The incircle of triangle $ABC$ touches the side $AB$ at point $C'$; the incircle of triangle $ACC'$ touches the sides $AB$ and $AC$ at points $C_1, B_1$; the incircle of triangle $BCC'$ touches the sides $AB$ and $BC$ at points $C_2$, $A_2$. Prove that the lines $B_1C_1$, $A_2C_2$, and $CC'$ concur.

Let $ABC$ be a triangle for which $AB+BC = 3AC$. Let $D$ and $E$ be the points of tangency of the incircle with the sides $AB$ and $BC$ respectively, and let $K$ and $L$ be the other endpoints of the diameters originating from $D$ and $E.$ Prove that $C , A, L$, and $K$ lie on a circle.

Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by 20% on every pair of shoes. Carlos also knew that he had to pay a 7.5% sales tax on the discounted price. He had 43 dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy? A)$46$ B)$50$ C)$48$ D)$47$ E)$49$

Let $X$ and $Y$ be independent, identically distributed random points on the unit sphere in $\mathbb{R}^3$. For which distribution of $X$ will the expectation of the (Euclidean) distance of $X$ and $Y$ be maximal?

Is it possible to fill a $40 \times 41$ table with integers so that each integer equals the number of adjacent (by an edge) cells with the same integer? Alexandr Gribalko

For a positive integer $n$, let $\omega(n)$ denote the number of positive prime divisors of $n$. Find the smallest positive tinteger $k$ such that $2^{\omega(n)}\leq k\sqrt[4]{n}\forall n\in\mathbb{N}$.

Let $G$ be a finite group of order $n$ generated by $a$ and $b$. Prove or disprove: there is a sequence \[ g_1, g_2, g_3, \cdots, g_{2n} \] such that: $(1)$ every element of $G$ occurs exactly twice, and $(2)$ $g_{i+1}$ equals $g_{i}a$ or $g_ib$ for $ i = 1, 2, \cdots, 2n $. (interpret $g_{2n+1}$ as $g_1$.)

Iris does not know what to do with her 1-kilogram pie, so she decides to share it with her friend Rosabel. Starting with Iris, they take turns to give exactly half of total amount of pie (by mass) they possess to the other person. Since both of them prefer to have as few number of pieces of pie as possible, they use the following strategy: During each person's turn, she orders the pieces of pie that she has in a line from left to right in increasing order by mass, and starts giving the pieces of pie to the other person beginning from the left. If she encounters a piece that exceeds the remaining mass to give, she cuts it up into two pieces with her sword and gives the appropriately sized piece to the other person. When the pie has been cut into a total of 2017 pieces, the largest piece that Iris has is $\frac{m}{n}$ kilograms, and the largest piece that Rosabel has is $\frac{p}{q}$ kilograms, where $m,n,p,q$ are positive integers satisfying $\gcd(m,n)=\gcd(p,q)=1$. Compute the remainder when $m+n+p+q$ is divided by 2017. [i]Proposed by Yannick Yao[/i]

The sequence $(a_n)$ is defined by $a_1=1$ and $a_n=a_{n-1}+\frac{1}{n^3}$ for $n>1.$ (a) Prove that $a_n<\frac{5}{4}$ for all $n.$ (b) Given $\epsilon>0$, find the smallest natural number $n_0$ such that ${\mid a_{n+1}-a_n}\mid<\epsilon$ for all $n>n_0.$

The number of triples $(a,b,c)$ of positive integers which satisfy the simultaneous equations \begin{align*} ab+bc &= 44,\\ ac+bc &= 23, \end{align*} is $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }3\qquad \textbf{(E) }4$

Let $x, y$ and $z$ be positive real numbers such that $x \geq y+z$. Proof that $$\frac{x+y}{z} + \frac{y+z}{x} +\frac{z+x}{y} \geq 7$$ When does equality occur? (Walther Janous)

Let $ABCD$ be a cyclic quadrilateral with center $O$. Suppose the circumcircles of triangles $AOB$ and $COD$ meet again at $G$, while the circumcircles of triangles $AOD$ and $BOC$ meet again at $H$. Let $\omega_1$ denote the circle passing through $G$ as well as the feet of the perpendiculars from $G$ to $AB$ and $CD$. Define $\omega_2$ analogously as the circle passing through $H$ and the feet of the perpendiculars from $H$ to $BC$ and $DA$. Show that the midpoint of $GH$ lies on the radical axis of $\omega_1$ and $\omega_2$. [i]Proposed by Yang Liu[/i]

The vertices $V$ of a centrally symmetric hexagon in the complex plane are given by $$V=\left\{ \sqrt{2}i,-\sqrt{2}i, \frac{1}{\sqrt{8}}(1+i),\frac{1}{\sqrt{8}}(-1+i),\frac{1}{\sqrt{8}}(1-i),\frac{1}{\sqrt{8}}(-1-i) \right\}.$$ For each $j$, $1\leq j\leq 12$, an element $z_j$ is chosen from $V$ at random, independently of the other choices. Let $P={\prod}_{j=1}^{12}z_j$ be the product of the $12$ numbers selected. What is the probability that $P=-1$? $\textbf{(A) } \dfrac{5\cdot11}{3^{10}} \qquad \textbf{(B) } \dfrac{5^2\cdot11}{2\cdot3^{10}} \qquad \textbf{(C) } \dfrac{5\cdot11}{3^{9}} \qquad \textbf{(D) } \dfrac{5\cdot7\cdot11}{2\cdot3^{10}} \qquad \textbf{(E) } \dfrac{2^2\cdot5\cdot11}{3^{10}}$

Circles $\Omega_a$ and $\Omega_b$ are externally tangent at $D$, circles $\Omega_b$ and $\Omega_c$ are externally tangent at $E$, circles $\Omega_a$ and $\Omega_c$ are externally tangent at $F$. Let $P$ be an arbitrary point on $\Omega_a$ different from $D$ and $F$. Extend $PD$ to meet $\Omega_b$ again at $B$, extend $BE$ to meet $\Omega_c$ again at $C$ and extend $CF$ to meet $\Omega_a$ again at $A$. Show that $PA$ is a diameter of circle $\Omega_a$.

$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.

Let $p \geq 3$ be a prime number. Prove that there is a permutation $(x_1,\ldots, x_{p-1})$ of $(1,2,\ldots,p-1)$ such that $x_1x_2 + x_2x_3 + \cdots + x_{p-2}x_{p-1} \equiv 2 \pmod p$.

If $a,b,c$ are the sides of a triangle and $r$ the inradius of the triangle, prove that \[\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\le \frac{1}{4r^2} \]

Two fixed circles $\omega_1$ and $\omega_2$ pass through point $O$. A circle of an arbitrary radius $R$ centered at $O$ meets $\omega_1$ at points $A$ and $B$, and meets $\omega_2$ at points $C$ and $D$. Let $X$ be the common point of lines $AC$ and $BD$. Prove that all the points X are collinear as $R$ changes.

At what $a$ does the graph of the function $y = x^4+x^3+ax$ have an axis of symmetry parallel to the axis $Oy$?

Egor and Igor take turns (Igor starts) replacing the coefficients of the polynomial \[a_{99}x^{99} + \cdots + a_1x + a_0\]with non-zero integers. Egor wants the polynomial to have as many different integer roots as possible. What is the largest number of roots he can always achieve?

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

Nine people have attended four different meetings sitting around a circular table. Could they have done so in a way that no two people sat next to each other more than once? Justify your answer.

For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$ $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11$

In the grid below, fill each gray cell with one of the numbers from the provided bank, with each number used once, and fill each white cell with a positive one-digit number. The number in a gray cell must equal the sum of the numbers in all touching white cells, where two cells sharing a vertex are considered touching. All of the terms in each of these sums must be distinct, meaning that two white cells with the same digit may not touch the same gray cell. Bank: 15, 23, 28, 35, 36, 38, 40, 42, 44 [asy] unitsize(1.2cm); defaultpen(fontsize(30pt)); int[][] x = { {0,5,0,0,0,8,0}, {4,0,0,0,0,0,0}, {0,3,0,0,0,0,1}, {0,0,0,0,0,7,0}, {0,8,0,0,1,0,4}, {0,0,0,0,0,2,0}}; void square(int a, int b) { filldraw((a,b)--(a+1,b)--(a+1,b+1)--(a,b+1)--cycle,mediumgray); } square(1,0); square(3,1); square(5,1); square(1,2); square(3,3); square(5,3); square(1,4); square(5,4); square(3,5); for(int i = 0; i < 8; ++i) { draw((i,0)--(i,6)); } for(int i = 0; i < 7; ++i) { draw((0,i)--(7,i)); } for(int k = 0; k<6; ++k){ for(int l = 0; l<7; ++l){ if(x[k][l]!=0){ label(string(x[k][l]),(l+0.5,-k+5.5)); } } } [/asy] There is a unique solution, but you do not need to prove that your answer is the only one possible. You merely need to find an answer that satisfies the constraints above. (Note: in any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)

Suppose the curve $ C: y \equal{} ax^3 \plus{} 4x\ (a\neq 0)$ has a common tangent line at the point $ P$ with the hyperbola $ xy \equal{} 1$ in the first quadrant. (1) Find the value of $ a$ and the coordinate of the point $ P$. (2) Find the volume formed by the revolution of the solid of the figure bounded by the line segment $ OP$ and the curve $ C$ about the line $ OP$. [color=green][Edited.][/color]