Find the sum of the squares of all solutions to $$(x^2 + 7x + 9)^2 = 9.$$ [i]Lightning 1.1[/i]

Real numbers $w$, $x$, $y$, and $z$ satisfy $w+x+y = 3$, $x+y+z = 4,$ and $w+x+y+z = 5$. What is the value of $x+y$? $\textbf{(A) }-\frac{1}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }\frac{3}{2}\qquad\textbf{(D) }2\qquad\textbf{(E) }3$

Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let $d$ be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of $d$? $\textbf{(A) } (0,4) \qquad \textbf{(B) } (4,5) \qquad \textbf{(C) } (4,6) \qquad \textbf{(D) } (5,6) \qquad \textbf{(E) } (5,\infty) $

[b]p1.[/b] Suppose that $20\%$ of a number is $17$. Find $20\%$ of $17\%$ of the number. [b]p2.[/b] Let $A, B, C, D$ represent the numbers $1$ through $4$ in some order, with $A \ne 1$. Find the maximum possible value of $\frac{\log_A B}{C +D}$. Here, $\log_A B$ is the unique real number $X$ such that $A^X = B$. [b]p3. [/b]There are six points in a plane, no four of which are collinear. A line is formed connecting every pair of points. Find the smallest possible number of distinct lines formed. [b]p4.[/b] Let $a,b,c$ be real numbers which satisfy $$\frac{2017}{a}= a(b +c), \frac{2017}{b}= b(a +c), \frac{2017}{c}= c(a +b).$$ Find the sum of all possible values of $abc$. [b]p5.[/b] Let $a$ and $b$ be complex numbers such that $ab + a +b = (a +b +1)(a +b +3)$. Find all possible values of $\frac{a+1}{b+1}$. [b]p6.[/b] Let $\vartriangle ABC$ be a triangle. Let $X,Y,Z$ be points on lines $BC$, $CA$, and $AB$, respectively, such that $X$ lies on segment $BC$, $B$ lies on segment $AY$ , and $C$ lies on segment $AZ$. Suppose that the circumcircle of $\vartriangle XYZ$ is tangent to lines $AB$, $BC$, and $CA$ with center $I_A$. If $AB = 20$ and $I_AC = AC = 17$ then compute the length of segment $BC$. [b]p7. [/b]An ant makes $4034$ moves on a coordinate plane, beginning at the point $(0, 0)$ and ending at $(2017, 2017)$. Each move consists of moving one unit in a direction parallel to one of the axes. Suppose that the ant stays within the region $|x - y| \le 2$. Let N be the number of paths the ant can take. Find the remainder when $N$ is divided by $1000$. [b]p8.[/b] A $10$ digit positive integer $\overline{a_9a_8a_7...a_1a_0}$ with $a_9$ nonzero is called [i]deceptive [/i] if there exist distinct indices $i > j$ such that $\overline{a_i a_j} = 37$. Find the number of deceptive positive integers. [b]p9.[/b] A circle passing through the points $(2, 0)$ and $(1, 7)$ is tangent to the $y$-axis at $(0, r )$. Find all possible values of $ r$. [b]p10.[/b] An ellipse with major and minor axes $20$ and $17$, respectively, is inscribed in a square whose diagonals coincide with the axes of the ellipse. Find the area of the square. PS. You had better use hide for answers.

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

A quadrilateral $ABCD$ is inscribed in a circle $k$ where $AB$ $>$ $CD$,and $AB$ is not paralel to $CD$.Point $M$ is the intersection of diagonals $AC$ and $BD$, and the perpendicular from $M$ to $AB$ intersects the segment $AB$ at a point $E$.If $EM$ bisects the angle $CED$ prove that $AB$ is diameter of $k$. Proposed by Emil Stoyanov,Bulgaria

A sequence of vertices $v_1,v_2,\ldots,v_k$ in a graph, where $v_i=v_j$ only if $i=j$ and $k$ can be any positive integer, is called a $\textit{cycle}$ if $v_1$ is attached by an edge to $v_2$, $v_2$ to $v_3$, and so on to $v_k$ connected to $v_1$. Rotations and reflections are distinct: $A,B,C$ is distinct from $A,C,B$ and $B,C,A$. Supposed a simple graph $G$ has $2013$ vertices and $3013$ edges. What is the minimal number of cycles possible in $G$?

$A_1A_2A_3A_4$ is a cyclic quadrilateral inscribed in circle $\Omega$, with side lengths $A_1A_2 = 28$, $A_2A_3 =12\sqrt3$, $A_3A_4 = 28\sqrt3$, and $A_4A_1 = 8$. Let $X$ be the intersection of $A_1A_3, A_2A_4$. Now, for $i = 1, 2, 3, 4$, let $\omega_i$ be the circle tangent to segments$ A_iX$, $A_{i+1}X$, and $\Omega$, where we take indices cyclically (mod $4$). Furthermore, for each $i$, say $\omega_i$ is tangent to $A_1A_3$ at $X_i $, $A_2A_4$ at $Y_i$ , and $\Omega$ at $T_i$ . Let $P_1$ be the intersection of $T_1X_1$ and $T_2X_2$, and $P_3$ the intersection of $T_3X_3$ and $T_4X_4$. Let $P_2$ be the intersection of $T_2Y_2$ and $T_3Y_3$, and $P_4$ the intersection of $T_1Y_1$ and $T_4Y_4$. Find the area of quadrilateral $P_1P_2P_3P_4$.

Let $n$ be an integer greater than $1$. Let $x\in\mathbb R$. (a) Evaluate $S(x,n)=\sum(x+p)(x+q)$, where the summation is over all pairs $(p,q)$ of different numbers from $\{1,2,\ldots,n\}$. (b) Do there exist integers $x,n$ for which $S(x,n)=0$?

Let $OABC$ be a tetrahedon with $\angle BOC=\alpha,\angle COA =\beta$ and $\angle AOB =\gamma$. The angle between the faces $OAB$ and $OAC$ is $\sigma$ and the angle between the faces $OAB$ and $OBC$ is $\rho$. Show that $\gamma > \beta \cos\sigma + \alpha \cos\rho$.

In a $2 \times 3$ rectangle there is a polyline of length $36$, which can have self-intersections. Show that there exists a line parallel to two sides of the rectangle, which intersects the other two sides in their interior points and intersects the polyline in fewer than $10$ points. [i]Proposed by Josef Tkadlec, Czechia and Vojtech Bálint, Slovakia[/i]

Let $(a_n)_{n\ge 1}$ be an increasing sequence of positive integers. Assume that there is a constant $M>0$ satisfying$$0<a_{n+1}-a_n<M.a_n^{5/8},\forall n\ge 1.$$ Prove that: there exists a real number $A$ such that for each $k\in \mathbb{Z}^+,[A^{3^k}]$ is an element of $(a_n)_{n\ge 1}.$

The polynomial $P (x)$ is such that the polynomials $P (P (x))$ and $P (P (P (x)))$ are strictly monotone on the whole real axis. Prove that $P (x)$ is also strictly monotone on the whole real axis.

a) Find an integer $a$ for which $(x - a)(x - 10) + 1$ factors in the product $(x + b)(x + c)$ with integers $b$ and $c$. b) Find nonzero and nonequal integers $a, b, c$ so that $x(x - a)(x - b)(x - c) + 1$ factors into the product of two polynomials with integer coefficients.

A number is called [i]winning[/i] if it can be expressed in the form $\frac{a}{20}+\frac{b}{23}$ where $a$ and $b$ are positive integers. How many [i]winning[/i] numbers are less than 1? [i]Proposed by Andy Xu[/i]

For which real $a$ are there distinct reals $x$, $y$ such that $$\begin{cases} x = a - y^2 \\ y = a - x^2 \,\,\, ? \end {cases}$$

Suppose that $x$ and $y$ are nonzero real numbers such that \[\frac{3x+y}{x-3y}= -2.\] What is the value of \[\frac{x+3y}{3x-y}?\] $\textbf{(A) } {-3} \qquad \textbf{(B) } {-1} \qquad \textbf{(C) } 1 \qquad \textbf{(D) }2 \qquad \textbf{(E) } 3$

Let $ABCD$ be a convex quadrilateral. $DA$ and $CB$ meet at $F$ and $AB$ and $DC$ meet at $E$. The bisectors of the angles $DFC$ and $AED$ are perpendicular. Prove that these angle bisectors are parallel to the bisectors of the angles between the lines $AC$ and $BD.$

A point $K$ is chosen on the side $CD$ of a rectangle $ABCD$. From the vertex $B$, the perpendicular $BH$ is dropped to the segment $AK$. The segments $AK$ and $BH$ divide the rectangle into three parts such that each of them has the inscribed circle (see figure). Prove that if the circles tangent to $CD$ are equal then the third circle is also equal to them.

The figure below shows a regular $ABCDE$ pentagon inscribed in an equilateral triangle $MNP$ . Determine the measure of the angle $CMD$. [img]http://4.bp.blogspot.com/-LLT7hB7QwiA/Xp9fXOsihLI/AAAAAAAAL14/5lPsjXeKfYwIr5DyRAKRy0TbrX_zx1xHQCK4BGAYYCw/s200/2012%2Bobm%2Bl2.png[/img]

Let $a$ and $b$ be real numbers for which the following is true: $acscx + b cot x \ge 1$, for all $0 <x < \pi$ Find the least value of $a^2 + b$.

Eight points are chosen on the circumference of a circle, labelled $P_1$, $P_2$, ..., $P_8$ in clockwise order. A route is a sequence of at least two points $P_{a_1}$, $P_{a_2}$, $...$, $P_{a_n}$ such that if an ant were to visit these points in their given order, starting at $P_{a_1}$ and ending at $P_{a_n}$, by following $n-1$ straight line segments (each connecting each $P_{a_i}$ and $P_{a_{i+1}}$), it would never visit a point twice or cross its own path. Find the number of routes.

A function $ f: R^3\rightarrow R$ for all reals $ a,b,c,d,e$ satisfies a condition: \[ f(a,b,c)\plus{}f(b,c,d)\plus{}f(c,d,e)\plus{}f(d,e,a)\plus{}f(e,a,b)\equal{}a\plus{}b\plus{}c\plus{}d\plus{}e\] Show that for all reals $ x_1,x_2,\ldots,x_n$ ($ n\geq 5$) equality holds: \[ f(x_1,x_2,x_3)\plus{}f(x_2,x_3,x_4)\plus{}\ldots \plus{}f(x_{n\minus{}1},x_n,x_1)\plus{}f(x_n,x_1,x_2)\equal{}x_1\plus{}x_2\plus{}\ldots\plus{}x_n\]

Three faces of a regular tetrahedron are painted in white and the remaining one in black. Initially, the tetrahedron is positioned on a plane with the black face down. It is then tilted several times over its edges. After a while it returns to its original position. Can it now have a white face down?

Let $m$ and $n$ be two positive integers greater than $1$. Prove that there are $m$ positive integers $N_1$ , $\ldots$ , $N_m$ (some of them may be equal) such that \[\sqrt{m}=\sum_{i=1}^m{(\sqrt{N_i}-\sqrt{N_i-1})^{\frac{1}{n}}.}\]