Let $f$ be a function whose domain is $[1, 20]$ and whose range is a subset of $[-100, 100].$ Suppose $\tfrac{f(x)}{y} - \tfrac{f(y)}{x} \leq (x - y)^2$ for all $x$ and $y$ in $[1, 20].$ Compute the largest value of $f(x) - f(y)$ over all such functions $f$ and all $x$ and $y$ in the domain $[1, 20].$

Does there exist a triangle, whose side is equal to some of its altitudes, another side is equal to some of its bisectors, and the third is equal to some of its medians?

[b]p1.[/b] An $18$oz glass of apple juice is $6\%$ sugar and a $6$oz glass of orange juice is $12\%$ sugar. The two glasses are poured together to create a cocktail. What percent of the cocktail is sugar? [b]p2.[/b] Find the number of positive numbers that can be expressed as the difference of two integers between $-2$ and $2012$ inclusive. [b]p3.[/b] An annulus is defined as the region between two concentric circles. Suppose that the inner circle of an annulus has radius $2$ and the outer circle has radius $5$. Find the probability that a randomly chosen point in the annulus is at most $3$ units from the center. [b]p4.[/b] Ben and Jerry are walking together inside a train tunnel when they hear a train approaching. They decide to run in opposite directions, with Ben heading towards the train and Jerry heading away from the train. As soon as Ben finishes his $1200$ meter dash to the outside, the front of the train enters the tunnel. Coincidentally, Jerry also barely survives, with the front of the train exiting the tunnel as soon as he does. Given that Ben and Jerry both run at $1/9$ of the train’s speed, how long is the tunnel in meters? [b]p5.[/b] Let $ABC$ be an isosceles triangle with $AB = AC = 9$ and $\angle B = \angle C = 75^o$. Let $DEF$ be another triangle congruent to $ABC$. The two triangles are placed together (without overlapping) to form a quadrilateral, which is cut along one of its diagonals into two triangles. Given that the two resulting triangles are incongruent, find the area of the larger one. [b]p6.[/b] There is an infinitely long row of boxes, with a Ditto in one of them. Every minute, each existing Ditto clones itself, and the clone moves to the box to the right of the original box, while the original Ditto does not move. Eventually, one of the boxes contains over $100$ Dittos. How many Dittos are in that box when this first happens? [b]p7.[/b] Evaluate $$26 + 36 + 998 + 26 \cdot 36 + 26 \cdot 998 + 36 \cdot 998 + 26 \cdot 36 \cdot 998.$$ [b]p8. [/b]There are $15$ students in a school. Every two students are either friends or not friends. Among every group of three students, either all three are friends with each other, or exactly one pair of them are friends. Determine the minimum possible number of friendships at the school. [b]p9.[/b] Let $f(x) = \sqrt{2x + 1 + 2\sqrt{x^2 + x}}$. Determine the value of $$\frac{1}{f(1)}+\frac{1}{f(1)}+\frac{1}{f(3)}+...+\frac{1}{f(24)}.$$ [b]p10.[/b] In square $ABCD$, points $E$ and $F$ lie on segments $AD$ and $CD$, respectively. Given that $\angle EBF = 45^o$, $DE = 12$, and $DF = 35$, compute $AB$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Assume that $X$ is a seperable metric space. Prove that if $f: X\longrightarrow\mathbb R$ is a function that $\lim_{x\rightarrow a}f(x)$ exists for each $a\in\mathbb R$. Prove that set of points in which $f$ is not continuous is countable.

Prove that for $n, p$ integers, $n \geq 4$ and $p \geq 4$, the proposition $\mathcal{P}(n, p)$ \[\sum_{i=1}^{n}\frac{1}{{x_{i}}^{p}}\geq \sum_{i=1}^{n}{x_{i}}^{p}\quad \textrm{for}\quad x_{i}\in \mathbb{R}, \quad x_{i}> 0 , \quad i=1,\ldots,n \ ,\quad \sum_{i=1}^{n}x_{i}= n,\] is false. [i]Dan Schwarz[/i] [hide="Remark"]In the competition, the students were informed (fact that doesn't actually relate to the problem's solution) that the propositions $\mathcal{P}(4, 3)$ are $\mathcal{P}(3, 4)$ true.[/hide]

One thousand people are in a tennis tournament where each person plays against each other person exactly once, and there are no ties. Prove that it is possible to put all the competitors in a line so that each of the $998$ people who are not at an end of the line either defeated both their neighbors or lost to both their neighbors.

Let $a,b$ and $c$ be positive real numbers such that $a+b+c+3abc\geq (ab)^2+(bc)^2+(ca)^2+3$. Show that the following inequality hold, $$\frac{a^3+b^3+c^3}{3}\geq\frac{abc+2021}{2022}.$$

Let $ S \equal{} \{2^0,2^1,2^2,\ldots,2^{10}\}$. Consider all possible positive differences of pairs of elements of $ S$. Let $ N$ be the sum of all of these differences. Find the remainder when $ N$ is divided by $ 1000$.

Is there a solution of the Cauchy problem \[x(x^2+y^2)\frac{\partial u}{\partial x}+y^3\frac{\partial u}{\partial y}=0,\;u|_{y=0}=1\] in a neighbourhood of the point $(x_0,0)$ of the $x$-axis? Is it unique?

Let $ABC$ be a triangle and $ABDE, BCFZ, CAKL$ be three similar rectangles constructed externally of the triangle. Let $A'$ be the intersection of $EF$ and $ZK, B'$ the intersection of $KZ$ and $DL$, and $C'$ the intersection of $DL$ and $EF$. Prove that $AA'$ passes through the midpoint of the line segment $B'C'$. Kostas Vittas

Let $a,b,c$ be sides of a triangle. Prove that \[ 2 < \frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b} - \frac{a^3+b^3+c^3}{abc}\leq 3 \]

Solve in integers $x$ and $y$ the equation $x^3-y^3=2xy+8$.

The sequence $ u_n$, $ n\equal{}0,1,2,...$ is defined by $ u_0\equal{}0, u_1\equal{}1$ and for each $ n \ge 1$, $ u_{n\plus{}1}$ is the smallest positive integer greater than $ u_n$ such that $ \{ u_0,u_1,...,u_{n\plus{}1} \}$ contains no three elements in arithmetic progression. Find $ u_{100}$.

Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy: $f(xf(y))=yf(x)$ for all $x,y$; and $f(x)\to0$ as $x\to\infty$.

A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied: (i) All the squares are congruent. (ii) If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares. (iii) Each square touches exactly three other squares. How many positive integers $n$ are there with $2018\leq n \leq 3018$, such that there exists a collection of $n$ squares that is tri-connected?

Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property: \[ n!\mid a^n \plus{} 1 \] [i]Proposed by Carlos Caicedo, Colombia[/i]

Let $ABC$ be an acute angled triangle with orthocenter $H$ and $AB<AC$. The point $T$ lies on line $BC$ so that $AT$ is a tangent to the circumcircle of $ABC$. Let lines $AH$ and $BC$ meet at point $D$ and let $M$ be the midpoint of $HC$. Let the circumcircle of $AHT$ meets $CH$ in $P \not=H$ and the circumcircle of $PDM$ meet $BC$ in $Q \not=D$. Prove that $QT=QA$.

Mr. Vader gave out a multiple choice test, and every question had an answer that was one of A, B, or C. After the test, he curved the test so that everybody got +50 (so a person who got $x\%$ right would get a score of $x+50$). In the class, a score in the range $\left[90,\infty\right)$ gets an A, a score in the range $\left[80,90\right)$ gets a B, and a score in the range $\left[70,80\right)$ gets a C. After the curve, Mr. Vader makes this statement: ``Guess A, get an A. Guess B, get a B. Guess C, get a C.'' That is, answering every question with the answer choice X would give, with the curve, a score receiving a grade of X, where X is one of A, B, C. Luke, a student in Mr. Vader's class, was told ahead of time that there were either $5$ or $6$ answers as A on the test. Find the sum of all possible values of the number of questions on the test, given this information. [i]2017 CCA Math Bonanza Tiebreaker Round #4[/i]

The first number in the following sequence is $1$. It is followed by two $1$'s and two $2$'s. This is followed by three $1$'s, three $2$'s, and three $3$'s. The sequence continues in this fashion. \[1,1,1,2,2,1,1,1,2,2,2,3,3,3,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,\dots.\] Find the $2014$th number in this sequence.

Suppose there are $m$ Martians and $n$ Earthlings at an intergalactic peace conference. To ensure the Martians stay peaceful at the conference, we must make sure that no two Martians sit together, such that between any two Martians there is always at least one Earthling. (a) Suppose all $m + n$ Martians and Earthlings are seated in a line. How many ways can the Earthlings and Martians be seated in a line? (b) Suppose now that the $m+n$ Martians and Earthlings are seated around a circular round-table. How many ways can the Earthlings and Martians be seated around the round-table?

Let $S = \{(x,y) | x = 1, 2, \ldots, 1993, y = 1, 2, 3, 4\}$. If $T \subset S$ and there aren't any squares in $T.$ Find the maximum possible value of $|T|.$ The squares in T use points in S as vertices.

$32$ stones, with pairwise different weights, and lever scales without weights are given. How to determine by $35$ scaling, which stone is the heaviest and which is the second by weight?

Let $x,y$ and $z$ be positive real numbers. Show that $x^2+xy^2+xyz^2\ge 4xyz-4$.

Let $a, b, c$ be the sides and $R$ be the circumradius of a triangle. Prove that \[\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\ge\frac{1}{R^2}.\]

In acute $\triangle ABC,$ let $D$ be the foot of perpendicular from $A$ on $BC$. Consider points $K, L, M$ on segment $AD$ such that $AK= KL= LM= MD$. Suppose the sum of the areas of the shaded region equals the sum of the areas of the unshaded regions in the following picture. Prove that $BD= DC$. [img]http://s27.postimg.org/a0d0plr4z/Untitled.png[/img]