[b]p1.[/b] The following number is the product of the divisors of $n$. $$2^63^3$$ What is $n$? [b]p2.[/b] Let a right triangle have the sides $AB =\sqrt3$, $BC =\sqrt2$, and $CA = 1$. Let $D$ be a point such that $AD = BD = 1$. Let $E$ be the point on line $BD$ that is equidistant from $D$ and $A$. Find the angle $\angle AEB$. [b]p3.[/b] There are twelve indistinguishable blackboards that are distributed to eight different schools. There must be at least one board for each school. How many ways are there of distributing the boards? [b]p4.[/b] A Nishop is a chess piece that moves like a knight on its first turn, like a bishop on its second turn, and in general like a knight on odd-numbered turns and like a bishop on even-numbered turns. A Nishop starts in the bottom-left square of a $3\times 3$-chessboard. How many ways can it travel to touch each square of the chessboard exactly once? [b]p5.[/b] Let a Fibonacci Spiral be a spiral constructed by the addition of quarter-circles of radius $n$, where each $n$ is a term of the Fibonacci series: $$1, 1, 2, 3, 5, 8,...$$ (Each term in this series is the sum of the two terms that precede it.) What is the arclength of the maximum Fibonacci spiral that can be enclosed in a rectangle of area $714$, whose side lengths are terms in the Fibonacci series? [b]p6.[/b] Suppose that $a_1 = 1$ and $$a_{n+1} = a_n -\frac{2}{n + 2}+\frac{4}{n + 1}-\frac{2}{n}$$ What is $a_{15}$? [b]p7.[/b] Consider $5$ points in the plane, no three of which are collinear. Let $n$ be the number of circles that can be drawn through at least three of the points. What are the possible values of $n$? [b]p8.[/b] Find the number of positive integers $n$ satisfying $\lfloor n /2014 \rfloor =\lfloor n/2016 \rfloor$. [b]p9.[/b] Let $f$ be a function taking real numbers to real numbers such that for all reals $x \ne 0, 1$, we have $$f(x) + f \left( \frac{1}{1 - x}\right)= (2x - 1)^2 + f\left( 1 -\frac{1}{ x}\right)$$ Compute $f(3)$. [b]p10.[/b] Alice and Bob split $5$ beans into piles. They take turns removing a positive number of beans from a pile of their choice. The player to take the last bean loses. Alice plays first. How many ways are there to split the piles such that Alice has a winning strategy? [b]p11.[/b] Triangle $ABC$ is an equilateral triangle of side length $1$. Let point $M$ be the midpoint of side $AC$. Another equilateral triangle $DEF$, also of side length $1$, is drawn such that the circumcenter of $DEF$ is $M$, point $D$ rests on side $AB$. The length of $AD$ is of the form $\frac{a+\sqrt{b}}{c}$ , where $b$ is square free. What is $a + b + c$? [b]p12.[/b] Consider the function $f(x) = \max \{-11x- 37, x - 1, 9x + 3\}$ defined for all real $x$. Let $p(x)$ be a quadratic polynomial tangent to the graph of $f$ at three distinct points with x values $t_1$, $t_2$ and $t_3$ Compute the maximum value of $t_1 + t_2 + t_3$ over all possible $p$. [b]p13.[/b] Circle $J_1$ of radius $77$ is centered at point $X$ and circle $J_2$ of radius $39$ is centered at point $Y$. Point $A$ lies on $J1$ and on line $XY$ , such that A and Y are on opposite sides of $X$. $\Omega$ is the unique circle simultaneously tangent to the tangent segments from point $A$ to $J_2$ and internally tangent to $J_1$. If $XY = 157$, what is the radius of $\Omega$ ? [b]p14.[/b] Find the smallest positive integer $n$ so that for any integers $a_1, a_2,..., a_{527}$,the number $$\left( \prod^{527}_{j=1} a_j\right) \cdot\left( \sum^{527}_{j=1} a^n_j\right)$$ is divisible by $527$. [b]p15.[/b] A circle $\Omega$ of unit radius is inscribed in the quadrilateral $ABCD$. Let circle $\omega_A$ be the unique circle of radius $r_A$ externally tangent to $\Omega$, and also tangent to segments $AB$ and $DA$. Similarly define circles $\omega_B$, $\omega_C$, and $\omega_D$ and radii $r_B$, $r_C$, and $r_D$. Compute the smallest positive real $\lambda$ so that $r_C < \lambda$ over all such configurations with $r_A > r_B > r_C > r_D$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f : \mathbb R\to\mathbb R $ such that \[f(x+yf(x^2))=f(x)+xf(xy)\] for all real numbers $x$ and $y$.

Let $a_n = 1 + \frac{2}{n} - \frac{2}{n^3} - \frac{1}{n^4}$. For which smallest positive integer $n$ does the value of $P_n = a_2a_3a_4 \ldots a_n$ exceed $100$?

Let $ABC$ be a triangle such that it's circumcircle radius is equal to the radius of outer inscribed circle with respect to $A$. Suppose that the outer inscribed circle with respect to $A$ touches $BC,AC,AB$ at $M,N,L$. Prove that $O$ (Center of circumcircle) is the orthocenter of $MNL$.

Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $P$. The circumscribed circles of triangles $APD$ and $BPC$ intersect the line $AB$ at points $E, F$ correspondingly. $Q$ and $R$ are the projections of $P$ onto the lines $FC, DE$ correspondingly. Show that $AB \parallel QR$. [i](Proposed by Mykhailo Shtandenko)[/i]

A parallelogram is constructed on the coordinate plane, the coordinates of which are integers. It is known that inside the parallelogram and on its contour there are other (except vertices) points with integer coordinates. Prove that the area of ​​the parallelogram is not less than $3/2$.

For some constant $k$ the polynomial $p(x) = 3x^2 + kx + 117$ has the property that $p(1) = p(10)$. Evaluate $p(20)$.

$f(n)$ is the sum of all integers less than $n$ and relatively prime to $n$. Find all integers $n$ such that there exist integers $k$ and $\ell$ such that $f(n^k) = n^{\ell}$.

A $7 \times 7$ grid is given. Julián colors $29$ cells black. Pilar must then place an $L$-shaped piece, covering exactly three cells (oriented in any direction, as shown in the figure). Pilar wins if all three cells covered by the $L$-shaped piece are black. Can Julián color the grid in such a way that it is impossible for Pilar to win? [asy] size(1.5cm); draw((0,1)--(1,1)--(1,2)--(0,2)--(0,1)--(0,0)--(1,0)--(2,0)--(2,1)--(1,1)--(1,0)); [/asy]

Determine the real number $a$ having the property that $f(a)=a$ is a relative minimum of $f(x)=x^4-x^3-x^2+ax+1$.

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

Consider the isosceles right triangle $ABC$ with $\angle BAC = 90^\circ$. Let $\ell$ be the line passing through $B$ and the midpoint of side $AC$. Let $\Gamma$ be the circumference with diameter $AB$. The line $\ell$ and the circumference $\Gamma$ meet at point $P$, different from $B$. Show that the circumference passing through $A,\ C$ and $P$ is tangent to line $BC$ at $C$.

Let $ABC$ be a triangle with sides 3, 4, and 5, and $DEFG$ be a 6-by-7 rectangle. A segment is drawn to divide triangle $ABC$ into a triangle $U_1$ and a trapezoid $V_1$ and another segment is drawn to divide rectangle $DEFG$ into a triangle $U_2$ and a trapezoid $V_2$ such that $U_1$ is similar to $U_2$ and $V_1$ is similar to $V_2$. The minimum value of the area of $U_1$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $O$ be a point with three other points $A,B,C$ and $\angle AOB=\angle BOC=\angle AOC=2\pi/3$. Consider the average area of the set of triangles $ABC$ where $OA,OB,OC\in\{3,4,5\}$. The average area can be written in the form $m\sqrt n$ where $m,n$ are integers and $n$ is not divisible by a perfect square greater than $1$. Find $m+n$.

Consider the two functions \[f(x) = x^2+2bx+1\quad\text{and}\quad g(x) = 2a(x+b),\] where the variable $x$ and the constants $a$ and $b$ are real numbers. Each such pair of the constants $a$ and $b$ may be considered as a point $(a,b)$ in an $ab-$plane. Let $S$ be the set of such points $(a,b)$ for which the graphs of $y = f(x)$ and $y = g(x)$ do NOT intersect (in the $xy-$ plane.). The area of $S$ is $\textbf{(A)} \ 1 \qquad \textbf{(B)} \ \pi \qquad \textbf{(C)} \ 4 \qquad \textbf{(D)} \ 4 \pi \qquad \textbf{(E)} \ \text{infinite}$

The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$, and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only \[a\le\cos A+\sqrt3\sin A.\]

[b]p1.[/b] Consider the system of simultaneous equations $$(x+y)(x+z)=a^2$$ $$(x+y)(y+z)=b^2$$ $$(x+z)(y+z)=c^2$$ , where $abc \ne 0$. Find all solutions $(x,y,z)$ in terms of $a$,$b$, and $c$. [b]p2.[/b] Shown in the figure is a triangle $PQR$ upon whose sides squares of areas $13$, $25$, and $36$ sq. units have been constructed. Find the area of the hexagon $ABCDEF$ . [img]https://cdn.artofproblemsolving.com/attachments/b/6/ab80f528a2691b07430d407ff19b60082c51a1.png[/img] [b]p3.[/b] Suppose $p,q$, and $r$ are positive integers no two of which have a common factor larger than $1$. Suppose $P,Q$, and $R$ are positive integers such that $\frac{P}{p}+\frac{Q}{q}+\frac{R}{r}$ is an integer. Prove that each of $P/p$, $Q/q$, and $R/r$ is an integer. [b]p4.[/b] An isosceles tetrahedron is a tetrahedron in which opposite edges are congruent. Prove that all face angles of an isosceles tetrahedron are acute angles. [img]https://cdn.artofproblemsolving.com/attachments/7/7/62c6544b7c3651bba8a9d210cd0535e82a65bd.png[/img] [b]p5.[/b] Suppose that $p_1$, $p_2$, $p_3$ and $p_4$ are the centers of four non-overlapping circles of radius $1$ in a plane and that, $p$ is any point in that plane. Prove that $$\overline{p_1p}^2+\overline{p_2p}^2+\overline{p_3p}^2+\overline{p_4p}^2 \ge 6.$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We call a positive integer $q$ a $convenient \quad denominator$ for a real number $\alpha$ if $\displaystyle |\alpha - \dfrac{p}{q}|<\dfrac{1}{10q}$ for some integer $p$. Prove that if two irrational numbers $\alpha$ and $\beta$ have the same set of convenient denominators then either $\alpha+\beta$ or $\alpha- \beta$ is an integer.

Find all monic polynomials $P(x)$ such that the polynomial $P(x)^2-1$ is divisible by the polynomial $P(x+1)$.

There are $N$ cities in a country. Any two of them are connected either by a road or by an airway. A tourist wants to visit every city exactly once and return to the city at which he started the trip. Prove that he can choose a starting city and make a path, changing means of transportation at most once.

How many four-digit whole numbers are there such that the leftmost digit is odd, the second digit is even, and all four digits are different? $\text{(A)}\ 1120 \qquad \text{(B)}\ 1400 \qquad \text{(C)}\ 1800 \qquad \text{(D)}\ 2025 \qquad \text{(E)}\ 2500$

Given an integer $n \geqslant 3$ the polynomial $f(x_1, \ldots, x_n)$ with integer coefficients is called [i]good[/i] if $f(0,\ldots, 0) = 0$ and \[f(x_1, \ldots, x_n)=f(x_{\pi_1}, \ldots, x_{\pi_n}),\]for any permutation of $\pi$ of the numbers $1,\ldots, n$. Denote by $\mathcal{J}$ the set of polynomials of the form \[p_1q_1+\cdots+p_mq_m,\]where $m$ is a positive integer and $q_1,\ldots , q_m$ are polynomials with integer coefficients, and $p_1,\ldots , p_m$ are good polynomials. Find the smallest natural number $D{}$ such that each monomial of degree $D{}$ lies in the set $\mathcal{J}$.

Every two of the $n$ cities of Ruritania are connected by a direct flight of one from two airlines. Promonopoly Committee wants at least $k$ flights performed by one company. To do this, he can at least every day to choose any three cities and change the ownership of the three flights connecting these cities each other (that is, to take each of these flights from a company that performs it, and pass the other). What is the largest $k$ committee knowingly will be able to achieve its goal in no time, no matter how the flights are distributed hour?

Find all such functions $f :\mathbb{R}\to \mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following holds : \[ f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y) \]

If $n_1, n_2, \cdots, n_k$ are natural numbers and $n_1+n_2+\cdots+n_k = n$, show that \[max(n_1n_2\cdots n_k)=(t + 1)^rt^{k-r},\] where $t =\left[\frac{n}{k}\right]$ and $r$ is the remainder of $n$ upon division by $k$; i.e., $n = tk + r, 0 \le r \le k- 1$.