Suppose the function $f(x)-f(2x)$ has derivative $5$ at $x=1$ and derivative $7$ at $x=2$. Find the derivative of $f(x)-f(4x)$ at $x=1$.

If $a$ and $b$ are real numbers such that $$\begin{cases} a^3-3ab^2 = 8 \\ b^3-3a^2b = 11 \end{cases}$$ then what is $a^2+b^2$?

The number $a$ is such that both fractions $$(1-3a)/(2a + 3) \,\,\, and \,\,\, (17 + 4a)/(7 + a)$$ are positive. Which one is closer to $\sqrt5$?

Let the circles $S_1$ and $S_2$ meet at the points $A$ and $B$. A line through $B$ meets $S_1$ at a point $D$ other than $B$ and meets $S_2$ at a point $C$ other than $B$. The tangent to $S_1$ through $D$ and the tangent to $S_2$ through $C$ meet at $E$. If $|AD|=15$, $|AC|=16$, $|AB|=10$, what is $|AE|$? $ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 24 \qquad\textbf{(C)}\ 25 \qquad\textbf{(D)}\ 26 \qquad\textbf{(E)}\ 31 $

Let $a, b, c$ be positive reals, and let $\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P$. Prove that \[\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).\][i]Proposed by David Stoner[/i]

Triangle $ABC$ has a right angle at $C$, $AC = 3$ and $BC = 4$. Triangle $ABD$ has a right angle at $A$ and $AD = 12$. Points $C$ and $D$ are on opposite sides of $\overline{AB}$. The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$. If $\frac{DE}{DB} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, then $m + n = $ [asy] size(170); defaultpen(fontsize(10pt)+linewidth(.8pt)); pair C=origin, A=(0,3), B=(4,0), D=(7.2,12.6), E=(7.2,0); draw(A--C--B--A--D--B--E--D); label("$A$",A,W); label("$B$",B,S); label("$C$",C,SW); label("$D$",D,NE); label("$E$",E,SE); [/asy] $ \textbf{(A)}\ 25\qquad\textbf{(B)}\ 128\qquad\textbf{(C)}\ 153\qquad\textbf{(D)}\ 243\qquad\textbf{(E)}\ 256 $

Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.

[b]a)[/b] prove that the function $\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$ that is defined on the area $Re(s)>1$, is an analytic function. [b]b)[/b] prove that the function $\zeta(s)-\frac{1}{s-1}$ can be spanned to an analytic function over $\mathbb C$. [b]c)[/b] using the span of part [b]b[/b] show that $\zeta(1-n)=-\frac{B_n}{n}$ that $B_n$ is the $n$th bernoli number that is defined by generating function $\frac{t}{e^t-1}=\sum_{n=0}^{\infty}B_n\frac{t^n}{n!}$.

[u]Round 1[/u] [b]1.1.[/b] What is the area of a circle with diameter $2$? [b]1.2.[/b] What is the slope of the line through $(2, 1)$ and $(3, 4)$? [b]1.3.[/b] What is the units digit of $2^2 \cdot 4^4 \cdot 6^6$ ? [u]Round 2[/u] [b]2.1.[/b] Find the sum of the roots of $x^2 - 5x + 6$. [b]2. 2.[/b] Find the sum of the solutions to $|2 - x| = 1$. [b]2.3.[/b] On April $1$, $2018$, Mr. Dospinescu, Mr. Phaovibul and Mr. Pohoata all go swimming at the same pool. From then on, Mr. Dospinescu returns to the pool every 4th day, Mr. Phaovibul returns every $7$th day and Mr. Pohoata returns every $13$th day. What day will all three meet each other at the pool again? Give both the month and the day. [u]Round 3[/u] [b]3. 1.[/b] Kendall and Kylie are each selling t-shirts separately. Initially, they both sell t-shirts for $\$ 33$ each. A week later, Kendall marks up her t-shirt price by $30 \%$, but after seeing a drop in sales, she discounts her price by $30\%$ the following week. If Kim wants to buy $360$ t-shirts, how much money would she save by buying from Kendall instead of Kylie? Write your answer in dollars and cents. [b]3.2.[/b] Richard has English, Math, Science, Spanish, History, and Lunch. Each class is to be scheduled into one distinct block during the day. There are six blocks in a day. How many ways could he schedule his classes such that his lunch block is either the $3$rd or $4$th block of the day? [b]3.3.[/b] How many lattice points does $y = 1 + \frac{13}{17}x$ pass through for $x \in [-100, 100]$ ? (A lattice point is a point where both coordinates are integers.) [u]Round 4[/u] [b]4. 1.[/b] Unsurprisingly, Aaron is having trouble getting a girlfriend. Whenever he asks a girl out, there is an eighty percent chance she bursts out laughing in his face and walks away, and a twenty percent chance that she feels bad enough for him to go with him. However, Aaron is also a player, and continues asking girls out regardless of whether or not previous ones said yes. What is the minimum number of girls Aaron must ask out for there to be at least a fifty percent chance he gets at least one girl to say yes? [b]4.2.[/b] Nithin and Aaron are two waiters who are working at the local restaurant. On any given day, they may be fired for poor service. Since Aaron is a veteran who has learned his profession well, the chance of him being fired is only $\frac{2}{25}$ every day. On the other hand, Nithin (who never paid attention during job training) is very lazy and finds himself constantly making mistakes, and therefore the chance of him being fired is $\frac{2}{5}$. Given that after 1 day at least one of the waiters was fired, find the probability Nithin was fired. [b]4.3.[/b] In a right triangle, with both legs $4$, what is the sum of the areas of the smallest and largest squares that can be inscribed? An inscribed square is one whose four vertices are all on the sides of the triangle. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784569p24468582]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$9$ different pieces of cheese are placed on a plate. Is it always possible to cut one of them into two parts so that the $10$ pieces obtained were divisible into two portions of equal mass of $5$ pieces each?

Assume that $2n+1$ positive integers satisfy the following: If we remove any of these integers, the remaining $2n$ integers can be partitioned in two groups of $n$ numbers in each, such that the sum of the numbers in one group is equal to the sum of the numbers in the other. Prove that all of these numbers must be equal.

Inside triangle $ABC$, the point $P$ is chosen such that $\angle PAB = \angle PCB =\frac14 (\angle A+ \angle C)$. Let $BL$ be the bisector of $\vartriangle ABC$. Line $PL$ intersects the circumcircle of $\vartriangle APC$ at point $Q$. Prove that the line $QB$ is the bisector of $\angle AQC$.

Consider all ordered pairs $(a, b)$ of positive integers such that $\frac{a^2 + b^2 + 2}{ab}$ is an integer and $a\le b$. We label all such pairs in increasing order by their distance from the origin. (It is guaranteed that no ties exist.) Thus $P_1 = (1, 1), P_2 = (1, 3)$, and so on. If $P_{2020} = (x, y),$ then compute the remainder when $x + y$ is divided by $2017$. [i]Proposed by Ashwin Sah[/i]

Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

Find the smallest positive integer $n$ for which one can select $n$ distinct real numbers such that each of them is equal to the sum of some two other selected numbers. [i]Proposed by Anton Trygub[/i]

The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.

Let $ \{A_n\}_{n \in \mathbb{N}}$ be a sequence of measurable subsets of the real line which covers almost every point infinitely often. Prove, that there exists a set $ B \subset \mathbb{N}$ of zero density, such that $ \{A_n\}_{n \in B}$ also covers almost every point infinitely often. (The set $ B \subset \mathbb{N}$ is of zero density if $ \lim_{n \to \infty} \frac {\#\{B \cap \{0, \dots, n \minus{} 1\}\}}{n} \equal{} 0$.)

Solve for real $x$: \[ \frac{1}{[x]} + \frac{1}{[2x]} = x - [x] + \frac{1}{3}. \]

If $ \frac {x}{y} \equal{} \frac {3}{4}$, then the incorrect expression in the following is: $ \textbf{(A)}\ \frac {x \plus{} y}{y} \equal{} \frac {7}{4} \qquad\textbf{(B)}\ \frac {y}{y \minus{} x} \equal{} \frac {4}{1} \qquad\textbf{(C)}\ \frac {x \plus{} 2y}{x} \equal{} \frac {11}{3}$ $ \textbf{(D)}\ \frac {x}{2y} \equal{} \frac {3}{8} \qquad\textbf{(E)}\ \frac {x \minus{} y}{y} \equal{} \frac {1}{4}$

A square with side length $6$ is rotated by $90^\circ$ about its center. What is the area of the region swept out by the perimeter of the square (that is, the four line segments forming the boundary of the square)?

For each pair of integers $(i,k)$ such that $1\le i\le k$, the linear transformation $P_{i,k}:\mathbb{R}^k\to\mathbb{R}^k$ is defined as: $P_{i,k}(a_1,\cdots,a_{i-1},a_i,a_{i+1},\cdots,a_k)=(a_1,\cdots,a_{i-1},0,a_{i+1},\cdots,a_k)$ Prove that for all $n\ge2$ and for every set of $n-1$ linearly independent vectors $v_1,\cdots,v_{n-1}$ in $\mathbb{R}^n$, there is an integer $k$ such that $1\le k\le n$ and such that the vectors $P_{k,n}(v_1),\cdots,P_{k,n}(v_{n-1})$ are linearly independent.

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is 7. Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\frac{m}n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $ABC$ be a 90-degree triangle with hypotenuse $BC$. Let $D$ and $E$ distinct points on segment $BC$ and $P, Q$ be the foot of the perpendicular from $D$ to $AB$ and $E$ to $AC$, respectively. $DP$ and $EQ$ intersect at $R$. Lines $CR$ and $AB$ intersect at $M$ and lines $BR$ and $AC$ intersect at $N$. Prove that $MN \parallel BC$ if and only if $BD=CE$.