A circle with center $O$ is tangent to the sides of the angle with the vertex $A$ at the points B and C. Let M be a point on the larger of the two arcs $BC$ of this circle (different from $B$ and $C$) such that $M$ does not lie on the line $AO$. Lines $BM$ and $CM$ intersect the line $AO$ at the points $P$ and $Q$ respectively. Let $K$ be the foot of the perpendicular drawn from $P$ to $AC$ and $L$ be the foot of the perpendicular drawn from $Q$ to $AB$. Prove that the lines $OM$ and $KL$ are perpendicular.

Find all continuous positive functions $f(x)$, for $0\leq x \leq 1$, such that $$\int_{0}^{1} f(x)\; dx =1, $$ $$\int_{0}^{1} xf(x)\; dx =\alpha,$$ $$\int_{0}^{1} x^2 f(x)\; dx =\alpha^2, $$ where $\alpha$ is a given real number.

Let $1+x+x^2+...+x^{n-1}=F(x)G(x)$, where $n>1$ and where $F$ and $G$ are polynomials whose coefficients are zeroes and units. Prove that one of the polynomials $F$ and $G$ can be represented in the form $(1+x+x^2+...x^{k-1})T(x),$ where $k>1$ and $T$ is a polynomial whose coefficients are zeroes and units.

Line $\ell$ is perpendicular to one of the medians of the triangle. The perpendicular bisectors of the sides of this triangle intersect line $\ell$ at three points. Prove that one of them is the midpoint of the segment formed by the remaining two.

[b]p1. [/b]Ed, Fred and George are playing on a see-saw that is slightly off center. When Ed sits on the left side and George, who weighs $100$ pounds, on the right side, they are perfectly balanced. Similarly, if Fred, who weighs $400$ pounds, sits on the left and Ed sits on the right, they are also perfectly balanced. Assuming the see-saw has negligible weight, what is the weight of Ed, in pounds? [b]p2.[/b] How many digits does the product $2^{42}\cdot 5^{38}$ have? [b]p3.[/b] Square $ABCD$ has equilateral triangles drawn external to each side, as pictured. If each triangle is folded upwards to meet at a point $E$, then a square pyramid can be made. If the center of square $ABCD$ is $O$, what is the measure of $\angle OEA$? [img]https://cdn.artofproblemsolving.com/attachments/9/a/39c0096ace5b942a9d3be1eafe7aa7481fbb9f.png[/img] [b]p4.[/b] How many solutions $(x, y)$ in the positive integers are there to $3x + 7y = 1337$ ? [b]p5.[/b] A trapezoid with height $12$ has legs of length $20$ and $15$ and a larger base of length $42$. What are the possible lengths of the other base? [b]p6.[/b] Let $f(x) = 6x + 7$ and $g(x) = 7x + 6$. Find the value of a such that $g^{-1}(f^{-1}(g(f(a)))) = 1$. [b]p7.[/b] Billy and Cindy want to meet at their favorite restaurant, and they have made plans to do so sometime between $1:00$ and $2:00$ this Sunday. Unfortunately, they didn’t decide on an exact time, so they both decide to arrive at a random time between $1:00$ and $2:00$. Silly Billy is impatient, though, and if he has to wait for Cindy, he will leave after $15$ minutes. Cindy, on the other hand, will happily wait for Billy from whenever she arrives until $2:00$. What is the probability that Billy and Cindy will be able to dine together? [b]p8.[/b] As pictured, lines are drawn from the vertices of a unit square to an opposite trisection point. If each triangle has legs with ratio $3 : 1$, what is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/e/9/35a6340018edcddfcd7e085f8f6e56686a8e07.png[/img] [b]p9.[/b] For any positive integer $n$, let $f_1(n)$ denote the sum of the squares of the digits of $n$. For $k \ge 2$, let $f_k(n) = f_{k-1}(f_1(n))$. Then, $f_1(5) = 25$ and $f_3(5) = f_2(25) = 85$. Find $f_{2012}(15)$. [b]p10.[/b] Given that $2012022012$ has $ 8$ distinct prime factors, find its largest prime factor. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let the two tangents from a point $A$ outside a circle $k$ touch $k$ at $M$ and $N$. A line $p$ through $A$ intersects $k$ at $B$ and $C$, and $D$ is the midpoint of $MN$. Prove that $MN$ bisects the angle $BDC$

Given the following system of equations: $$\begin{cases} R I +G +SP = 50 \\ R I +T + M = 63 \\ G +T +SP = 25 \\ SP + M = 13 \\ M +R I = 48 \\ N = 1 \end{cases}$$ Find the value of L that makes $LMT +SPR I NG = 2023$ true.

Compute the number of ways to arrange the numbers $1, 2, 3, 4, 5, 6,$ and $7$ around a circle such that the product of every pair of adjacent numbers on the circle is at most $20.$ (Rotations and reflections count as different arrangements.)

Let $ APD $ be an acute-angled triangle and let $ B,C $ be two points on the segments (excluding their endpoints) $ AP,PD, $ respectively. The diagonals of $ ABCD $ meet at $ Q. $ Denote by $ H_1,H_2 $ the orthocenters of $ APD,BPC, $ respectively. The circumcircles of $ ABQ $ and $ CDQ $ intersect at $ X\neq Q, $ and the circumcircles of $ ADQ,BCQ $ meet at $ Y\neq Q. $ Prove that if the line $ H_1H_2 $ passes through $ X, $ then it also passes through $ Y. $

a) Consider the positive integers $a, b, c$ so that $a < b < c$ and $a^2+b^2 = c^2$. If $a_1 = a^2$, $a_2 = ab$, $a_3 = bc$, $a_4 = c^2$, prove that $a_1^2+a_2^2+a_3^2=a_4^2$ and $a_1 < a_2 < a_3 < a_4$. b) Show that for any $n \in N$, $n\ge 3$, there exist the positive integers $a_1, a_2,..., a_n$ so that $a_1^2+a_2^2+...+ a_{n-1}^2=a_n^2$ and $a_1 < a_2 < ...< a_{n-1} < a_n$

Let $m$ and $n$ be fixed distinct positive integers. A wren is on an infinite board indexed by $\mathbb Z^2$, and from a square $(x,y)$ may move to any of the eight squares $(x\pm m, y\pm n)$ or $(x\pm n, y \pm m)$. For each $\{m,n\}$, determine the smallest number $k$ of moves required to travel from $(0,0)$ to $(1,0)$, or prove that no such $k$ exists. [i]Proposed by Michael Ren

Consider the convex lattice quadrilateral $PQRS$ whose diagonals intersect at $E$. Prove that if $\angle P+\angle Q<180^\circ$, then the $\triangle PQE$ contains inside it or on one of its sides a lattice point other than $P$ and $Q$.

Let $X$ be a finite set of positive integers. Prove that for every subset $A$ of $X$, there is a subset $B$ of $X$, with the following property: For each element $ e$ of $X$, $e$ divides an odd number of elements of $B$, if and only if $e$ is an element of $A$.

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. [i]Carl Schildkraut, USA[/i]

Points $A, S$ are given in plane such that $AS = a > 0$ as well as positive numbers $b, c$ satisfying $b < a < c$. Construct an equilateral triangle $ABC$ with the property $BS = b$, $CS = c$. Discuss conditions of solvability.

Let $a>1,\ 0\leq x\leq \frac{\pi}{4}$. Find the volume of the solid generated by a rotation of the part bounded by two curves $y=\frac{\sqrt{2}\sin x}{\sqrt{\sin 2x+a}},\ y=\frac{1}{\sqrt{\sin 2x+a}}$ about the $x$-axis. [i]1993 Hiroshima Un iversity entrance exam/Science[/i]

A number $\overline{abcd}$ is called [i]balanced[/i] if $a+b=c+d$. Find all balanced numbers with 4 digits that are the sum of two palindrome numbers.

Higher Secondary P10 $X$ is a set of $n$ elements. $P_m(X)$ is the set of all $m$ element subsets (i.e. subsets that contain exactly $m$ elements) of $X$. Suppose $P_m(X)$ has $k$ elements. Prove that the elements of $P_m(X)$ can be ordered in a sequence $A_1, A_2,...A_i,...A_k$ such that it satisfies the two conditions: (A) each element of $P_m(X)$ occurs exactly once in the sequence, (B) for any $i$ such that $0<i<k$, the size of the set $A_i \cap A_{i+1}$ is $m-1$.

Each of eight boxes contains six balls. Each ball has been colored with one of $n$ colors, such that no two balls in the same box are the same color, and no two colors occur together in more than one box. Determine, with justification, the smallest integer $n$ for which this is possible.

Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.

Find the circumradius of the triangle with side lengths $104$, $112$, and $120$.

Points $A, B, C, D$ lie on a line $l$, in that order. Find the locus of points $P$ in the plane for which $\angle{APB}=\angle{CPD}$.

Let $n \ge 5$ be an integer. Show that $n$ is prime if and only if $n_{i} n_{j} \neq n_{p} n_{q}$ for every partition of $n$ into $4$ integers, $n=n_{1}+n_{2}+n_{3}+n_{4}$, and for each permutation $(i, j, p, q)$ of $(1, 2, 3, 4)$.

In convex hexagonal pyramid 11 edges are equal to 1.Find all possible values of 12th edge.

Prove that there is a similarity between a triangle $ABC$ and the triangle having as sides the medians of the triangle $ABC$ if and only if the squares of the lengths of the sides of triangle $ABC$ form an arithmetic sequence. [i]Marian Teler & Marin Ionescu[/i]