finite , then there exists a natural number m such that A<sup>m</sup> = B<sup>m</sup>.
2024 LMT Fall, 19
Let $P(n)$ denote the product of digits of $n$. Find the number of positive integers $n \leq 2024$ where $P(n)$ is divisible by $n$.
2010 Sharygin Geometry Olympiad, 16
A circle touches the sides of an angle with vertex $A$ at points $B$ and $C.$ A line passing through $A$ intersects this circle in points $D$ and $E.$ A chord $BX$ is parallel to $DE.$ Prove that $XC$ passes through the midpoint of the segment $DE.$
1992 AMC 12/AHSME, 25
In triangle $ABC$, $\angle ABC = 120^{\circ}$, $AB = 3$ and $BC = 4$. If perpendiculars constructed to $\overline{AB}$ at $A$ and to $\overline{BC}$ at $C$ meet at $D$, then $CD = $
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ \frac{8}{\sqrt{3}}\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ \frac{11}{2}\qquad\textbf{(E)}\ \frac{10}{\sqrt{3}} $
2020 Brazil Cono Sur TST, 2
A number n is called charming when $ 4k^2 + n $ is a prime number for every $ 0 \leq k <n $ integer, find all charming numbers.
2003 Croatia National Olympiad, Problem 3
In a tetrahedron $ABCD$, all angles at vertex $D$ are equal to $\alpha$ and all dihedral angles between faces having $D$ as a vertex are equal to $\phi$. Prove that there exists a unique $\alpha$ for which $\phi=2\alpha$.
2006 Stanford Mathematics Tournament, 10
What is the square root of the sum of the first 2006 positive odd integers?
2011 Danube Mathematical Competition, 3
Determine all positive integer numbers $n$ satisfying the following condition:
the sum of the squares of any $n$ prime numbers greater than $3$ is divisible by $n$.