Points $ A$ and $ B$ are on the parabola $ y \equal{} 4x^2 \plus{} 7x \minus{} 1$, and the origin is the midpoint of $ \overline{AB}$. What is the length of $ \overline{AB}$? $ \textbf{(A)}\ 2\sqrt5 \qquad \textbf{(B)}\ 5\plus{}\frac{\sqrt2}{2} \qquad \textbf{(C)}\ 5\plus{}\sqrt2 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 5\sqrt2$

Rectangle $ABCD$ is inscribed in circle $O$. Let the projections of a point $P$ on minor arc $CD$ onto $AB,AC,BD$ be $K,L,M$, respectively. Prove that $\angle LKM=45$if and only if $ABCD$ is a square.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that$$f(x+f(y^2)) + f(xy) = f(x) + yf(x+y)$$ for all $x, y \in \mathbb{R}$. [i]Proposed by Md. Fuad Al Alam[/i]

A king decides to reward one of his knights by making a game. He sits the knights at a round table and has them call out $1,2,3,1,2,3,\dots$ around the circle (that is, clockwise, and each person says a number). The people who say $2$ or $3$ immediately lose, and this continues until the last knight is left, the winner. Numbering the knights initially as $1,2,\dots,n$, find all values of $n$ such that knight $2008$ is the winner.

Note that $8^3 - 7^3 = 169 = 13^2$ and $13 = 2^2 + 3^2.$ Prove that if the difference between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

The number $ \left (2+2^{96} \right )!$ has $2^{93}$ trailing zeroes when expressed in base $B$. [b] a)[/b] Find the minimum possible $B$. [b]b)[/b] Find the maximum possible $B$. [b]c)[/b] Find the total number of possible $B$. [i]Proposed by Lewis Chen[/i]

Let $l,m$ be two parallel lines in the plane. Let $P$ be a fixed point between them. Let $E,F$ be variable points on $l,m$ such that the angle $EPF$ is fixed to a number like $\alpha$ where $0<\alpha<\frac{\pi}2$. (By angle $EPF$ we mean the directed angle) Show that there is another point (not $P$) such that it sees the segment $EF$ with a fixed angle too.

Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram. Show that $\triangle THN$ is similar to $\triangle PBC$. [i]Proposed by Sammy Luo[/i]

Let $n\geq3$ be a positive integer. Let $C_1,C_2,C_3,\ldots,C_n$ be unit circles in the plane, with centres $O_1,O_2,O_3,\ldots,O_n$ respectively. If no line meets more than two of the circles, prove that \[ \sum\limits^{}_{1\leq i<j\leq n}{1\over O_iO_j}\leq{(n-1)\pi\over 4}. \]

It is given acute triangle $ABC$ with orthocenter at point $H$. Prove that $$AH \cdot h_a+BH \cdot h_b+CH \cdot h_c=\frac{a^2+b^2+c^2}{2}$$ where $a$, $b$ and $c$ are sides of a triangle, and $h_a$, $h_b$ and $h_c$ altitudes of $ABC$

How many $5-$digit positive numbers which contain only odd numbers are there such that there is at least one pair of consecutive digits whose sum is $10$? $ \textbf{(A)}\ 3125 \qquad\textbf{(B)}\ 2500 \qquad\textbf{(C)}\ 1845 \qquad\textbf{(D)}\ 1190 \qquad\textbf{(E)}\ \text{None of the preceding} $

Calculate with $10\%$ relative error \[\int_{-\infty}^{\infty}\cos(100(x^4-x))dx\]

The operation $\oslash$, called "reciprocal sum," is useful in many areas of physics. If we say that $x=a\oslash b$, this means that $x$ is the solution to $$\frac{1}{x}=\frac{1}{a}+\frac{1}{b}$$ Compute $4\oslash2\oslash4\oslash3\oslash4\oslash4\oslash2\oslash3\oslash2\oslash4\oslash4\oslash3$.

Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral? $ \textbf{(A)}\ \sqrt{15} \qquad\textbf{(B)}\ \sqrt{21} \qquad\textbf{(C)}\ 2\sqrt{6} \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 2\sqrt{7} $

Consider the cube with the vertices $A(1,1,1)$, $B(-1,1,1)$, $C(-1,-1,1)$, $D(1,-1,1)$ and $A',B',C',D'$ symmetric to $A,B,C,D$ respectively with respect to the origin $O$. Let $T$ be a point not on the circumsphere of the cube and let $OT=d$. Denote $\alpha=\angle ATA'$, $\beta=\angle BTB'$, $\gamma=\angle CTC'$, $\delta=\angle DTD'$. Prove that $$\tan^2\alpha+\tan^2\beta+\tan^2\gamma+\tan^2\delta=\frac{32d^2}{\left(d^2-3\right)^2}.$$

In the middle cell of the $1 \times 2005$ strip there is a chip. Two players each queues move it: first, the first player moves the piece one cell in any direction, then the second one moves it $2$ cells, the $1$st - by $4$ cells, the 2nd by $8$, etc. (the $k$-th shift occurs by $2^{k-1}$ cells). That, whoever cannot make another move loses. Who can win regardless of the opponent's play?

For how many positive integers $x$ less than $4032$ is $x^2-20$ divisible by $16$ and $x^2-16$ divisible by $20$? [i] Proposed by Tristan Shin [/i]

Two positive integers $n,m\ge 2$ are called [i]allies[/i] if when written as a product of primes (not necessarily different): $n=p_1p_2...p_s$ and $m=q_1q_2...q_t$, turns out that: $$p_1 + p_2 + ... + p_s = q_1 + q_2 + ... + q_t$$ (a) Show that the biggest ally of any positive integer has to have only $2$ and $3$ in its prime factorization. (b) Find the biggest number which is allied of $2021$ .

Points $A, B, C, D, E,$ and $F$ lie on a sphere with center $O$ and radius $R$ such that $\overline{AB}, \overline{CD},$ and $\overline{EF}$ are pairwise perpendicular and all meet at a point $X$ inside the sphere. If $AX=1, CX=\sqrt{2}, EX=2,$ and $OX=\tfrac{\sqrt{2}}{2},$ compute the sum of all possible values of $R^2.$

The distance between cities $A$ and $B$ is $30$ km. Three tourists went from $A$ to $B$. The three of them have two bicycles: a racing bike, on which each of them rides at a speed of $30$ km/h, and a tourist bike, on which they can travel at a speed of $20$ km/h. Each of them can walk at a speed of $6$ km/h. Any bicycle can be left on the road, where it will lie until another tourist can use it. Tourists want to get to $B$ in the shortest time possible, with the end time of the trip corresponding to the moment the last of them arrives at $B$. What is this shortest time?

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Explain whether it possible that the sum of two squares of positive whole numbers has all digits equal to $1$: $$n^2 + m^2 = 111...111$$ [b]p3. [/b]Two players play the following game on an $8 \times 8$ chessboard. The first player can put a rook on an arbitrary square. Then the second player can put another rook on a free square that is not controlled by the first rook. Then the first player can put a new rook on a free square that is not controlled by the rooks on the board. Then the second player can do the same, etc. A player who cannot put a new rook on the board loses the game. Who has a winning strategy? [b]p4.[/b] Show that the difference $9^{2008} - 7^{2008}$ is divisible by $10$. [b]p5.[/b] Is it possible to find distict positive whole numbers $a, b, c, d, e$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1?$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P$ be a point inside or on the boundary of a square $ABCD$. Find the minimum and maximum values of $f(P ) = \angle ABP + \angle BCP + \angle CDP + \angle DAP$.

Evaluate the double series $$\sum_{j=0}^{\infty} \sum_{k=0}^{\infty} 2^{-3k -j -(k+j)^{2}}.$$

Find the largest possible positive integer $n$ , so that there exist$n$ distinct positive real numbers $x_1,x_2,...,x_n$ satisfying the following inequality : for any $1\le i,j \le n,$ $(3x_i-x_j) (x_i-3x_j)\geq (1-x_ix_j)^2$