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} $

The kid cut out of grid paper with the side of the cell $1$ rectangle along the grid lines and calculated its area and perimeter. Carlson snatched his scissors and cut out of this rectangle along the lines of the grid a square adjacent to the boundary of the rectangle. - My rectangle ... - kid sobbed. - There is something strange about this figure! - Nonsense, do not mention it - Carlson said - waving his hand carelessly. - Here you see, in this figure the perimeter is the same as the area of ​​the rectangle was, and the area is the same as was the perimeter! What size square did Carlson cut out?

Find all functions $f:\mathbb{Q}^+\to\mathbb{Q}^+$ such that \[xy(f(x)-f(y))|x-f(f(y))\] holds for all positive rationals $x$, $y$ (we define that $a|b$ if and only if exist $n \in \mathbb{Z}$ such that $b=an$) [i]Proposed by supercarry & windleaf1A[/i]

A rectangular piece of paper has the side lengths $12$ and $15$. A corner is bent about as shown in the figure. Determine the area of the gray triangle. [img]https://1.bp.blogspot.com/-HCfqWF0p_eA/XzcIhnHS1rI/AAAAAAAAMYg/KfY14frGPXUvF-H6ZVpV4RymlhD_kMs-ACLcBGAsYHQ/s0/1993%2BMohr%2Bp2.png[/img]

Five real numbers of absolute values not greater than $1$ and having the sum equal to $1$ are written on the circumference of a circle. Prove that one can choose three consecutively disposed numbers $a, b, c$, such that all the sums $a + b,b + c$ and $a + b + c$ are nonnegative.

Acute triangle $ABC$ has circumcircle $\Gamma$. Let $M$ be the midpoint of $BC.$ Points $P$ and $Q$ lie on $\Gamma$ so that $\angle APM = 90^{\circ}$ and $Q \neq A$ lies on line $AM.$ Segments $PQ$ and $BC$ intersect at $S$. Suppose that $BS = 1, CS = 3, PQ = 8\sqrt{\tfrac{7}{37}},$ and the radius of $\Gamma$ is $r$. If the sum of all possible values of $r^2$ can be expressed as $\tfrac ab$ for relatively prime positive integers $a$ and $b,$ compute $100a + b$.