In a room there are several children and a pile of 1000 sweets. The children come to the pile one after another in some order. Upon reaching the pile each of them divides the current number of sweets in the pile by the number of children in the room, rounds the result if it is not integer, takes the resulting number of sweets from the pile and leaves the room. All the boys round upwards and all the girls round downwards. The process continues until everyone leaves the room. Prove that the total number of sweets received by the boys does not depend on the order in which the children reach the pile. [i]Maxim Didin[/i]

Let acute scalene triangle $ABC$ have orthocenter $H$ and altitude $AD$ with $D$ on side $BC$. Let $M$ be the midpoint of side $BC$, and let $D'$ be the reflection of $D$ over $M$. Let $P$ be a point on line $D'H$ such that lines $AP$ and $BC$ are parallel, and let the circumcircles of $\triangle AHP$ and $\triangle BHC$ meet again at $G \neq H$. Prove that $\angle MHG = 90^\circ$. [i]Proposed by Daniel Hu.[/i]

Three faces of a rectangular prism have diagonal lengths of $7$, $8$, and $9$ inches. How many cubic inches are in the volume of the rectangular prism? $\text{(A) }48\sqrt{11}\qquad\text{(B) }160\qquad\text{(C) }14\sqrt{95}\qquad\text{(D) }35\sqrt{15}\qquad\text{(E) }504$

Eleven distinct points $ P_1,P_2,...,P_{11}$ are given on a line so that $ P_i P_j \le 1$ for every $ i,j$. Prove that the sum of all distances $ P_i P_j, 1 \le i <j \le 11$, is smaller than $ 30$.

What is the largest positive integer $n$ less than $10, 000$ such that in base $4$, $n$ and $3n$ have the same number of digits; in base $8$, $n$ and $7n$ have the same number of digits; and in base $16$, $n$ and $15n$ have the same number of digits? Express your answer in base $10$.

Find all integers $n$ with $n \geq 4$ for which there exists a sequence of distinct real numbers $x_1, \ldots, x_n$ such that each of the sets $$\{x_1, x_2, x_3\}, \{x_2, x_3, x_4\},\ldots,\{x_{n-2}, x_{n-1}, x_n\}, \{x_{n-1}, x_n, x_1\},\text{ and } \{x_n, x_1, x_2\}$$ forms a 3-term arithmetic progression when arranged in increasing order.

Let $ABCD$ be a cyclic quadrilateral, and angle bisectors of $\angle BAD$ and $\angle BCD$ meet at point $I$. Show that if $\angle BIC = \angle IDC$, then $I$ is the incenter of triangle $ABD$.

Show that for any two distinct odd primes $p, q$, there exists a positive integer $n$ such that $$\{d(n), d(n + 2) \} = \{p, q\}$$ where $d(n)$ is the smallest prime factor of $n$. [i]Proposed By - ltf0501[/i]

Find all functions $f : R- \{-1\} \to R$ such that $$f(x)f \left( f \left(\frac{1 - y}{1 + y} \right)\right) = f\left(\frac{x + y}{xy + 1}\right) $$ for all $x, y \in R$ that satisfy $(x + 1)(y + 1)(xy + 1) \ne 0$.

Prove that if $m$ is a natural number and $P,Q,R$ polynomials of degrees less than $m$ satisfying $$x^{2m}P(x,y)+y^{2m}Q(x,y) = (x+y)^{2m}R(x,y),$$ then each of the polynomials is zero.

A container is shaped like a right circular cone with base diameter $18$ and height $12$. The vertex of the container is pointing down, and the container is open at the top. Four spheres, each with radius $3$, are placed inside the container as shown. The first sphere sits at the bottom and is tangent to the cone along a circle. The second, third, and fourth spheres are placed so they are each tangent to the cone and tangent to the rst sphere, and the second and fourth spheres are each tangent to the third sphere. The volume of the tetrahedron whose vertices are at the centers of the spheres is $K$. Find $K^2$. [img]https://cdn.artofproblemsolving.com/attachments/9/c/648ec2cf0f0c2f023cd00b1c0595a9396d0ddc.png[/img]

The six-digit number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ is prime for only one digit $A.$ What is $A?$ $(\textbf{A})\: 1\qquad(\textbf{B}) \: 3\qquad(\textbf{C}) \: 5 \qquad(\textbf{D}) \: 7\qquad(\textbf{E}) \: 9$

a) How many ways are there to pair up the elements of $\{1,2,\dots,14\}$ into seven pairs so that each pair has sum at least $15$? b) How many ways are there to pair up the elements of $\{1,2,\dots,14\}$ into seven pairs so that each pair has sum at least $13$? c) How many ways are there to pair up the elements of $\{1,2,\dots,2024\}$ into $1012$ pairs so that each pair has sum at least $2001$?

Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest possible number of black edges is $\textbf{(A) }2\qquad \textbf{(B) }3\qquad \textbf{(C) }4\qquad \textbf{(D) }5\qquad \textbf{(E) }6\qquad$

Bisector of one angle of triangle $ABC$ is equal to the bisector of its external angle at the same vertex (see figure). Find the difference between the other two angles of the triangle. [img]https://cdn.artofproblemsolving.com/attachments/c/3/d2efeb65544c45a15acccab8db05c8314eb5f2.png[/img]

Prove that for every positive integer $n \geq 2$ the following inequality holds: $e^{n-1}n!<n^{n+\frac{1}{2}}$

Five people are standing in a straight line, and the distance between any two people is a unique positive integer number of units. Find the least possible distance between the leftmost and rightmost people in the line in units.

In an $m\times n$ table there is a policeman in cell $(1,1)$, and there is a thief in cell $(i,j)$. A move is going from a cell to a neighbor (each cell has at most four neighbors). Thief makes the first move, then the policeman moves and ... For which $(i,j)$ the policeman can catch the thief?

Let $ABCD$ be a convex quadrilateral with non-parallel sides $BC$ and $AD$. Assume that there is a point $E$ on the side $BC$ such that the quadrilaterals $ABED$ and $AECD$ are circumscribed. Prove that there is a point $F$ on the side $AD$ such that the quadrilaterals $ABCF$ and $BCDF$ are circumscribed if and only if $AB$ is parallel to $CD$.

Find the largest real number $x$ such that \[\left(\dfrac{x}{x-1}\right)^2+\left(\dfrac{x}{x+1}\right)^2=\dfrac{325}{144}.\]

For a given positive integer $n$, find the greatest positive integer $k$, such that there exist three sets of $k$ non-negative distinct integers, $A=\{x_1,x_2,\cdots,x_k\}, B=\{y_1,y_2,\cdots,y_k\}$ and $C=\{z_1,z_2,\cdots,z_k\}$ with $ x_j\plus{}y_j\plus{}z_j\equal{}n$ for any $ 1\leq j\leq k$. [size=85][color=#0000FF][Moderator edit: LaTeXified][/color][/size]

Find the sum of the first 5 positive integers $n$ such that $n^2 - 1$ is the product of 3 distinct primes.

For real number $ a$ with $ |a|>1$, evaluate $ \int_0^{2\pi} \frac{d\theta}{(a\plus{}\cos \theta)^2}$.

Consider triangle $ABC$ with $\angle A=2\angle B$. The angle bisectors from $A$ and $C$ intersect at $D$, and the angle bisector from $C$ intersects $\overline{AB}$ at $E$. If $\dfrac{DE}{DC}=\dfrac13$, compute $\dfrac{AB}{AC}$.

A regular polygon of $n$ sides is inscribed in a circle of radius $R$. The area of the polygon is $3R^2$. Then $n$ equals: ${{ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15}\qquad\textbf{(E)}\ 18} $