Let $ABC$ be an isosceles triangle with $AB = AC$. Let $AD$ be the diameter of the circumcircle of $ABC$ and let $P$ be a point on the smaller arc $BD$. The line $DP$ intersects the rays $AB$ and $AC$ at points $M$ and $N$, respectively. The line $AD$ intersects the lines $BP$ and $CP$ at points $Q$ and $R$, respectively. Prove that the midpoint of $MN$ lies on the circumcircle of $PQR$

The Fibonacci sequence is defined as $f_1=f_2=1$, $f_{n+2}=f_{n+1}+f_n$ ($n\in\mathbb{N}$). Suppose that $a$ and $b$ are positive integers such that $\frac ab$ lies between the two fractions $\frac{f_n}{f_{n-1}}$ and $\frac{f_{n+1}}{f_{n}}$. Show that $b\ge f_{n+1}$.

A car has an engine which delivers a constant power. It accelerates from rest at time $t = 0$, and at $t = t_\text{0}$ its acceleration is $a_\text{0}$. What is its acceleration at $t = 2t_\text{0}$? Ignore energy loss due to friction. (a) $\frac{1}{2}a_\text{0}$ (b) $\frac{1}{\sqrt{2}}a_\text{0}$ (c) $a_\text{0}$ (d) $\sqrt{2}a_\text{0}$ (e) $2a_\text{0}$

On a circle $k$, we marked four points $(A, B, C, D)$ and drew pairwise their connecting segments. We denoted angles as seen on the diagram. We know that $\alpha_1 : \alpha_2 = 2 : 5$, $\beta_1 : \beta_2 = 7 : 11$, and $\gamma_1 : \gamma_2 = 10 : 3$. If $\delta_1 : \delta_2 = p : q$, where $p$ and $q$ are coprime positive integers, then what is $p$? [img]https://cdn.artofproblemsolving.com/attachments/c/e/b532dd164a7cf99cea7b3b7d98f81796622da5.png[/img]

Prove that $\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\le \frac{a+b+c}{2}$, where $a,b,c\in\mathbb{R}^{+}$.

Call a positive integer $n\geq 2$ [i]junk[/i] if there exist two distinct $n$ digit binary strings $a_1a_2\cdots a_n$ and $b_1b_2\cdots b_n$ such that [list] [*] $a_1+a_2=b_1+b_2,$ [*] $a_{i-1}+a_i+a_{i+1}=b_{i-1}+b_i+b_{i+1}$ for all $2\leq i\leq n-1,$ and [*] $a_{n-1}+a_n=b_{n-1}+b_n$. [/list] Find the number of junk positive integers less than or equal to $2016$. [i]Proposed by Nathan Ramesh

Six unit disks $C_1$, $C_2$, $C_3$, $C_4$, $C_5$, $C_6$ are in the plane such that they don't intersect each other and $C_i$ is tangent to $C_{i+1}$ for $1 \le i \le 6$ (where $C_7 = C_1$). Let $C$ be the smallest circle that contains all six disks. Let $r$ be the smallest possible radius of $C$, and $R$ the largest possible radius. Find $R - r$.

We have a 7x7 board. We want to color some 1x1 squares such that any 3x3 sub-board have more painted 1x1 than no painted 1x1. What is the smallest number of 1x1 that we need to color?

Prove that if for angles $A,B,C$ of a triangle holds $$\sin^2 A+\sin^2 B +\sin^2 C=2$$ iff the triangle $ABC$ is right.

Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x,y$, we have $$f(xf(x)+2y)=f(x)^2+x+2f(y)$$

The function $f$ is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$. It satisfies the conditions $f(1) = 2$ and $f(xy) = f(x)f(y) - f(x+y) + 1$ for all $x,y \in \mathbb{Q}$. Determine $f$.

Let $ABC$ be an acute-angled triangle with $AB=AC$, let $D$ be the foot of perpendicular, of the point $C$, to the line $AB$ and the point $M$ is the midpoint of $AC$. Finally, the point $E$ is the second intersection of the line $BC$ and the circumcircle of $\triangle CDM$. Prove that the lines $AE, BM$ and $CD$ are concurrents if and only if $CE=CM$.

Solve in integers the equation \[{x^3}{y^2}\left( {2y - x} \right) = {x^2}{y^4} - 36\]

Let $n$ be a positive odd integer and let $\theta$ be a real number such that $\theta/\pi$ is irrational. Set $a_{k}=\tan(\theta+k\pi/n),\ k=1,2\dots,n.$ Prove that \[\frac{a_{1}+a_{2}+\cdots+a_{n}}{a_{1}a_{2}\cdots a_{n}}\] is an integer, and determine its value.

For any real $ a$ define $ f_a : \mathbb{R} \rightarrow \mathbb{R}^2$ by the law $ f_a(t) \equal{} \left( \sin(t), \cos(at) \right)$. a) Prove that $ f_{\pi}$ is not periodic. b) Determine the values of the parameter $ a$ for which $ f_a$ is periodic. [b]Remark[/b]. L. Euler proved in $ 1737$ that $ \pi$ is irrational.

There are $23$ balls on a table, all of which are either red or blue, such that the probability that there are $n$ red balls and $23-n$ blue balls on the table ($1 \le n \le 22$) is proportional to $n$. (e.g. the probability that there are $2$ red balls and $21$ blue balls is twice the probability that there are $1$ red ball and $22$ blue balls.) Given that the probability that the red balls and blue balls can be arranged in a line such that there is a blue ball on each end, no two red balls are next to each other, and an equal number of blue balls can be placed between each pair of adjacent red balls is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find $a+b$. Note: There can be any nonzero number of consecutive blue balls at the ends of the line. [i]Proposed by Ada Tsui[/i]

$2021$ points are given on a circle. Each point is colored by one of the $1,2, \cdots ,k$ colors. For all points and colors $1\leq r \leq k$, there exist an arc such that at least half of the points on it are colored with $r$. Find the maximum possible value of $k$.

Let $P$ denote the set of all ordered pairs $ \left(p,q\right)$ of nonnegative integers. Find all functions $f: P \rightarrow \mathbb{R}$ satisfying \[ f(p,q) \equal{} \begin{cases} 0 & \text{if} \; pq \equal{} 0, \\ 1 \plus{} \frac{1}{2} f(p+1,q-1) \plus{} \frac{1}{2} f(p-1,q+1) & \text{otherwise} \end{cases} \] Compare IMO shortlist problem 2001, algebra A1 for the three-variable case.

Let $ABC$ be a triangle with incenter $I$, and suppose that $AI$, $BI$, and $CI$ intersect $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. Let the circumcircles of $BDF$ and $CDE$ intersect at $D$ and $P$, and let $H$ be the orthocenter of $DEF$. Prove that $HI=HP$.

Let $ G $ be the center of mass of a triangle $ ABC, $ and the points $ M,N,P $ on the segments $ AB,BC, $ respectively, $ CA $ (excluding the extremities) such that $$ \frac{AM}{MB} =\frac{BN}{NC} =\frac{CP}{PA} . $$ $ G_1,G_2,G_3 $ are the centers of mass of the triangles $ AMP, BMN, $ respectively, $ CNP. $ Pove that: [b]a)[/b] The centers of mas of $ ABC $ and $ G_1G_2G_3 $ are the same. [b]b)[/b] For any planar point $ D, $ the inequality $$ 3\cdot DG< DG_1+DG_2+DG_3<DA+DB+DC $$ holds.

An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first move. For all moves after the first move, the ant does not return to its previous vertex, but chooses to move to one of the other two adjacent vertices. All choices are selected at random so that each of the possible moves is equally likely. The probability that after exactly 8 moves that ant is at a vertex of the top face on the cube is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

A sequence with first two terms equal $1$ and $24$ respectively is defined by the following rule: each subsequent term is equal to the smallest positive integer which has not yet occurred in the sequence and is not coprime with the previous term. Prove that all positive integers occur in this sequence.

Suppose that there are 100 seats in a saloon for 100 students. All students except one know their seat. First student (which is the one who doesn't know his seat) comes to the saloon and sits randomly somewhere. Then others enter the saloon one by one. Every student that enters the saloon and finds his seat vacant, sits there and if he finds his seat occupied he sits somewhere else randomly. Find the probability that last two students sit on their seats.

Let $n$ be a positive integer. If $r\hspace{0.25mm} \equiv \hspace{1mm} n\hspace{1mm} (mod\hspace{1mm} 2)$ and $r\hspace{0.10mm} \in \hspace{0.10mm} \{ 0,\hspace{0.10mm} 1 \} $, find the number of integer solutions to the system of equations $\left\{\begin{array}{l}x+y+z = r \\ \mid x \mid + \mid y \mid + \mid z \mid = n \end{array}\right.$