All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$? $\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16$

There are $1998$ rectangular pieces $2$ cm wide and $3$ cm long and with them squares are assembled (without overlapping or gaps). What is the greatest number of different squares that can be had at the same time?

Let $ABCD$ be a parallelogram. Construct squares $ABC_1D_1, BCD_2A_2, CDA_3B_3, DAB_4C_4$ on the outer side of the parallelogram. Construct a square having $B_4D_1$ as one of its sides and it is on the outer side of $AB_4D_1$ and call its center $O_A$. Similarly do it for $C_1A_2, D_2B_3, A_3C_4$ to obtain $O_B, O_C, O_D$. Prove that $AO_A = BO_B = CO_C = DO_D$.

Ten distinct numbers are chosen at random from the set $\{1, 2, 3, ... , 37\}$. Show that one can select four distinct numbers out of those ten so that the sum of two of them is equal to the sum of the other two.

Let $n$ be a natural number. A sequence $x_1,x_2, \cdots ,x_{n^2}$ of $n^2$ numbers is called $n-\textit{good}$ if each $x_i$ is an element of the set $\{1,2,\cdots ,n\}$ and the ordered pairs $\left(x_i,x_{i+1}\right)$ are all different for $i=1,2,3,\cdots ,n^2$ (here we consider the subscripts modulo $n^2$). Two $n-$good sequences $x_1,x_2,\cdots ,x_{n^2}$ and $y_1,y_2,\cdots ,y_{n^2}$ are called $\textit{similar}$ if there exists an integer $k$ such that $y_i=x_{i+k}$ for all $i=1,2,\cdots,n^2$ (again taking subscripts modulo $n^2$). Suppose that there exists a non-trivial permutation (i.e., a permutation which is different from the identity permutation) $\sigma$ of $\{1,2,\cdots ,n\}$ and an $n-$ good sequence $x_1,x_2,\cdots,x_{n^2}$ which is similar to $\sigma\left(x_1\right),\sigma\left(x_2\right),\cdots ,\sigma\left(x_{n^2}\right)$. Show that $n\equiv 2\pmod{4}$.

Let $ a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $ a^2_1 \plus{} a^2_2 \plus{} \ldots \plus{} a^2_{100} \equal{} 1.$ Prove that \[ a^2_1 \cdot a_2 \plus{} a^2_2 \cdot a_3 \plus{} \ldots \plus{} a^2_{100} \cdot a_1 < \frac {12}{25}. \] [i]Author: Marcin Kuzma, Poland[/i]

Let $\mathbb{R^+}$ denote the set of all positive real numbers. Find all functions $f: \mathbb{R^+}\rightarrow \mathbb{R^+}$ such that $$xf(x + y) + f(xf(y) + 1) = f(xf(x))$$ for all $x, y \in\mathbb{R^+}.$ [i]Proposed by Amadej Kristjan Kocbek, Jakob Jurij Snoj[/i]

Let $ M \equal{} \lbrace 2, 3, 4, \ldots\, 1000 \rbrace$. Find the smallest $ n \in \mathbb{N}$ such that any $ n$-element subset of $ M$ contains 3 pairwise disjoint 4-element subsets $ S, T, U$ such that [b]I.[/b] For any 2 elements in $ S$, the larger number is a multiple of the smaller number. The same applies for $ T$ and $ U$. [b]II.[/b] For any $ s \in S$ and $ t \in T$, $ (s,t) \equal{} 1$. [b]III.[/b] For any $ s \in S$ and $ u \in U$, $ (s,u) > 1$.

Suppose that $a > 0; b > 0$ and $a + b \leq 1$. Determine the minimum value of $M=\frac{1}{ab} +\frac{1}{a^2+ab}+\frac{1}{ab+b^2}+\frac{1}{a^2+b^2}$.

Prove that if $\alpha, \beta, \gamma$ are angles of a triangle, then $\frac{1}{\sin \alpha}+ \frac{1}{\sin \beta} \ge \frac{8}{ 3+2 \ cos\gamma}$ .

Let $f(x)$ be a polynomial with integer co-efficients. Assume that $3$ divides the value $f(n)$ for each integer $n$. Prove that when $f(x)$ is divided by $x^3-x$ , the remainder is of the form $3r(x)$ where $r(x)$ is a polynomial with integer coefficients.

A [i]complex set[/i], along with its [i]complexity[/i], is defined recursively as the following: [list] [*]The set $\mathbb{C}$ of complex numbers is a complex set with complexity $1$. [*]Given two complex sets $C_1, C_2$ with complexity $c_1, c_2$ respectively, the set of all functions $f:C_1\rightarrow C_2$ is a complex set denoted $[C_1, C_2]$ with complexity $c_1 + c_2$. [/list] A [i]complex expression[/i], along with its [i]evaluation[/i] and its [i]complexity[/i], is defined recursively as the following: [list] [*]A single complex set $C$ with complexity $c$ is a complex expression with complexity $c$ that evaluates to itself. [*]Given two complex expressions $E_1, E_2$ with complexity $e_1, e_2$ that evaluate to $C_1$ and $C_2$ respectively, if $C_1 = [C_2, C]$ for some complex set $C$, then $(E_1, E_2)$ is a complex expression with complexity $e_1+e_2$ that evaluates to $C$. [/list] For a positive integer $n$, let $a_n$ be the number of complex expressions with complexity $n$ that evaluate to $\mathbb{C}$. Let $x$ be a positive real number. Suppose that \[a_1+a_2x+a_3x^2+\dots = \dfrac{7}{4}.\] Then $x=\frac{k\sqrt{m}}{n}$, where $k$,$m$, and $n$ are positive integers such that $m$ is not divisible by the square of any integer greater than $1$, and $k$ and $n$ are relatively prime. Compute $100k+10m+n$. [i]Proposed by Luke Robitaille and Yannick Yao[/i]

Let $S$ be a set of $n>0$ elements, and let $A_1 , A_2 , \ldots A_k$ be a family of distinct subsets such that any two have a non-empty intersection. Assume that no other subset of $S$ intersects all of the $A_i.$ Prove that $ k=2^{n-1}.$

Consider the acute triangle $ABC$. Let $C_1$ and $C_2$ be semicircles with diameters $AB$ and $AC$, respectively, positioned outside triangle $ABC$. The altitude passing through $C$ intersects $C_1$ at $P$, and similarly, $Q$ is the intersection of $C_2$ with the extension of the altitude passing through $B$. Prove that $AP = AQ$.

A number $N$ has three digits when expressed in base $7$. When $N$ is expressed in base $9$ the digits are reversed. Then the middle digit is: $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$

The integers $a_1,a_2,...,a_{2012}$ are given. Exactly $29$ of them are divisible by $3$. Prove that the sum $a_1^2+a_2^2+...+a_{2012}^2$ is divisible by $3$.

All the species of plants existing in Russia are catalogued (numbered by integers from $2$ to $2000$ ; one after another, without omissions or repetitions). For any pair of species the gcd of their catalogue numbers was calculated and recorded but the catalogue numbers themselves were lost. Is it possible to restore the catalogue numbers from the data in hand?

Let $ABC$ be a triangle such that the altitude $CD$ is equal to $AB$. The squares $DBEF$ and $ADGH$ are constructed with $F,G$ on $CD$. Show that the segments $CD,AE$ and $BH$ are concurrent.

A scalene triangle $ABC$ is inscribed in a circle $\Gamma$. The bisector of angle $A$ meets $BC$ at $E$. Let $M$ be the midpoint of the arc $BAC$. The line $ME$ intersects $\Gamma$ again at $D$. Show that the circumcentre of triangle $AED$ coincides with the intersection point of the tangent to $\Gamma$ at $D$ and the line $BC$.

What is the largest integer that is a divisor of \[ (n\plus{}1)(n\plus{}3)(n\plus{}5)(n\plus{}7)(n\plus{}9) \]for all positive even integers $ n$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 165$

Let $ABC$ be an acute triangle. Let $M_A, M_B$ and $M_C$ be the midpoints of sides $BC,CA$, respectively $AB$. Let $M'_A , M'_B$ and $M'_C$ be the the midpoints of the arcs $BC, CA$ and $AB$ respectively of the circumscriberd circle of triangle $ABC$. Let $P_A$ be the intersection of the straight line $M_BM_C$ and the perpendicular to $M'_BM'_C$ through $A$. Define $P_B$ and $P_C$ similarly. Show that the straight line $M_AP_A, M_BP_B$ and $M_CP_C$ intersect at one point.

Mike rides a bike for $30$ minutes, traveling $8$ miles. He started riding at $20$ miles per hour, but by the end of his journey he was only traveling at $10$ miles per hour. What was his average speed, in miles per hour? [i]Proposed by Nathan Ramesh

Consider the function $ f:\mathbb{R}\longrightarrow\mathbb{R} ,\quad f(x)=\max (x-3,2) . $ Find the perimeter and the area of the figure delimited by the lines $ x=-3,x=1, $ the $ Ox $ axis, and the graph of $ f. $

Let $X$ be a set which consists from $8$ consecutive positive integers. Set $X$ is divided on two disjoint subsets $A$ and $B$ with equal number of elements. If sum of squares of elements from set $A$ is equal to sum of squares of elements from set $B$, prove that sum of elements of set $A$ is equal to sum of elements of set $B$.

From an $n\times n$ square divided into $n^2$ unit squares, one corner unit square is cut off. Find all positive integers $n$ for which it is possible to tile the remaining part of the square with $L$-trominos. [img]https://cdn.artofproblemsolving.com/attachments/0/4/d13e6e7016d943b867f44375a2205b10ccf552.png[/img]