[b]p1.[/b] It’s currently $6:00$ on a $12$ hour clock. What time will be shown on the clock $100$ hours from now? Express your answer in the form hh : mm. [b]p2.[/b] A tub originally contains $10$ gallons of water. Alex adds some water, increasing the amount of water by 20%. Barbara, unhappy with Alex’s decision, decides to remove $20\%$ of the water currently in the tub. How much water, in gallons, is left in the tub? Express your answer as an exact decimal. [b]p3.[/b] There are $2000$ math students and $4000$ CS students at Berkeley. If $5580$ students are either math students or CS students, then how many of them are studying both math and CS? [b]p4.[/b] Determine the smallest integer $x$ greater than $1$ such that $x^2$ is one more than a multiple of $7$. [b]p5.[/b] Find two positive integers $x, y$ greater than $1$ whose product equals the following sum: $$9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29.$$ Express your answer as an ordered pair $(x, y)$ with $x \le y$. [b]p6.[/b] The average walking speed of a cow is $5$ meters per hour. If it takes the cow an entire day to walk around the edges of a perfect square, then determine the area (in square meters) of this square. [b]p7.[/b] Consider the cube below. If the length of the diagonal $AB$ is $3\sqrt3$, determine the volume of the cube. [img]https://cdn.artofproblemsolving.com/attachments/4/d/3a6fdf587c12f2e4637a029f38444914e161ac.png[/img] [b]p8.[/b] I have $18$ socks in my drawer, $6$ colored red, $8$ colored blue and $4$ colored green. If I close my eyes and grab a bunch of socks, how many socks must I grab to guarantee there will be two pairs of matching socks? [b]p9.[/b] Define the operation $a @ b$ to be $3 + ab + a + 2b$. There exists a number $x$ such that $x @ b = 1$ for all $b$. Find $x$. [b]p10.[/b] Compute the units digit of $2017^{(2017^2)}$. [b]p11.[/b] The distinct rational numbers $-\sqrt{-x}$, $x$, and $-x$ form an arithmetic sequence in that order. Determine the value of $x$. [b]p12.[/b] Let $y = x^2 + bx + c$ be a quadratic function that has only one root. If $b$ is positive, find $\frac{b+2}{\sqrt{c}+1}$. [b]p13.[/b] Alice, Bob, and four other people sit themselves around a circular table. What is the probability that Alice does not sit to the left or right of Bob? [b]p14.[/b] Let $f(x) = |x - 8|$. Let $p$ be the sum of all the values of $x$ such that $f(f(f(x))) = 2$ and $q$ be the minimum solution to $f(f(f(x))) = 2$. Compute $p \cdot q$. [b]p15.[/b] Determine the total number of rectangles ($1 \times 1$, $1 \times 2$, $2 \times 2$, etc.) formed by the lines in the figure below: $ \begin{tabular}{ | l | c | c | r| } \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline \end{tabular} $ [b]p16.[/b] Take a square $ABCD$ of side length $1$, and let $P$ be the midpoint of $AB$. Fold the square so that point $D$ touches $P$, and let the intersection of the bottom edge $DC$ with the right edge be $Q$. What is $BQ$? [img]https://cdn.artofproblemsolving.com/attachments/1/1/aeed2c501e34a40a8a786f6bb60922b614a36d.png[/img] [b]p17.[/b] Let $A$, $B$, and $k$ be integers, where $k$ is positive and the greatest common divisor of $A$, $B$, and $k$ is $1$. Define $x\# y$ by the formula $x\# y = \frac{Ax+By}{kxy}$ . If $8\# 4 = \frac12$ and $3\# 1 = \frac{13}{6}$ , determine the sum $A + B + k$. [b]p18.[/b] There are $20$ indistinguishable balls to be placed into bins $A$, $B$, $C$, $D$, and $E$. Each bin must have at least $2$ balls inside of it. How many ways can the balls be placed into the bins, if each ball must be placed in a bin? [b]p19.[/b] Let $T_i$ be a sequence of equilateral triangles such that (a) $T_1$ is an equilateral triangle with side length 1. (b) $T_{i+1}$ is inscribed in the circle inscribed in triangle $T_i$ for $i \ge 1$. Find $$\sum^{\infty}_{i=1} Area (T_i).$$ [b]p20.[/b] A [i]gorgeous [/i] sequence is a sequence of $1$’s and $0$’s such that there are no consecutive $1$’s. For instance, the set of all gorgeous sequences of length $3$ is $\{[1, 0, 0]$,$ [1, 0, 1]$, $[0, 1, 0]$, $[0, 0, 1]$, $[0, 0, 0]\}$. Determine the number of gorgeous sequences of length $7$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We define \[f(x,y,z)=|xy|\sqrt{x^2+y^2}+|yz|\sqrt{y^2+z^2}+|zx|\sqrt{z^2+x^2}.\] Find the best constants $c_1,c_2\in\mathbb{R}$ such that \[c_1(x^2+y^2+z^2)^{3/2}\leq f(x,y,z)\leq c_1(x^2+y^2+z^2)^{3/2}\] hold for all reals $x,y,z$ satisfying $x+y+z=0$. [i]Proposed by Untro368.[/i]

Determine all pairs of functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ such that for any $x,y\in \mathbb{R}$, \[f(x)f(y)=g(x)g(y)+g(x)+g(y).\]

Let the four real solutions to the equation $x^2 + \frac{144}{x^2} = 25$ be $r_1, r_2, r_3$, and $r_4$. Find $|r_1| +|r_2| +|r_3| +|r_4|$.

Let $Q$ be a quadratic polynomial. If the sum of the roots of $Q^{100}(x)$ (where $Q^i(x)$ is defined by $Q^1(x)=Q(x)$, $Q^i(x)=Q(Q^{i-1}(x))$ for integers $i\geq 2$) is $8$ and the sum of the roots of $Q$ is $S$, compute $|\log_2(S)|$.

Francisco wrote a sequence of numbers starting with $25$. From the fourth term of the sequence onwards, each term of the sequence is the average of the previous three. Given that the first six terms of the sequence are natural numbers and that the sixth number written was $8$, what is the fifth term of the sequence?

Find all real numbers $y$, for which there exists at least one real number $x$ such that $y=\frac{\sqrt{x^2+4}}{\sqrt{x^2+1}+\sqrt{x^2+9}}.$

Determine all real numbers that satisfy the equation $$(x+1)^{21}+(x+1)^{20}(x-1)+(x+1)^{19}(x-1)^2+...+(x-1)^{21}=0$$ (RM Kuznec)

Let $r(x)$ be a polynomial of odd degree with real coefficients. Prove that there exist only finitely many (or none at all) pairs of polynomials $p(x) $ and $q(x)$ with real coefficients satisfying the equation $(p(x))^3 + q(x^2) = r(x)$.

Find all functions $f : R_{>0} \to R_{>0}$ with the following property: $$f \left( \frac{x}{y + 1}\right) = 1 - xf(x + y)$$ for all $x > y > 0$ .

Every nonnegative periodic decimal fraction represents a rational number, also in the form $\frac{p}{q}$ can be represented ($p$ and $q$ are natural numbers and coprime, $p\ge 0$, $q > 0)$. Now let $a_1$, $a_2$, $a_3$ and $a_4$ be digits to represent numbers in the decadic system. Let $a_1 \ne a_3$ or $a_2 \ne a_4$.Prove that it for the numbers: $z_1 = 0, \overline{a_1a_2a_3a_4} = 0,a_1a_2a_3a_4a_1a_2a_3a_4...$ $z_2 = 0, \overline{a_4a_1a_2a_3}$ $z_3 = 0, \overline{a_3a_4a_1a_2}$ $z_4 = 0, \overline{a_2a_3a_4a_1}$ In the above representation $p/q$ always have the same denominator. [hide=original wording]Jeder nichtnegative periodische Dezimalbruch repr¨asentiert eine rationale Zahl, die auch in der Form p/q dargestellt werden kann (p und q nat¨urliche Zahlen und teilerfremd, p >= 0, q > 0). Nun seien a1, a2, a3 und a4 Ziffern zur Darstellung von Zahlen im dekadischen System. Dabei sei a1 $\ne$ a3 oder a2 $\ne$ a4. Beweisen Sie! Die Zahlen z1 = 0, a1a2a3a4 = 0,a1a2a3a4a1a2a3a4... z2 = 0, a4a1a2a3 z3 = 0, a3a4a1a2 z4 = 0, a2a3a4a1 haben in der obigen Darstellung p/q stets gleiche Nenner.[/hide]

Let $a_1,a_2,a_3, \ldots$ be an infinite sequence of nonnegative real numbers such that for all positive integers $k$ the following conditions hold: $i)$ $a_k-2a_{k+1}+a_{k+2} \geq 0$; $ii)$ $\sum_{j=1}^{k} a_j \leq 1$. Prove that for all positive integer $k$ holds: $0 \leq a_k - a_{k+1} < \frac{2}{k^2}$

Consider all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying $f(1-f(x))=x$ for all $x \in \mathbb{R}$. a) By giving a concrete example, show that such a function exists. b) For each such function define the sum \[S_f=f(-2017)+f(-2016)+\dots+f(-1)+f(0)+f(1)+\dots+f(2017)+f(2018).\] Determine all possible values of $S_f$.

Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that $ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$ (Proposed by Gerhard Woeginger, Austria)

Prove the inequality $$\sum_{k=1}^{n}\frac{\sin (k+1)x}{\sin kx}< 2\frac{\cos x}{\sin^2x}$$ where $0 < nx < \pi/2$, $n \in N$.

What condition must be satisfied by the coefficients $u,v,w$ if the roots of the polynomial $x^3 -ux^2+vx-w$ are the sides of a triangle

For which $n$ do the equations have a solution in integers: \begin{eqnarray*}x_1 ^2 + x_2 ^2 + 50 &=& 16x_1 + 12x_2 \\ x_2 ^2 + x_3 ^2 + 50 &=& 16x_2 + 12x_3 \\ \cdots \quad \cdots \quad \cdots & \cdots & \cdots \quad \cdots \\ x_{n-1} ^2 + x_n ^2 + 50 &=& 16x_{n-1} + 12x_n \\ x_n ^2 + x_1 ^2 + 50 &=& 16x_n + 12x_1 \end{eqnarray*}

$(SWE 2)$ For each $\lambda (0 < \lambda < 1$ and $\lambda = \frac{1}{n}$ for all $n = 1, 2, 3, \cdots)$, construct a continuous function $f$ such that there do not exist $x, y$ with $0 < \lambda < y = x + \lambda \le 1$ for which $f(x) = f(y).$

Let be given natural numbers $a_1,a_2,...,a_n$ and a function $f : Z \to R$ such that $f(x) = 1$ for all integers $x < 0$ and $f(x) = 1- f(x-a_1)f(x-a_2)... f(x-a_n)$ for all integers $x \ge 0$. Prove that there exist natural numbers $s$ and $t$ such that for all integers $x > s$ it holds that $f(x+t) = f(x)$.

Evaluate the sum: $1^3 + 3^3 + 5^3 +... + (2n - 1)^3$.

Compute the smallest positive integer $n$ for which $$\sqrt{100+\sqrt{n}}+\sqrt{100-\sqrt{n}}$$ is an integer.

Let $P(x)$ be a nonconstant complex coefficient polynomial and let $Q(x,y)=P(x)-P(y).$ Suppose that polynomial $Q(x,y)$ has exactly $k$ linear factors unproportional two by tow (without counting repetitons). Let $R(x,y)$ be factor of $Q(x,y)$ of degree strictly smaller than $k$. Prove that $R(x,y)$ is a product of linear polynomials. [b]Note: [/b] The [i]degree[/i] of nontrivial polynomial $\sum_{m}\sum_{n}c_{m,n}x^{m}y^{n}$ is the maximum of $m+n$ along all nonzero coefficients $c_{m,n}.$ Two polynomials are [i]proportional[/i] if one of them is the other times a complex constant. [i]Proposed by Navid Safaie[/i]

Let $n\in \mathbb{N}$. Prove that the polynom $p(x)=x^{2n}-2x^{2n-1}+3x^{2n-2}-...-2nx+2n+1$ doesn't have real roots.

Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $\boxed{1} \ f(1) = 1$ $\boxed{2} \ f(m+n)(f(m)-f(n)) = f(m-n)(f(m)+f(n)) \ \forall \ m,n \in \mathbb{Z}$

How many times changes the sign of the function \[ f(x)\equal{}\cos x\cos\frac{x}{2}\cos\frac{x}{3}\cdots\cos\frac{x}{2009}\] at the interval $ \left[0, \frac{2009\pi}{2}\right]$?