Let $ABC$ be an equilateral triangle. Let $A_1,B_1,C_1$ be interior points of $ABC$ such that $BA_1=A_1C$, $CB_1=B_1A$, $AC_1=C_1B$, and $$\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ$$ Let $BC_1$ and $CB_1$ meet at $A_2,$ let $CA_1$ and $AC_1$ meet at $B_2,$ and let $AB_1$ and $BA_1$ meet at $C_2.$ Prove that if triangle $A_1B_1C_1$ is scalene, then the three circumcircles of triangles $AA_1A_2, BB_1B_2$ and $CC_1C_2$ all pass through two common points. (Note: a scalene triangle is one where no two sides have equal length.) [i]Proposed by Ankan Bhattacharya, USA[/i]

There are $100$ school students from two clubs $A$ and $B$ standing in circle. Among them $62$ students stand next to at least one student from club $A$, and $54$ students stand next to at least one student from club $B$. 1) How many students stand side-by-side with one friend from club $A$ and one friend from club $B$? 2) What is the number of students from club $A$?

Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could [i]not[/i] be the total value of the four coins, in cents? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 55$

Real numbers $ a$ and $ b$ satisfy the equations $ 3^{a} \equal{} 81^{b \plus{} 2}$ and $ 125^{b} \equal{} 5^{a \minus{} 3}$. What is $ ab$? $ \textbf{(A)} \minus{} \!60 \qquad \textbf{(B)} \minus{} \!17 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 60$

Let $ABC$ be a triangle with $\angle BCA=20.$ Let points $D\in(BC), F\in(AC)$ be such that $CD=DF=FB=BA.$ Find $\angle ADF.$

Bob draws the graph of $y = x^3 - 13x^2 + 40x + 25$ and is dismayed to find out that it only has one root. Alice comes to the rescue, translating (without rotating or dilating) the axes so that the origin is at the point that used to be $(-20, 16)$. This new graph has three $x$-intercepts; compute their sum.

Is it true that for any polynomial $P(x)$ with real coefficients of degree $2023$, there exists a natural number $n$ such that the equation $P(x) = n^{-100}$ has no rational root?

The sequence $(a_n)$ is defined with the recursion $a_{n + 1} = 5a^6_n + 3a^3_{n-1} + a^2_{n-2}$ for $n\ge 2$ and the set of initial values $\{a_0, a_1, a_2\} = \{2013, 2014, 2015\}$. (That is, the initial values are these three numbers in any order.) Show that the sequence contains no sixth power of a natural number.

Let $(a_n)_{n\in\mathbb N}$ be a sequence, given by the recurrence: $$ma_{n+1}+(m-2)a_n-a_{n-1}=0$$ where $m\in\mathbb R$ is a parameter and the first two terms of $a_n$ are fixed known real numbers. Find $m\in\mathbb R$, so that $$\lim_{n\to\infty}a_n=0$$ [i]Proposed by Laurențiu Modan[/i]

There are $2$ boys for every $3$ girls in Ms. Johnson's math class. If there are $30$ students in her class, what percent of them are boys? $\text{(A)}\ 12\% \qquad \text{(B)}\ 20\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 60\% \qquad \text{(E)}\ 66\frac{2}{3}\% $

Let $S$ be a set and let $\circ$ be a binary operation on $S$ satisfying two laws $$x\circ x=x \text{ for all } x \text{ in } S, \text{ and}$$ $$(x \circ y) \circ z= (y\circ z) \circ x \text{ for all } x,y,z \text{ in } S.$$ Show that $\circ$ is associative and commutative.

Let $(a_n)_{n\geq 1}$ be a sequence for real numbers given by $a_1=1/2$ and for each positive integer $n$ \[ a_{n+1}=\frac{a_n^2}{a_n^2-a_n+1}. \] Prove that for every positive integer $n$ we have $a_1+a_2+\cdots + a_n<1$.

[b]p1.[/b] Two perpendicular chords intersect in a circle. The lengths of the segments of one chord are $3$ and $4$. The lengths of the segments of the other chord are $6$ and $2$. Find the diameter of the circle. [b]p2.[/b] Determine the greatest integer that will divide $13,511$, $13,903$ and $14,589$ and leave the same remainder. [b]p3.[/b] Suppose $A, B$ and $C$ are the angles of the triangle. Show that $\cos^2 A + \cos^2 B + \cos^2 C + 2 \cos A \cos B \cos C = 1$ [b]p4.[/b] Given the linear fractional transformation $f_1(x) =\frac{2x - 1}{x + 1}$. Define $f_{n+1}(x) = f_1(f_n(x))$ for $n = 1, 2, 3,...$ . It can be shown that $f_{35} = f_5$. (a) Find a function $g$ such that $f_1(g(x)) = g(f_1(x)) = x$. (b) Find $f_{28}$. [b]p5.[/b] Suppose $a$ is a complex number such that $a^{10} + a^5 + 1 = 0$. Determine the value of $a^{2005} + \frac{1}{a^{2005}}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let the sum of a set of numbers be the sum of its elements. Let $S$ be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of $S$ have the same sum. What is the largest sum a set $S$ with these properties can have?

Pål has more chickens than he can manage to keep track of. Therefore, he keeps an index card for each chicken. He keeps the cards in ten boxes, each of which has room for $2021$ cards. Unfortunately, Pål is quite disorganized, so he may lose some of his boxes. Therefore, he makes several copies of each card and distributes them among different boxes, so that even if he can only find seven boxes, no matter which seven, these seven boxes taken together will contain at least one card for each of his chickens. What is the largest number of chickens Pål can keep track of using this system?

Let $a$ and $b$ be two fixed positive real numbers. Find all real numbers $x$, such that inequality holds $$\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{a+b-x}} < \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}$$

Prove that for all reas $a,b,c,d\in(0,1)$ we have $$\left(ab-cd\right)\left(ac+bd\right)\left(ad-bc\right)+\min{\left(a,b,c,d\right)} < 1.$$

Show that system of equations $2ab=6(a+b)-13$ $a^2+b^2=4$ has not solutions in set of real numbers.

In a $m\times{n}$ grid are there are token. Every token [i]dominates [/i] every square on its same row ($\leftrightarrow$), its same column ($\updownarrow$), and diagonal ($\searrow\hspace{-4.45mm}\nwarrow$)(Note that the token does not \emph{dominate} the diagonal ($\nearrow\hspace{-4.45mm}\swarrow$), determine the lowest number of tokens that must be on the board to [i]dominate [/i] all the squares on the board.

Let $ABC$ be an acute-angled triangle. The circle $k$ with diameter $AB$ intersects $AC$ and $BC$ again at $P$ and $Q$, respectively. The tangents to $k$ at $A$ and $Q$ meet at $R$, and the tangents at $B$ and $P$ meet at $S$. Show that $C$ lies on the line $RS$.

There are 2016 points near a line such that the distance from each point to the line is less than 1 cm, and the distance between any two points is always greater than 2 cm. Prove that there exist two points whose distance is at least 17 meters.

Find all prime numbers $p$ such that there exists a unique $a \in \mathbb{Z}_p$ for which $a^3 - 3a + 1 = 0.$

$p$ is a prime and $m$ is a non-negative integer $< p-1$. Show that $ \sum_{j=1}^p j^m$ is divisible by $p$.

Find all triplets $(k, m, n)$ of positive integers having the following property: Square with side length $m$ can be divided into several rectangles of size $1\times k$ and a square with side length $n$.

In a country, there are $2018$ cities, some of which are connected by roads. Each city is connected to at least three other cities. It is possible to travel from any city to any other city using one or more roads. For each pair of cities, consider the shortest route between these two cities. What is the greatest number of roads that can be on such a shortest route?