Determine all functions $ f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfy: $ f(x\plus{}f(y))\equal{}f(x)\plus{}y$ for all $ x,y \in \mathbb{N}$.

In triangle $ABC$, $K$ and $L$ are points on the side $BC$ ($K$ being closer to $B$ than $L$) such that $BC \cdot KL = BK \cdot CL$ and $AL$ bisects $\angle KAC$. Show that $AL \perp AB.$

Let \( n > 2 \) be an integer, \( k > 1 \) a real number, and \( x_1, x_2, \ldots, x_n \) be positive real numbers such that \( x_1 \cdot x_2 \cdots x_n = 1 \). Prove that: \[ \frac{1 + x_1^k}{1 + x_2} + \frac{1 + x_2^k}{1 + x_3} + \cdots + \frac{1 + x_n^k}{1 + x_1} \geq n. \] When does equality hold?

Triangle $ABC$ has incenter $I$. Let $D$ be the foot of the perpendicular from $A$ to side $BC$. Let $X$ be a point such that segment $AX$ is a diameter of the circumcircle of triangle $ABC$. Given that $ID = 2$, $IA = 3$, and $IX = 4$, compute the inradius of triangle $ABC$.

A semicircle is inscribed in a semicircle of radius $2$ as shown. Find the radius of the smaller semicircle. [img]https://cdn.artofproblemsolving.com/attachments/c/1/c60cd40eaecfe417aca46ce4fd386fe22af85b.png[/img]

ABC is a triangle with A=90 and C=30.Let M be the midpoint of BC. Let W be a circle passing through A tangent in M to BC. Let P be the circumcircle of ABC. W is intersecting AC in N and P in M. prove that MN is perpendicular to BC.

Let $a$ be a positive integer, $a<100$, and $a^3+23$ is a multiple of $24$. Then, the number of such $a$ is $\text{(A)}4\qquad\text{(B)}5\qquad\text{(C)}9\qquad\text{(D)}10$

[color=darkred]The sides $AD$ and $CD$ of a tangent quadrilateral $ABCD$ touch the incircle $\varphi$ at $P$ and $Q,$ respectively. If $M$ is the midpoint of the chord $XY$ determined by $\varphi$ on the diagonal $BD,$ prove that $\angle AMP = \angle CMQ.$[/color]

$A$ and $B$ are two cylindrical containers that contain water. The height of the water at$ A$ is $1000$ cm and at $B$, $350$ cm. Using a pump, water is transferred from $A$ to $B$. It is noted that, in container $A$, the height of the water decreases $4$ cm per minute and in $B$ it increases $9$ cm per minute. After how much time, since the pump was started, will the heights at $A$ and $B$ be the same?

Alice, Bob, and Carol are playing a game. Each turn, one of them says one of the $3$ players' names, chosen from {Alice, Bob, Carol} uniformly at random. Alice goes first, Bob goes second, Carol goes third, and they repeat in that order. Let $E$ be the expected number of names that are have been said when, for the first time, all $3$ names have been said twice. If $E = \tfrac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$. (Include the last name to be said twice in your count.)

Jake, Jonathan, and Joe are playing a dice game involving polyhedron dice. The dice are as follows: 4 sides, 6 sides, 12 sides, and 20 sides. An n-sided dice has the numbers 1 through n labeled on the sides. Jake starts by selecting a 4-sided die and a 20-sided die. The amount of points that a player gets is the sum of the numbers on the rolled dice. Jonathan then selects a 12-sided die and an 20-sided die. Finally, Joe selects a 20-sided die and a 6-sided die. [list=a] [*]What is the probability that Joe places last? [*]What is the probability that Joe places second? [*]What is the probability that Joe places first? [*]What is the probability that there is a three-way tie? [/list] [i]Proposed by beanielove2[/i]

Lunasa, Merlin, and Lyrica are performing in a concert. Each of them will perform two different solos, and each pair of them will perform a duet, for nine distinct pieces in total. Since the performances are very demanding, no one is allowed to perform in two pieces in a row. In how many different ways can the pieces be arranged in this concert? [i]Proposed by Yannick Yao[/i]

The CMU Kiltie Band is attempting to crash a helicopter via grappling hook. The helicopter starts parallel (angle $0$ degrees) to the ground. Each time the band members pull the hook, they tilt the helicopter forward by either $x$ or $x+1$ degrees, with equal probability, if the helicopter is currently at an angle $x$ degrees with the ground. Causing the helicopter to tilt to $90$ degrees or beyond will crash the helicopter. Find the expected number of times the band must pull the hook in order to crash the helicopter. [i]Proposed by Justin Hsieh[/i]

Square $ABCD$ has side length of $2$. Quarter-circle arcs $BD$ (centered at $C$) and $AC$ (centered at $D$) divide $ABCD$ into four sections. The area of the smallest of the four sections that are formed can be expressed as $a - \frac{b\pi }{c} - \sqrt{d}$. Find abcd, where $a, b, c$ and $d$ are integers, $ \sqrt{d}$ is a written in simplestradical form, and $\frac{b}{c}$ is written in simplest form.

Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.

Let $\triangle{ABC}$ be an equilateral triangle. Let $E_{AB}$ be the ellipse with foci $A, B$ passing through $C,$ and in the parallel manner define $E_{BC}, E_{AC}.$ Let $\triangle{GHI}$ be a (nondegenerate) triangle with vertices where two ellipses intersect such that the edges of $\triangle{GHI}$ do not intersect those of $\triangle{ABC}.$ Compute the ratio of the largest sides of $\triangle{GHI}$ and $\triangle{ABC}.$

Let $z\ne0$ be a complex number such that $z^8=\overline z$. What are the possible values of $z^{2001}$?

Find all value of $ n$ for which there are nonzero real numbers $ a, b, c, d$ such that after expanding and collecting similar terms, the polynomial $ (ax \plus{} b)^{100} \minus{} (cx \plus{} d)^{100}$ has exactly $ n$ nonzero coefficients.

For an arbitrary point $M$ inside a given square $ABCD$, let $T_1,T_2,T_3$ be the centroids of triangles $ABM,BCM$, and $DAM$, respectively. Let $OM$ be the circumcenter of triangle $T_1T_2T_3$. Find the locus of points $OM$ when $M$ takes all positions within the interior of the square.

Given are a prime $p$ and a positive integer $a$. Let $q$ be a prime divisor of $\frac{a^{p^3}-1}{a^{p^2}-1}$ and $q\neq p$. Prove that $q\equiv 1 ( \mod p^3)$.

Find the sum of all integer values of $n$ such that the equation $\frac{x}{(yz)^2} + \frac{y}{(zx)^2} + \frac{z}{(xy)^2} = n$ has a solution in positive integers.

Aaron takes a square sheet of paper, with one corner labeled $A$. Point $P$ is chosen at random inside of the square and Aaron folds the paper so that points $A$ and $P$ coincide. He cuts the sheet along the crease and discards the piece containing $A$. Let $p$ be the probability that the remaining piece is a pentagon. Find the integer nearest to $100p$. [i]Proposed by Aaron Lin[/i]

Let two circles $C_1$ and $C_2$ intersect in points $A$ and $B$. The tangent to $C_1$ at $A$ intersects $C_2$ in $P$ and the line $PB$ intersects $C_1$ for the second time in $Q$ (suppose that $Q$ is outside $C_2$). The tangent to $C_2$ from $Q$ intersects $C_1$ and $C_2$ in $C$ and $D$, respectively. (The points $A$ and $D$ lie on different sides of the line $PQ$.) Show that $AD$ is the bisector of $\angle CAP$. [i]Proposed by Iman Maghsoudi[/i]

Find all functions $f:\mathbb {R}^{+} \rightarrow \mathbb{R}^{+}$, such that $$f(af(b)+a)(f(bf(a))+a)=1$$ for any positive reals $a, b$.

Let be given positive reals $a$, $b$, $c$. Prove that: $\frac{a^{3}}{\left(a+b\right)^{3}}+\frac{b^{3}}{\left(b+c\right)^{3}}+\frac{c^{3}}{\left(c+a\right)^{3}}\geq \frac{3}{8}$.