A piece of paper is folded in half. A second fold is made at an angle $ \phi$ ($ 0^\circ < \phi < 90^\circ$) to the first, and a cut is made as shown below. [img]12881[/img] When the piece of paper is unfolded, the resulting hole is a polygon. Let $ O$ be one of its vertices. Suppose that all the other vertices of the hole lie on a circle centered at $ O$, and also that $ \angle XOY \equal{} 144^\circ$, where $ X$ and $ Y$ are the the vertices of the hole adjacent to $ O$. Find the value(s) of $ \phi$ (in degrees).

Consider an increasing sequence of real numbers $a_1<a_2<\ldots<a_{2023}$ such that all pairwise sums of the elements in the sequence are different. For such a sequence, denote by $M$ the number of pairs $(a_i,a_j)$ such that $a_i<a_j$ and $a_i+a_j<a_2+a_{2022}$. Find the minimal and the maximal possible value of $M$.

Do there exist a infinite sequence of positive integers $(a_{n})$ ,such that for any $n\ge 1$ the relation $ a_{n+2}=\sqrt{a_{n+1}}+a_{n} $?

An urn contains balls of $k$ different colors; there are $n_i$ balls of $i-th$ color. Balls are selected at random from the urn, one by one, without replacement, until among the selected balls $m$ balls of the same color appear. Find the greatest number of selections.

The medians of a right triangle $ABC$ ($\angle C = 90^{\circ}$) intersect at $M$. Point $L$ lies on the $AC$ such that $\angle ABL=\angle CBL$. It turned out that $\angle BML = 90^{\circ}$. Find the ration $AB : BC$.

a) Show that the number $(2k + 1)^3 - (2k - 1)^3$, $k \in Z$, is the sum of three perfect squares. b) Represent the number $(2n + 1)^3 -2$, $n \in N^*$, as the sum of $3n- 1$ perfect squares greater than $1$.

In a $99\times 101$ table , cubes of natural numbers, as shown in figure . Prove that the sum of all numbers in the table are divisible by $200$. [img]https://cdn.artofproblemsolving.com/attachments/3/e/dd3d38ca00a36037055acaaa0c2812ae635dcb.png[/img]

Let $R_+=(0,+\infty)$. Find all functions $f: R_+ \to R_+$ such that $f(xf(y))+f(yf(z))+f(zf(x))=xy+yz+zx$, for all $x,y,z \in R_+$. by Athanasios Kontogeorgis (aka socrates)

Circle $\omega$ is inscribed in quadrilateral $ABCD$ such that $AB$ and $CD$ are not parallel and intersect at point $O.$ Circle $\omega_1$ touches the side $BC$ at $K$ and touches line $AB$ and $CD$ at points which are located outside quadrilateral $ABCD;$ circle $\omega_2$ touches side $AD$ at $L$ and touches line $AB$ and $CD$ at points which are located outside quadrilateral $ABCD.$ If $O,K,$ and $L$ are collinear$,$ then show that the midpoint of side $BC,AD,$ and the center of circle $\omega$ are also collinear.

Let $ G$ be a finite group. For arbitrary sets $ U, V, W \subset G$, denote by $ N_{UVW}$ the number of triples $ (x, y, z) \in U \times V \times W$ for which $ xyz$ is the unity . Suppose that $ G$ is partitioned into three sets $ A, B$ and $ C$ (i.e. sets $ A, B, C$ are pairwise disjoint and $ G = A \cup B \cup C$). Prove that $ N_{ABC}= N_{CBA}.$

p1. A quadratic equation has the natural roots $a$ and $ b$. Another quadratic equation has roots $ b$ and $c$ with $a\ne c$. If $a$, $ b$, and $c$ are prime numbers less than $15$, how many triplets $(a,b,c)$ that might meet these conditions are there (provided that the coefficient of the quadratic term is equal to $ 1$)? p2. In Indonesia, was formerly known the "Archipelago Fraction''. The [i]Archipelago Fraction[/i] is a fraction $\frac{a}{b}$ such that $a$ and $ b$ are natural numbers with $a < b$. Find the sum of all Archipelago Fractions starting from a fraction with $b = 2$ to $b = 1000$. p3. Look at the following picture. The letters $a, b, c, d$, and $e$ in the box will replaced with numbers from $1, 2, 3, 4, 5, 6, 7, 8$, or $9$, provided that $a,b, c, d$, and $e$ must be different. If it is known that $ae = bd$, how many arrangements are there? [img]https://cdn.artofproblemsolving.com/attachments/f/2/d676a57553c1097a15a0774c3413b0b7abc45f.png[/img] p4. Given a triangle $ABC$ with $A$ as the vertex and $BC$ as the base. Point $P$ lies on the side $CA$. From point $A$ a line parallel to $PB$ is drawn and intersects extension of the base at point $D$. Point $E$ lies on the base so that $CE : ED = 2 :3$. If $F$ is the midpoint between $E$ and $C$, and the area of ​​triangle ABC is equal with $35$ cm$^2$, what is the area of ​​triangle $PEF$? p5. Each side of a cube is written as a natural number. At the vertex of each angle is given a value that is the product of three numbers on three sides that intersect at the vertex. If the sum of all the numbers at the points of the angle is equal to $1001$, find the sum of all the numbers written on the sides of the cube.

Let be a prime number $ p, $ the quotient ring $ R=\mathbb{Z}[X,Y]/(pX,pY), $ and a prime ideal $ I\supset pA $ that is not maximal. Show that the ring $ \left\{ r/i|r\in R, i\in I \right\} $ is factorial.

Problem 4 Let $ABCD$ be a rectangle with diagonals $AC$ and $BD$. Let $E$ be the intersection of the bisector of $\angle CAD$ with segment $CD$, $F$ on $CD$ such that $E$ is midpoint of $DF$, and $G$ on $BC$ such that $BG = AC$ (with $C$ between $B$ and $G$). Prove that the circumference through $D$, $F$ and $G$ is tangent to $BG$.

Let $p$ be a prime number of the form $4k+1$. Show that \[\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right \rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) = \frac{p-1}{2}.\]

For how many pairs $(a,b)$ from $(1,2)$, $(3,5)$, $(5,7)$, $(7,11)$, the polynomial $P(x)=x^5+ax^4+bx^3+bx^2+ax+1$ has exactly one real root? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 0 $

Let $a$ and $b$ be real numbers for which the equation $2x^4 + 2ax^3 + bx^2 + 2ax + 2 = 0$ has at least one real solution. For all such pairs $(a, b)$, find the minimum value of $8a^2 + b^2$.

Find all functions $f : R \to R$ satisfying $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.

a) Find : $A=\{(a,b,c) \in \mathbb{R}^{3} | a+b+c=3 , (6a+b^2+c^2)(6b+c^2+a^2)(6c+a^2+b^2) \neq 0\}$ b) Prove that for any $(a,b,c) \in A$ next inequality hold : \begin{align*} \frac{a}{6a+b^2+c^2}+\frac{b}{6b+c^2+a^2}+\frac{c}{6c+a^2+b^2} \le \frac{3}{8} \end{align*}

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals, such that $\angle BAC = \angle ADB$, $\angle CBD = \angle DCA$, $AB = 15$, $CD = 8$. Show that $ABCD$ is cyclic and find the distance between its circumcenter and the intersection point of its diagonals.

Let $d_i$ be the first decimal digit of $2^i$ for every non-negative integer $i$. Prove that for each positive integer $n$ there exists a decimal digit other than $0$ which can be found in the sequence $d_0, d_1, \dots, d_{n-1}$ strictly less than $\frac{n}{17}$ times.

The lengths of the altitudes of $\triangle{ABC}$ are the roots of the polynomial $x^3-34x^2+360x-1200.$ Find the area of $\triangle{ABC}.$

How many distinct four-tuples $ (a,b,c,d)$ of rational numbers are there with $ a \log_{10} 2 \plus{} b \log_{10} 3 \plus{} c \log_{10} 5 \plus{} d \log_{10} 7 \equal{} 2005$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 2004\qquad \textbf{(E)}\ \text{infinitely many}$

A mushroom is called [i]bad [/i] if it contains at least $10$ worms. A basket contains $90$ bad and $10$ good mushrooms. Can all mushrooms become good after some worms crawl from bad mushrooms to good ones? [hide=original wording]Гриб называется плохим, если в нем не менее 10 червей. В лукошке 90 плохих и 10 хороших грибов. Могут ли все грибы стать хорошими после того, как некоторые черви переползут из плохих грибов в хорошие?[/hide]

Let $ABC$ be an acute triangle, and let $M$ and $N$ be two points on the line $AC$ such that the vectors $MN$ and $AC$ are identical. Let $X$ be the orthogonal projection of $M$ on $BC$, and let $Y$ be the orthogonal projection of $N$ on $AB$. Finally, let $H$ be the orthocenter of triangle $ABC$. Show that the points $B$, $X$, $H$, $Y$ lie on one circle.

$p(x)$ is a polynomial of degree $n$ with leading coefficient $c$, and $q(x)$ is a polynomial of degree $m$ with leading coefficient $c$, such that \[ p(x)^2 = \left(x^2 - 1\right)q(x)^2 + 1 \] Show that $p'(x) = nq(x)$.