Let $P_n(x)=1+2x+3x^2+\cdots+nx^{n-1}.$ Prove that the polynomials $P_j(x)$ and $P_k(x)$ are relatively prime for all positive integers $j$ and $k$ with $j\ne k.$

A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white? $ \displaystyle \textbf{(A)} \ \frac {3}{10} \qquad \textbf{(B)} \ \frac {2}{5} \qquad \textbf{(C)} \ \frac {1}{2} \qquad \textbf{(D)} \ \frac {3}{5} \qquad \textbf{(E)} \ \frac {7}{10}$

Given that \[ \frac{((3!)!)!}{3!} = k \cdot n!, \] where $k$ and $n$ are positive integers and $n$ is as large as possible, find $k + n$.

Find all positive integer pairs $(a,n)$ such that $\frac{(a+1)^n-a^n}{n}$ is an integer.

A $\textit{cubic sequence}$ is a sequence of integers given by $a_n =n^3 + bn^2 + cn + d$, where $b, c$ and $d$ are integer constants and $n$ ranges over all integers, including negative integers. $\textbf{(a)}$ Show that there exists a cubic sequence such that the only terms of the sequence which are squares of integers are $a_{2015}$ and $a_{2016}$. $\textbf{(b)}$ Determine the possible values of $a_{2015} \cdot a_{2016}$ for a cubic sequence satisfying the condition in part $\textbf{(a)}$.

Let \(ABC\) be an acute scalene triangle with orthocenter \(H\). Line \(BH\) intersects \(\overline{AC}\) at \(E\) and line \(CH\) intersects \(\overline{AB}\) at \(F\). Let \(X\) be the foot of the perpendicular from \(H\) to the line through \(A\) parallel to \(\overline{EF}\). Point \(B_1\) lies on line \(XF\) such that \(\overline{BB_1}\) is parallel to \(\overline{AC}\), and point \(C_1\) lies on line \(XE\) such that \(\overline{CC_1}\) is parallel to \(\overline{AB}\). Prove that points \(B\), \(C\), \(B_1\), \(C_1\) are concyclic. [i]Proposed by Luke Robitaille[/i]

For the positive real numbers $ a,b,c$ prove that \[ \frac c{a \plus{} 2b} \plus{} \frac d{b \plus{} 2c} \plus{} \frac a{c \plus{} 2d} \plus{} \frac b{d \plus{} 2a} \geq \frac 43.\]

The centers of the three circles A, B, and C are collinear with the center of circle B lying between the centers of circles A and C. Circles A and C are both externally tangent to circle B, and the three circles share a common tangent line. Given that circle A has radius $12$ and circle B has radius $42,$ find the radius of circle C.

Each of $2k+1$ distinct 7-element subsets of the 20 element set intersects with exactly $k$ of them. Find the maximum possible value of $k$.

Let $ABC$ be an acute triangle and $D \in (BC) , E \in (AD)$ be mobile points. The circumcircle of triangle $CDE$ meets the median from $C$ of the triangle $ABC$ at $F$ Prove that the circumcenter of triangle $AEF$ lies on a fixed line.

Tom is chasing Jerry on the coordinate plane. Tom starts at $(x, y)$ and Jerry starts at $(0, 0)$. Jerry moves to the right at $1$ unit per second. At each positive integer time $t$, if Tom is within $1$ unit of Jerry, he hops to Jerry’s location and catches him. Otherwise, Tom hops to the midpoint of his and Jerry’s location. [i]Example. If Tom starts at $(3, 2)$, then at time $t = 1$ Tom will be at $(2, 1)$ and Jerry will be at $(1, 0)$. At $t = 2$ Tom will catch Jerry.[/i] Assume that Tom catches Jerry at some integer time $n$. (a) Show that $x \geq 0$. (b) Find the maximum possible value of $\frac{y}{x+1}$.

Vertex $B$ of the $\angle ABC$ lies out the circle, and the $[BA)$ and $[BC)$ beams intersect it. Point $K$ belongs to the intersection of the $[BA)$ beam and the circumference. Chord $KP$ is orthogonal to the angle bisector of $\angle ABC$ . Line $(KP)$ intersects the beam $BC$ in the point $M$. Prove that the segment $[PM]$ is twice as long as the distance from the circle centre to the angle bisector of $\angle ABC$ .

$(BUL 3)$ One hundred convex polygons are placed on a square with edge of length $38 cm.$ The area of each of the polygons is smaller than $\pi cm^2,$ and the perimeter of each of the polygons is smaller than $2\pi cm.$ Prove that there exists a disk with radius $1$ in the square that does not intersect any of the polygons.

Let $a_1,a_2,a_3,\ldots$ be an infinite sequence where for all positive integers $i$, $a_i$ is chosen to be a random positive integer between $1$ and $2016$, inclusive. Let $S$ be the set of all positive integers $k$ such that for all positive integers $j<k$, $a_j\neq a_k$. (So $1\in S$; $2\in S$ if and only if $a_1\neq a_2$; $3\in S$ if and only if $a_1\neq a_3$ and $a_2\neq a_3$; and so on.) In simplest form, let $\dfrac{p}{q}$ be the expected number of positive integers $m$ such that $m$ and $m+1$ are in $S$. Compute $pq$.

Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$, and let $q(x) \in \mathbb{F}_p[x]$ be given by $q(x) = \sum_{k=1}^{p-1} a_k x^k$ where $a_k = k^{(p-1)/2}$ mod $p$. Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\mathbb{F}_p[x]$.

Big Bad Brandon is assigning groups of his Big Bad Burglars to attack 7 different towers. Each Burglar can only belong to one attack group and Brandon takes over a tower if the number of Burglars attacking the tower strictly exceeds the number of knights guarding it. He knows there the total number of knights guarding the towers is 99 but does not know the exact number of knights guarding each tower. What is the minimum number of Burglars that Brandon needs to guarantee he can take over at least 4 of the 7 towers? [i]Proposed by Eric Wang[/i]

Find all positive real numbers $r<1$ such that there exists a set $\mathcal{S}$ with the given properties: i) For any real number $t$, exactly one of $t, t+r$ and $t+1$ belongs to $\mathcal{S}$; ii) For any real number $t$, exactly one of $t, t-r$ and $t-1$ belongs to $\mathcal{S}$.

For what pairs of natural numbers $(a, b)$ is the expression $$(a^6 + 21a^4b^2 + 35a^2b^4 + 7b^6) (b^6 + 21b^4a^2 + 35b^2a^4 + 7a^6)$$ the power of a prime number?

Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that for all integers $a$ and $b$, we have: $$f(a^2+ab)+f(b^2+ab)=(a+b)f(a+b).$$ Proposed by Heidar Shushtari

The Planar National Park is a subset of the Euclidean plane consisting of several trails which meet at junctions. Every trail has its two endpoints at two different junctions whereas each junction is the endpoint of exactly three trails. Trails only intersect at junctions (in particular, trails only meet at endpoints). Finally, no trails begin and end at the same two junctions. (An example of one possible layout of the park is shown to the left below, in which there are six junctions and nine trails.) [center] [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZS9mLzc1YmNjN2YxMWZhZTNhMTVkZTQ4NWE1ZDIyMDNhN2I5NzY0NTBlLnBuZw==&rn=Z3JhcGguUE5H[/img] [/center] A visitor walks through the park as follows: she begins at a junction and starts walking along a trail. At the end of that first trail, she enters a junction and turns left. On the next junction she turns right, and so on, alternating left and right turns at each junction. She does this until she gets back to the junction where she started. What is the largest possible number of times she could have entered any junction during her walk, over all possible layouts of the park?

Two points $A$ and $B$ are given in the plane. A point $X$ is called their [i]preposterous midpoint[/i] if there is a Cartesian coordinate system in the plane such that the coordinates of $A$ and $B$ in this system are non-negative, the abscissa of $X$ is the geometric mean of the abscissae of $A$ and $B$, and the ordinate of $X$ is the geometric mean of the ordinates of $A$ and $B$. Find the locus of all the [i]preposterous midpoints[/i] of $A$ and $B$. (K. Tyschu)

Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.

Let $P(n)$ be a polynomial of degree $m$ with integer coefficients, where $m \le 10$. Suppose that $P(0)=0$, $P(n)$ has $m$ distinct integer roots, and $P(n)+1$ can be factored as the product of two nonconstant polynomials with integer coefficients. Find the sum of all possible values of $P(2)$. [i]Proposed by Evan Chen[/i]

Let $ ABCD$ be a convex quadrilateral. Points $ P,Q$ and $ R$ are the feets of the perpendiculars from point $ D$ to lines $ BC, CA$ and $ AB$, respectively. Prove that $ PQ\equal{}QR$ if and only if the bisectors of the angles $ ABC$ and $ ADC$ meet on segment $ AC$.

Let $S$ be a set of $n\ge2$ positive integers. Prove that there exist at least $n^2$ integers that can be written in the form $x+yz$ with $x,y,z\in S$. [i]Proposed by Gerhard Woeginger, Austria[/i]