A polyhedron $W$ has the following properties: (i) It possesses a center of symmetry. (ii) The section of $W$ by a plane passing through the center of symmetry and one of its edges is always a parallelogram. (iii) There is a vertex of $W$ at which exactly three edges meet. Prove that $W$ is a parallelepiped.

Find all sequences of integer $x_1,x_2,..,x_n,...$ such that $ij$ divides $x_i+x_j$ for any distinct positive integer $i$, $j$.

Let $p$ be a prime. Find all positive integers $k$ such that the set $\{1,2, \cdots, k\}$ can be partitioned into $p$ subsets with equal sum of elements.

For 31 years, n (>6) tennis players have records of wins. It turns out that for every two players, there is a third player who has won over them before. Prove that for every integer $k,l$ such that $2^{2^k+1}-1>n, 1<l<2k+1$, there exist $l$ players ($A_1, A_2, ... , A_l$) such that every player $A_{i+1}$ won over $A_i$. ($A_{l+1}$ is same as $A_1$)

Let $ABC$ be a triangle with $AB < AC$ and $M$ the midpoint of the arc $BC$ containing $A$, plus $T$ the foot of the perpendicular from $M$ on side $AC$. Prove that $AB + AT = TC$. [img]https://cdn.artofproblemsolving.com/attachments/0/a/5c90d7001f73c2f8ff2b0e69078f9a2a5cd606.png[/img]

$n(>1)$ lotus leaves are arranged in a circle. A frog jumps from a particular leaf from another under the following rule: [list] [*]It always moves clockwise. [*]From starting it skips one leaf and then jumps to the next. After that it skips two leaves and jumps to the following. And the process continues. (Remember the frog might come back on a leaf twice or more.)[/list] Given that it reaches all leaves at least once. Show $n$ cannot be odd.

For a positive integer \( n \geq 6 \), find the smallest integer \( S(n) \) such that any graph with \( n \) vertices and at least \( S(n) \) edges must contain at least two disjoint cycles (cycles with no common vertices).

Determine, with proof, all possible values of $\gcd(a^2+b^2+c^2,abc)$ across all triples of positive integers $(a,b,c).$

Identical rectangular cardboard pieces are handed out to $30$ students, one to each. Each student cuts (parallel to the edges) his or her piece into equally large squares. Two different students’ squares do not necessarily have the same size. After all the cutting it turns out that the total number of squares is a prime. Prove that the original cardboard pieces must have been quadratic.

Two circles $k_1(O_1,R)$ and $k_2(O_2,r)$ are given in the plane such that $R \ge \sqrt2 r$ and $$O_1O_2 =\sqrt{R^2 +r^2 - r\sqrt{4R^2 +r^2}}.$$ Let $A$ be an arbitrary point on $k_1$. The tangents from $A$ to $k_2$ touch $k_2$ at $B$ and $C$ and intersect $k_1$ again at $D$ and $E$, respectively. Prove that $BD \cdot CE = r^2$

$S$ is a $(k+1) \times (k+1)$ array of lattice points. How many squares have their vertices in $S$?

In the figure above, the large triangle and all four shaded triangles are equilateral. If the areas of triangles $A, B,$ and $C$ are $1, 2,$ and $3,$ respectively, compute the smallest possible integer ratio between the area of the entire triangle to the area of triangle $D.$ [Insert Diagram] [i]Proposed by Alex Li[/i]

Let $N$ be the number of functions $f:\{1,2,3,4,5,6,7,8,9,10\} \rightarrow \{1,2,3,4,5\}$ that have the property that for $1\leq x\leq 5$ it is true that $f(f(x))=x$. Given that $N$ can be written in the form $5^a\cdot b$ for positive integers $a$ and $b$ with $b$ not divisible by $5$, find $a+b$. [i]Proposed by Nathan Ramesh

Let $\ell$ be a positive integer, and let $m,n$ be positive integers with $m\geq n$, such that $A_1,A_2,\cdots,A_m,B_1,\cdots,B_m$ are $m+n$ pairwise distinct subsets of the set $\{1,2,\cdots,\ell\}$. It is known that $A_i\Delta B_j$ are pairwise distinct, $1\leq i\leq m, 1\leq j\leq n$, and runs over all nonempty subsets of $\{1,2,\cdots,\ell\}$. Find all possible values of $m,n$.

Altitudes $AA_1,CC_1$ of acute-angles $ABC$ meet at point $H$ ; $B_0$ is the midpoint of $AC$. A line passing through $B$ and parallel to $AC$ meets $B_0A_1 , B_0C_1$ at points $A',C'$ respectively. Prove that $AA',CC'$ and $BH$ concur.

In triangle $ABC$ the angle $C$ is obtuse, $m(\angle A)=2m(\angle B)$ and the sidelengths are integers. Find the smallest possible perimeter of this triangle.

A treasure is buried at some point on a round island with a radius of $1$ km. On the coast of the island there is a mathematician with a device that indicates the direction to the treasure when the distance to the treasure does not exceed $500$ m. In addition, the mathematician has a map of the island, on which he can record all his movements, perform measurements and geometric constructions. The mathematician claims that he has an algorithm for how to get to the treasure after walking less than $4$ km. Could this be true? (B. Frenkin)

A triangle $\vartriangle A_0A_1A_2$ in the plane has sidelengths $A_0A_1 = 7$,$A_1A_2 = 8$,$A_2A_0 = 9$. For $i \ge 0$, given $\vartriangle A_iA_{i+1}A_{i+2}$, let $A_{i+3}$ be the midpoint of $A_iA_{i+1}$ and let Gi be the centroid of $\vartriangle A_iA_{i+1}A_{i+2}$. Let point $G$ be the limit of the sequence of points $\{G_i\}^{\infty}_{i=0}$. If the distance between $G$ and $G_0$ can be written as $\frac{a\sqrt{b}}{c}$ , where $a, b, c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime, find $a^2 + b^2 + c^2$.

For any positive integers $a$ and $b$, define $a \oplus b$ to be the result when adding $a$ to $b$ in binary (base $2$), neglecting any carry-overs. For example, $20 \oplus 14 = 10100_2 \oplus 1110_2 = 11010_2 = 26$. (The operation $\oplus$ is called the [i]exclusive or.[/i]) Compute the sum $$\sum^{2^{2014} -1}_{k=0} \left( k \oplus \left\lfloor \frac{k}{2} \right \rfloor \right).$$ Here $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.

Point A, which is within an acute, is reflected with respect to both sides of angle A to obtain the points B and C. the segment BC intersects the sides of angle A at points D and E respectively. Prove that BC/2>DE.

Find all positive integers $ n$ such that $ n^4\minus{}4n^3\plus{}22n^2\minus{}36n\plus{}18$ is a perfect square.

A standard deck of $52$ cards is shuffled and randomly arranged in a queue, with each card having a suit $(\diamondsuit,\clubsuit,\heartsuit,\spadesuit)$ and a rank $(\text{Ace},2,3,4,5,6,7,8,9,10,\text{Jack},\text{Queen},\text{ King})$. For example, a card with the $\diamondsuit$ suit and the $7$ rank would be denoted as $\diamondsuit7$, and a card with the $\spadesuit$ and the $\text{Ace}$ rank would be denoted as $\spadesuit\text{Ace}$. In the queue, there exists a card with a rank of $\text{Ace}$ that appears for the first time in the queue. Let the card immediately following the above card be denoted as card $C$. Is the probability that $C$ is a $\spadesuit\text{A}$ higher than, equal to, or lower than the probability that $C$ is a $\clubsuit2$?

For how many positive integer values of $n$ are both $\frac{n}{3}$ and $3n$ three-digit whole numbers? $\textbf{(A)}\ 12\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 27\qquad \textbf{(D)}\ 33\qquad \textbf{(E)}\ 34$

A student steps onto a stationary elevator and stands on a bathroom scale. The elevator then travels from the top of the building to the bottom. The student records the reading on the scale as a function of time. How tall is the building? $\textbf{(A) } 50 \text{ m}\\ \textbf{(B) } 80 \text{ m}\\ \textbf{(C) } 100 \text{ m}\\ \textbf{(D) } 150 \text{ m}\\ \textbf{(E) } 400 \text{ m}$

A number of $N$ children are at a party and they sit in a circle to play a game of Pass and Parcel. Because the host has no other form of entertainment, the parcel has infinitely many layers. On turn $i$, starting with $i=1$, the following two things happen in order: [b]$(1)$[/b] The parcel is passed $i^2$ positions clockwise; and [b]$(2)$[/b] The child currently holding the parcel unwraps a layer and claims the prize inside. For what values of $N$ will every chidren receive a prize? $Patrick \ Winter \, United \ Kingdom$