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 $1\le i,j \le n,$ $(3x_i-x_j) (x_i-3x_j)\geq (1-x_ix_j)^2$

Let us consider a matrix $T$ with n rows denoted $1,\ldots,n$ and $p$ columns $1,\ldots,p$. Its entries $a_{ik}~(1\le i\le n,1\le k\le p)$ are integers such that $1\le a_{ik}\le N$, where $N$ is a given natural number. Let $E_i$ be the set of numbers that appear on the $i$-th row. Answer question (a) or (b). (a) Assume $T$ satisfies the following conditions: $(1)$ $E_i$ has exactly $p$ elements for each $i$, and $(2)$ all $E_i$'s are mutually distinct. Let $m$ be the smallest value of $N$ that permits a construction of such an $n\times p$ table $T$. i. Compute $m$ if $n=p+1$. ii. Compute $m$ if $n=10^{30}$ and $p=1998$. iii. Determine $\lim_{n\to\infty}\frac{m^p}n$, where $p$ is fixed. (b) Assume $T$ satisfies the following conditions instead: $(1)$ $p=n$, $(2)$ whenever $i,k$ are integers with $i+k\le n$, the number $a_{ik}$ is not in the set $E_{i+k}$. i. Prove that all $E_i$'s are mutually distinct. ii. Prove that if $n\ge2^q$ for some integer $q>0$, then $N\ge q+1$. iii. Let $n=2^r-1$ for some integer $r>0$. Prove that $N\ge r$ and show that there is such a table with $N=r$.

A right cylinder is inscribed in a right circular cone with height $2$ and radius $2$ so that the cylinder’s bottom base sits on the cone’s base. What is the maximum possible surface area of the cylinder?

For a positive integer $k$, let $s(k)$ denote the number of $1$s in the binary representation of $k$. Prove that for any positive integer $n$, \[\sum_{i=1}^{n}(-1)^{s(3i)} > 0.\] [i]Holden Mui[/i]

Let $ABCDE$ be a cyclic pentagon such that $BC = DE$ and $AB$ is parallel to $DE$. Let $X, Y,$ and $Z$ be the midpoints of $BD, CE,$ and $AE$ respectively. Show that $AE$ is tangent to the circumcircle of the triangle $XYZ$. Proposed by [i]Nikola Velov, Macedonia[/i]

Let $S$ be a set of $2n+1$ points in the plane such that no three are collinear and no four concyclic. A circle will be called $\text{Good}$ if it has 3 points of $S$ on its circumference, $n-1$ points in its interior and $n-1$ points in its exterior. Prove that the number of good circles has the same parity as $n$.

If $a$ and $b$ are complex numbers such that $a^2 + b^2 = 5$ and $a^3 + b^3 = 7$, then their sum, $a + b$, is real. The greatest possible value for the sum $a + b$ is $\tfrac{m+\sqrt{n}}{2}$ where $m$ and $n$ are integers. Find $n.$

The altitude drawn to the base of an isosceles triangle is $ 8$, and the perimeter $ 32$. The area of the triangle is: $ \textbf{(A)}\ 56\qquad \textbf{(B)}\ 48\qquad \textbf{(C)}\ 40\qquad \textbf{(D)}\ 32\qquad \textbf{(E)}\ 24$

How many real pairs $(x,y)$ are there such that \[ x^2+2y = 2xy \\ x^3+x^2y = y^2 \] $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{None} $

For the positive integers $a, b, x_1$ we construct the sequence of numbers $(x_n)_{n=1}^{\infty}$ such that $x_n = ax_{n-1} + b$ for each $n \ge 2$. Specify the conditions for the given numbers $a, b$ and $x_1$ which are necessary and sufficient for all indexes $m, n$ to apply the implication $m | n \Rightarrow x_m | x_n$. (Jaromír Šimša)