An arbitrary triangle $ABC$ is given. Construct a straight line passing through vertex $B$ and dividing it into two triangles, the radii of the inscribed circles of which are equal. (M. Volchkevich)

Square $ABCD$ has sides of length 1. Points $E$ and $F$ are on $\overline{BC}$ and $\overline{CD}$, respectively, so that $\triangle AEF$ is equilateral. A square with vertex $B$ has sides that are parallel to those of $ABCD$ and a vertex on $\overline{AE}$. The length of a side of this smaller square is $\displaystyle \frac{a-\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b+c$.

Prove that there exists an integer $n$, $n\geq 2002$, and $n$ distinct positive integers $a_1,a_2,\ldots,a_n$ such that the number $N= a_1^2a_2^2\cdots a_n^2 - 4(a_1^2+a_2^2+\cdots + a_n^2) $ is a perfect square.

Find all positive integers \(k\) for which there is an irrational \(\alpha>1\) and a positive integer \(N\) such that \(\left\lfloor\alpha^{n}\right\rfloor\) is a perfect square minus \(k\) for every integer \(n\) with \(n>N\).

[u]Round 5[/u] [b]5.1.[/b] In a triangle $\vartriangle ABC$ with $AB = 48$, let the angle bisectors of $\angle BAC$ and $\angle BCA$ meet at $I$. Given $\frac{[ABI]}{[BCI]}=\frac{24}{7}$ and $\frac{[ACI]}{[ABI]}=\frac{25}{24}$ , find the area of $\vartriangle ABC$. [b]5.2.[/b] At a dinner party, $9$ people are to be seated at a round table. If person $A$ cannot be seated next to person $B$ and person $C$ cannot be next to person $D$, how many ways can the $9$ people be seated? Rotations of the table are indistinguishable. [b]5.3.[/b] Let $f(x)$ be a monic cubic polynomial such that $f(1) = f(7) = f(10) = a$ and $f(2) = f(5) = f(11) = b$. Find $|a - b|$. [u]Round 6[/u] [b]6.1.[/b] If $N$ has $16$ positive integer divisors and the sum of all divisors of $N$ that are multiples of $3$ is $39$ times the sum of divisors of $N$ that are not multiples of $3$, what is the smallest value of $N$? [b]6.2.[/b] In the two parabolas $y = x^2/16$ and $x = y^2/16$, the single line tangent to both parabolas intersects the parabolas at $A$ and $B$. If the parabolas intersect each other at $C$ which is not the origin, find the area of $\vartriangle ABC$. [b]6.3.[/b] Five distinguishable noncollinear points are drawn. How many ways are there to draw segments connecting the points, such that there are exactly two disjoint groups of connected points? Note that a single point can be considered a connected group of points. [u]Round 7[/u] [b]7.1.[/b] Let $a, b$ be positive integers, and $1 = d_1 < d_2 < d_3 < ... < d_n = a$ be the divisors of $a$, and $1 = e_1 < e_2 <e_3 < ... < e_m = b$ be the divisors of b. Given $gcd(a, b) = d_2 = e_6$, find the smallest possible value of $a + b$. [b]7.2.[/b] Let $\vartriangle ABC$ be a triangle such that $AB = 2$ and $AC = 3$. Let X be the point on $BC$ such that $m \angle BAX =\frac13 m\angle BAC$. Given that $AX = 1$, the sum of all possible values of $CX^2$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Find $a + b$. [b]7.3.[/b] Bob has a playlist of $6$ different songs in some order, and he listens to his playlist repeatedly. Every time he finishes listening to the third song in the playlist, he randomly shuffles his playlist and listens to the playlist starting with the new first song. The expected number of times Bob shuffles his songs before he listens each one of his $6$ songs at least once can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Find $a+b$. [u]Round 8[/u] [b]8.1.[/b] $\underline{A}, \underline{B}, \underline{C}, \underline{D}, \underline{E}, \underline{F}, \underline{G}, \underline{H}, \underline{I}$, and $\underline{J}$ represent distinct digits ($0$ to $9$) in the equation $\underline{FBGA} - \underline{ABAC} = \underline{DCE}$ (where $\underline{ABAC}$ and $\underline{F BGA}$ are four-digit numbers, and $\underline{DCE }$ is a three-digit number). If $\underline{A} < \underline{B} < \underline{C} < \underline{D}$ and $\underline{ABCDEF GHIJ}$ is minimized, find $\underline{ABCD} + \underline{EF G} + \underline{HI} + \underline{J}$. [b]8.2.[/b] $\underline{A}, \underline{B}, \underline{C}, \underline{D}, \underline{E}$,,, and $\underline{F}$ represent distinct digits ($0$ to $9$) in the equations $\underline{ABC} \cdot \underline{C} = \underline{DEA}, \underline{ABC} \cdot \underline{D} = \underline{BAF E}$, and $ \underline{DEA} + \underline{BAF E}0 = \underline{BF ACA}$ (where $\underline{ABC}$ and $\underline{DEA}$ are three-digit numbers, $\underline{BAF E}$ is a four-digit number, and $\underline{BF ACA}$ is a five-digit number). Find $\underline{ABC} + \underline{DE} + \underline{F}$. [b]8.3.[/b] $\underline{A}, \underline{B}, \underline{C}, \underline{D}, \underline{E}, \underline{F}, \underline{G}$, and $\underline{H}$ represent distinct digits ($0$ to $9$) in the equations $\underline{ABC } \cdot \underline{D} = \underline{AF GE}$, $\underline{ABC } \cdot \underline{C} = \underline{GHC}$, $\underline{GHC} + \underline{HF F} = \underline{AEHC}$, and $\underline{AF GE}0 + \underline{AEHC} = \underline{AEABC}$ (where $\underline{ABC}$, $\underline{GHC}$ and $\underline{HF F}$ are three-digit numbers, $\underline{AF GE}$ is a four-digit number, and $\underline{AEABC}$ is a five-digit number). Find $\underline{ABCD} + \underline{EF GH}$. [u]Round 9[/u] Estimate the arithmetic mean of all answers to this question. Only integer answers between $0$ to $100, 000$ will count for credit and count toward the average. Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05|I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3129699p28347299]here [/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that \[ f(x^2 + yf(z)) = xf(x) + zf(y) , \] for all $x, y, z \in \mathbb{R}$.

The polynomial $ x^3\minus{}2004x^2\plus{}mx\plus{}n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of $ n$ are possible? $ \textbf{(A)}\ 250,\!000 \qquad \textbf{(B)}\ 250,\!250 \qquad \textbf{(C)}\ 250,\!500 \qquad \textbf{(D)}\ 250,\!750 \qquad \textbf{(E)}\ 251,\!000$

Let $M$ and $N$ be two positive five-digit palindrome integers, such that $M <N$ and there is no other palindrome number between them. Determine the possible values ​​of $N-M$.

In the acute-angled triangle $ABC$, the altitudes $BP$ and $CQ$ were drawn, and the point $T$ is the intersection point of the altitudes of $\Delta PAQ$. It turned out that $\angle CTB = 90 {} ^ \circ$. Find the measure of $\angle BAC$. (Mikhail Plotnikov)

Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren? $\textbf{(A) }78\qquad\textbf{(B) }80\qquad\textbf{(C) }144\qquad\textbf{(D) }146\qquad\textbf{(E) }152$

Find all triplets (a, b, c) of positive integers so that $a^2b$, $b^2c$ and $c^2a$ devide $a^3+b^3+c^3$

Let $n\geq 2$ be an integer. Consider $2n$ points around a circle. Each vertex has been tagged with one integer from $1$ to $n$, inclusive, and each one of these integers has been used exactly two times. Isabel divides the points into $n$ pairs, and draws the segments joining them, with the condition that the segments do not intersect. Then, she assigns to each segment the greatest integer between its endpoints. a) Show that, no matter how the points have been tagged, Isabel can always choose the pairs in such a way that she uses exactly $\lceil n/2\rceil$ numbers to tag the segments. b) Can the points be tagged in such a way that, no matter how Isabel divides the points into pairs, she always uses exactly $\lceil n/2\rceil$ numbers to tag the segments? Note. For each real number $x$, $\lceil x\rceil$ denotes the least integer greater than or equal to $x$. For example, $\lceil 3.6\rceil=4$ and $\lceil 2\rceil=2$. [i]Proposed by Victor Domínguez[/i]

All positive integers from \( 1 \) to \( 2025 \) are written on a board. Mykhailo and Oleksii play the following game. They take turns, starting with Mykhailo, erasing one of the numbers written on the board. The game ends when exactly two numbers remain on the board. If their sum is a perfect square of an integer, Mykhailo wins; otherwise, Oleksii wins. Who wins if both players play optimally? [i]Proposed by Fedir Yudin[/i]

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 (with proof)

A robot is standing on the bottom left vertex $(0,0)$ of a $5\times5$ grid, and wants to go to $(5,5)$, only moving to the right $(a,b)\mapsto(a+1,b)$ or upward $(a,b)\mapsto(a,b+1)$. However this robot is not programmed perfectly, and sometimes takes the upper-left diagonal path $(a,b)\mapsto(a-1,b+1)$. As the grid is surrounded by walls, the robot cannot go outside the region $0\leq a,b\leq5$. Supposing that the robot takes the diagonal path exactly once, compute the number of different routes the robot can take.

Consider all products by $2, 4, 6, ..., 2000$ of the elements of the set $A =\left\{\frac12, \frac13, \frac14,...,\frac{1}{2000},\frac{1}{2001}\right\}$ . Find the sum of all these products.