[b]p1.[/b] Let $X,Y ,Z$ be nonzero real numbers such that the quadratic function $X t^2 - Y t + Z = 0$ has the unique root $t = Y$ . Find $X$. [b]p2.[/b] Let $ABCD$ be a kite with $AB = BC = 1$ and $CD = AD =\sqrt2$. Given that $BD =\sqrt5$, find $AC$. [b]p3.[/b] Find the number of integers $n$ such that $n -2016$ divides $n^2 -2016$. An integer $a$ divides an integer $b$ if there exists a unique integer $k$ such that $ak = b$. [b]p4.[/b] The points $A(-16, 256)$ and $B(20, 400)$ lie on the parabola $y = x^2$ . There exists a point $C(a,a^2)$ on the parabola $y = x^2$ such that there exists a point $D$ on the parabola $y = -x^2$ so that $ACBD$ is a parallelogram. Find $a$. [b]p5.[/b] Figure $F_0$ is a unit square. To create figure $F_1$, divide each side of the square into equal fifths and add two new squares with sidelength $\frac15$ to each side, with one of their sides on one of the sides of the larger square. To create figure $F_{k+1}$ from $F_k$ , repeat this same process for each open side of the smallest squares created in $F_n$. Let $A_n$ be the area of $F_n$. Find $\lim_{n\to \infty} A_n$. [img]https://cdn.artofproblemsolving.com/attachments/8/9/85b764acba2a548ecc61e9ffc29aacf24b4647.png[/img] [b]p6.[/b] For a prime $p$, let $S_p$ be the set of nonnegative integers $n$ less than $p$ for which there exists a nonnegative integer $k$ such that $2016^k -n$ is divisible by $p$. Find the sum of all $p$ for which $p$ does not divide the sum of the elements of $S_p$ . [b]p7. [/b] Trapezoid $ABCD$ has $AB \parallel CD$ and $AD = AB = BC$. Unit circles $\gamma$ and $\omega$ are inscribed in the trapezoid such that circle $\gamma$ is tangent to $CD$, $AB$, and $AD$, and circle $\omega$ is tangent to $CD$, $AB$, and $BC$. If circles $\gamma$ and $\omega$ are externally tangent to each other, find $AB$. [b]p8.[/b] Let $x, y, z$ be real numbers such that $(x+y)^2+(y+z)^2+(z+x)^2 = 1$. Over all triples $(x, y, z)$, find the maximum possible value of $y -z$. [b]p9.[/b] Triangle $\vartriangle ABC$ has sidelengths $AB = 13$, $BC = 14$, and $CA = 15$. Let $P$ be a point on segment $BC$ such that $\frac{BP}{CP} = 3$, and let $I_1$ and $I_2$ be the incenters of triangles $\vartriangle ABP$ and $\vartriangle ACP$. Suppose that the circumcircle of $\vartriangle I_1PI_2$ intersects segment $AP$ for a second time at a point $X \ne P$. Find the length of segment $AX$. [b]p10.[/b] For $1 \le i \le 9$, let Ai be the answer to problem i from this section. Let $(i_1,i_2,... ,i_9)$ be a permutation of $(1, 2,... , 9)$ such that $A_{i_1} < A_{i_2} < ... < A_{i_9}$. For each $i_j$ , put the number $i_j$ in the box which is in the $j$th row from the top and the $j$th column from the left of the $9\times 9$ grid in the bonus section of the answer sheet. Then, fill in the rest of the squares with digits $1, 2,... , 9$ such that $\bullet$ each bolded $ 3\times 3$ grid contains exactly one of each digit from $ 1$ to $9$, $\bullet$ each row of the $9\times 9$ grid contains exactly one of each digit from $ 1$ to $9$, and $\bullet$ each column of the $9\times 9$ grid contains exactly one of each digit from $ 1$ to $9$. PS. You had better use hide for answers.

Find the sum of the squares of the values $x$ that satisfy $\frac{1}{x} + \frac{2}{x+3}+\frac{3}{x+6} = 1$.

Determine all real tuples $\left(x_1,x_2,x_3,x_4,x_5,x_6\right)$ such that \begin{align*} x_1(x_6 + x_2) &= x_3 + x_5, \\ x_2(x_1 + x_3) &= x_4 + x_6, \\ x_3(x_2 + x_4) &= x_5 + x_1, \\ x_4(x_3 + x_5) &= x_6 + x_2, \\ x_5(x_4 + x_6) &= x_1 + x_3, \\ x_6(x_5 + x_1) &= x_2 + x_4. \end{align*}

On a line, there are $n$ closed intervals (none of which is a single point) whose union we denote by $S$. It's known that for every real number $d$, $0<d\le 1$, there are two points in $S$ that are a distance $d$ from each other. [list](a) Show that the sum of the lengths of the $n$ closed intervals is larger than $\frac{1}{n}$. (b) Prove that, for each positive integer $n$, the $\frac{1}{n}$ in the statement of part (a) cannot be replaced with a larger number.[/list]

Professor Oak is feeding his $100$ Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is $100$ kilograms. Professor Oak distributes $100$ kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of the bowl). The [i]dissatisfaction level[/i] of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $\lvert N-C\rvert$. Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute food in a way that the sum of the dissatisfaction levels over all the $100$ Pokémon is at most $D$. [i]Oleksii Masalitin, Ukraine[/i]

Let a polynomial $ P(x) \equal{} rx^3 \plus{} qx^2 \plus{} px \plus{} 1$ $ (r > 0)$ such that the equation $ P(x) \equal{} 0$ has only one real root. A sequence $ (a_n)$ is defined by $ a_0 \equal{} 1, a_1 \equal{} \minus{} p, a_2 \equal{} p^2 \minus{} q, a_{n \plus{} 3} \equal{} \minus{} pa_{n \plus{} 2} \minus{} qa_{n \plus{} 1} \minus{} ra_n$. Prove that $ (a_n)$ contains an infinite number of nagetive real numbers.

Points $(\sqrt{\pi}, a)$ and $(\sqrt{\pi}, b)$ are distinct points on the graph of $y^2+x^4=2x^2y+1$. What is $|a-b|$? $ \textbf{(A) }1\qquad\textbf{(B) }\dfrac{\pi}{2}\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{1+\pi}\qquad\textbf{(E) }1+\sqrt{\pi} $

Let $A$ be a set of $2025$ non-negative integers and $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ be a function with the following two properties: 1) For every two distinct positive integers $x,y$ there exists $a\in A$, such that $x-y$ divides $f(x+a) - f(y+a)$. 2) For every positive integer $N$ there exists a positive integer $t$ such that $f(x) \neq f(y)$ whenever $x,y \in [t, t+N]$ are distinct. Prove that there are infinitely many primes $p$ such that $p$ divides $f(x)$ for some positive integer $x$.

Solve the equation $$x^5 + (x + 1)^5 + (x + 2)^5 + ... + (x + 1998)^5 = 0.$$

Given is the function $ f\equal{} \lfloor x^2 \rfloor \plus{} \{ x \}$ for all positive reals $ x$. ( $ \lfloor x \rfloor$ denotes the largest integer less than or equal $ x$ and $ \{ x \} \equal{} x \minus{} \lfloor x \rfloor$.) Show that there exists an arithmetic sequence of different positive rational numbers, which all have the denominator $ 3$, if they are a reduced fraction, and don’t lie in the range of the function $ f$.

Calculate the sum $$ 2^n+2^{n-1}\cos\alpha +2^{n-2} \cos2\alpha +\cdots +2\cos (n-1)\alpha +\cos n\alpha , $$ where $ \alpha $ is a real number and $ n $ a natural one. [i]Dan Negulescu[/i]

Find all the functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$|f(x)-f(y)| = 2 |x - y| $$

[b]p1.[/b] (a) Brian has a big job to do that will take him two hours to complete. He has six friends who can help him. They all work at the same rate, somewhat slower than Brian. All seven working together can finish the job in $45$ minutes. How long will it take to do the job if Brian worked with only three of his friends? (b) Brian could do his next job in $N$ hours, working alone. This time he has an unlimited list of friends who can help him, but as he moves down the list, each friend works more slowly than those above on the list. The first friend would take $kN$ ($k > 1$) hours to do the job alone, the second friend would take $k^2N$ hours alone, the third friend would take $k^3N$ hours alone, etc. Theoretically, if Brian could get all his infinite number of friends to help him, how long would it take to complete the job? [b]p2.[/b] (a) The centers of two circles of radius $1$ are two opposite vertices of a square of side $1$. Find the area of the intersection of the two circles. (b) The centers of two circles of radius $1$ are two consecutive vertices of a square of side $1$. Find the area of the intersection of the two circles and the square. (c) The centers of four circles of radius $1$ are the vertices of a square of side $1$. Find the area of the intersection of the four circles. [b]p3.[/b] For any real number$ x$, $[x]$ denotes the greatest integer that does not exceed $x$. For example, $[7.3] = 7$, $[10/3] = 3$, $[5] = 5$. Given natural number $N$, denote as $f(N)$ the following sum of $N$ integers: $$f(N) = [N/1] + [N/2] + [N/3] + ... + [N/n].$$ (a) Evaluate $f(7) - f(6)$. (b) Evaluate $f(35) - f(34)$. (c) Evaluate (with explanation) $f(1996) - f(1995)$. [b]p4.[/b] We will say that triangle $ABC$ is good if it satisfies the following conditions: $AB = 7$, the other two sides are integers, and $\cos A =\frac27$. (a) Find the sides of a good isosceles triangle. (b) Find the sides of a good scalene (i.e. non-isosceles) triangle. (c) Find the sides of a good scalene triangle other than the one you found in (b) and prove that any good triangle is congruent to one of the three triangles you have found. [b]p5.[/b] (a) A bag contains nine balls, some of which are white, the others are black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the two balls are of the same color is the same as the probability that they are of different colors. How many of the nine balls were of one color and how many of the other color? (b) A bag contains $N$ balls, some of which are white, the others are black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the two balls are of the same color is the same as the probability that they are of different colors. It is also found that $180 < N < 220$. Find the exact value of $N$ and determine how many of the $N$ balls were of one color and how many of the other color. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x_0, x_1, \ldots$ be a sequence of real numbers such that $x_n = \frac{1 + x_{n -1}}{x_{n - 2}}$ for $n \geq 2$. Find the number of ordered pairs of positive integers $(x_0, x_1)$ such that the sequence gives $x_{2018} = \frac{1}{1000}$.

The sum of the squares of the roots of the equation $x^2+2hx=3$ is $10$. The absolute value of $h$ is equal to $\textbf{(A) }-1\qquad\textbf{(B) }\textstyle\frac{1}{2}\qquad\textbf{(C) }\textstyle\frac{3}{2}\qquad\textbf{(D) }2\qquad \textbf{(E) }\text{None of these}$

Given that $a$ and $b$ are real numbers such that for infinitely many positive integers $m$ and $n$, \[ \lfloor an + b \rfloor \ge \lfloor a + bn \rfloor \] \[ \lfloor a + bm \rfloor \ge \lfloor am + b \rfloor \] Prove that $a = b$.

Prove that for all positive real numbers $a,b,c$ the following inequality holds \[ \left( \frac ab + \frac bc + \frac ca \right)^2 \geq \frac 32 \cdot \left ( \frac{a+b}c + \frac{b+c}a + \frac{c+a} b \right) . \]

Let $P$ be a polynomial with each root is real and each coefficient is either $1$ or $-1$. The degree of $P$ can be at most ? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None} $

[b]p1.[/b] The length of the first side of a triangle is $1$, the length of the second side is $11$, and the length of the third side is an integer. Find that integer. [b]p2.[/b] Suppose $a, b$, and $c$ are positive numbers such that $a + b + c = 1$. Prove that $ab + ac + bc \le \frac13$. [b]p3.[/b] Prove that $1 +\frac12+\frac13+\frac14+ ... +\frac{1}{100}$ is not an integer. [b]p4.[/b] Find all of the four-digit numbers n such that the last four digits of $n^2$ coincide with the digits of $n$. [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $n=1681$ children, $a_1,a_2,...,a_{n}$ seated clockwise in a circle on the floor. The teacher walks behind the children in the clockwise direction with a box of $1000$ candies. She drops a candy behind the first child $a_1$. She then skips one child and drops a candy behind the third child, $a_3$. Now she skips two children and drops a candy behind the next child, $a_6$. She continues this way, at each stage skipping one child more than at the preceding stage before dropping a candy behind the next child. How many children will never receive a candy? Justify your answer.

Find the number of integer polynomials $P$ such that $P(x)^2 = P(P(x)) \forall x$.

Find all continuous functions $f(x)$ defined for all $x>0$ such that for every $x$, $y > 0$ \[ f\left(x+{1\over x}\right)+f\left(y+{1\over y}\right)= f\left(x+{1\over y}\right)+f\left(y+{1\over x}\right) . \] [i]Proposed by F. Petrov[/i]

The domain of the function $f(x) = \text{arcsin}(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$, where $m$ and $n$ are positive integers and $m > 1$. Find the remainder when the smallest possible sum $m+n$ is divided by $1000$.

Let $f(x)$ be a function such that every straight line has the same number of intersection points with the graph $y = f(x)$ and with the graph $y = x^2$. Prove that $f(x) = x^2.$

Consider the sequence $(a_n)_{n\in \mathbb{N}}$ with $a_0=a_1=a_2=a_3=1$ and $a_na_{n-4}=a_{n-1}a_{n-3} + a^2_{n-2}$. Prove that all the terms of this sequence are integer numbers.