On the board there are three real numbers $(a,b,c)$. During a $procedure$ the numbers are erased and in their place another three numbers a written, either $(c,b,a)$ or every time a nonzero real number $ d $ is chosen and the numbers $(a, 2ad+b, ad^2+bd+c)$ are written. 1) If we start with $(1,-2,-1)$ written on the board, can we have the numbers $(2,0,-1)$ on the board after a finite number of procedures? 2) If we start with $(1,-2,-1)$ written on the board, can we have the numbers $(2,-1,-1)$ on the board after a finite number of procedures?

Let $ O $ be the circumcenter of a triangle $ ABC $ with $ \angle BAC=60^{\circ } $ whose incenter is denoted by $ I. $ Let $ B_1,C_1 $ be the intersection of $ BI,CI $ with the circumcircle of $ ABC, $ respectively. Denote by $ O_1,O_2 $ the circumcenters of $ BIC_1,CIB_1, $ respectively. Show that $ O_1,I,O,O_2 $ are collinear. [i]Cătălin Țigăeru[/i]

Note that 1990 can be "turned into a square" by adding a digit on its right, and some digits on its left; i.e., $419904 = 648^2$. Prove that 1991 cannot be turned into a square by the same procedure; i.e., there are no digits $d,x,y,..$ such that $...yx1991d$ is a perfect square.

Let us use the word $ N$-measure for nonnegative, finitely additive set functions defined on all subsets of the positive integers, equal to $ 0$ on finite sets, and equal to $ 1$ on the whole set. We say that the system $ \Upsilon$ of sets determines the $ N$-measure $ \mu$ if any $ N$-measure coinciding with $ \mu$ on all elements of $ \Upsilon$ is necessarily identical with $ \mu$. Prove the existence of an $ N$-measure $ \mu$ that cannot be determined by a system of cardinality less than continuum. [i]I. Juhasz[/i]

Given a graph with $n$ vertices and a positive integer $m$ that is less than $ n$, prove that the graph contains a set of $m+1$ vertices in which the difference between the largest degree of any vertex in the set and the smallest degree of any vertex in the set is at most $m-1.$

Let $ABC$ be a triangle with $\angle ACB > 90^{\circ}$, and let $D$ be a point on $BC$ such that $AD$ is perpendicular to $BC$. Consider the circumference $\Gamma$ with with diameter $BC$. A line $\ell$ passes through $D$ and is tangent to $\Gamma$ at $P$, cuts $AC$ at $M$ (such that $M$ is in between $A$ and $C$), and cuts the side $AB$ at $N$. Prove that $M$ is the midpoint of $DP$ if and only if $N$ is the midpoint of $AB$.

Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20. [i]Proposed by Luxembourg[/i]

An infinitely large grid is filled such that each grid square contains exactly one of the digits $\{ 1,2,3,4\},$ each digit appears at least once, and the digit in each grid square equals the digit located $5$ squares above it as well as the digit located $5$ squares to the right. A group of $4$ horizontally adjacent digits or $4$ vertically adjacent digits is chosen randomly, and depending on its orientation is read left to right or top to bottom to form an $4$-digit integer. The expected value of this integer is also a $4$-digit integer $N$. Given this, find the last three digits of the sum of all possible values of $N$.

Let $ABC$ be a non-right isosceles triangle such that $AC = BC$. Let $D$ be such a point on the perpendicular bisector of $AB$, that $AD$ is tangent on the $ABC$ circumcircle. Let $E$ be such a point on $AB$, that $CE$ and $AD$ are perpendicular and let $F$ be the second intersection of line $AC$ and the circle $CDE$. Prove that $DF$ and $AB$ are parallel.

In the $xy$-coordinate plane, the horizontal line $y = k$ intersects the graph of the cubic $2x^3 + 6x^2 - 4x + 5$ in three points $P$, $Q$, and $R$. Given that $Q$ is the midpoint of $P$ and $R$, what is $k$?

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

Let $A,B\in\mathcal{M}_{2}(\mathbb{R})$ (real $2\times 2$ matrices), that satisfy $A^{2}+B^{2}=AB$. Prove that $(AB-BA)^{2}=O_{2}$.

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. $

A test consists of $30$ true or false questions. After the test (answering all $30$ questions), Victor gets his score: the number of correct answers. Victor is allowed to take the test (the same questions ) several times. Can Victor work out a strategy that insure him to get a perfect score after [b](a) [/b] $30$th attempt? [b](b)[/b] $25$th attempt? (Initially, Victor does not know any answer)

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ for which \[f(x + y) = f(x - y) + f(f(1 - xy))\] holds for all real numbers $x$ and $y$.

Denote by $\Omega$ the circumcircle of the acute triangle $ABC$. Point $D$ is the midpoint of the arc $BC$ of $\Omega$ not containing $A$. Circle $\omega$ centered at $D$ is tangent to the segment $BC$ at point $E$. Tangents to the circle $\omega$ passing through point $A$ intersect line $BC$ at points $K$ and $L$ such that points $B, K, L, C$ lie on the line $BC$ in that order. Circle $\gamma_1$ is tangent to the segments $AL$ and $BL$ and to the circle $\Omega$ at point $M$. Circle $\gamma_2$ is tangent to the segments $AK$ and $CK$ and to the circle $\Omega$ at point $N$. Lines $KN$ and $LM$ intersect at point $P$. Prove that $\sphericalangle KAP = \sphericalangle EAL$.

Let $m$ and $n$ be positive integers. Find all pairs of non-negative integers $a$ and $b$ that always satisfy the following condition: Given any configuration of $m$ white dots and $n$ black dots on a circle, there always exist a line cutting the circle into two arcs, one of which consists of exactly $a$ white dots and $b$ black dots. [i]Proposed by Tan Min Heng[/i]

Let $ABC$ an acutangle triangle with orthocenter $H$ and circumcenter $O$. Let $D$ the intersection of $AO$ and $BH$. Let $P$ be the point on $AB$ such that $PH=PD$. Prove that the points $B, D, O$ and $P$ lie on a circle.

Consider a $m\times n$ rectangular board consisting of $mn$ unit squares. Two of its unit squares are called [i]adjacent[/i] if they have a common edge, and a [i]path[/i] is a sequence of unit squares in which any two consecutive squares are adjacent. Two parths are called [i]non-intersecting[/i] if they don't share any common squares. Each unit square of the rectangular board can be colored black or white. We speak of a [i]coloring[/i] of the board if all its $mn$ unit squares are colored. Let $N$ be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let $M$ be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge. Prove that $N^{2}\geq M\cdot 2^{mn}$.

What is the remainder when $2^{202} +202$ is divided by $2^{101}+2^{51}+1$? $\textbf{(A) } 100 \qquad\textbf{(B) } 101 \qquad\textbf{(C) } 200 \qquad\textbf{(D) } 201 \qquad\textbf{(E) } 202$

In the isoceles triangle $ABC$ with $AB=AC$, $D$ and $E$ are points on $AB$ and $AC$ respectively such that $AD=CE$ and $DE=BC$. Suppose $\angle AED=18^{\circ}$. Find the size of $\angle BDE$ in degrees.

Given that $n$ elements $a_1, a_2,\dots, a_n$ are organized into $n$ pairs $P_1, P_2, \dots, P_n$ in such a way that two pairs $P_i, P_j$ share exactly one element when $(a_i, a_j)$ is one of the pairs, prove that every element is in exactly two of the pairs.

The REAL country has $n$ islands, and there are $n-1$ two-way bridges connecting these islands. Any two islands can be reached through a series of bridges. Arctan, the king of the REAL country, found that it is too difficult to manage $n$ islands, so he wants to bomb some islands and their connecting bridges to divide the country into multiple small areas. Arctan wants the number of connected islands in each group is less than $\delta n$ after bombing these islands, and the island he bomb must be a connected area. Besides, Arctan wants the number of islands to be bombed to be as less as possible. Find all real numbers $\delta$ so that for any positive integer $n$ and the layout of the bridge, the method of bombing the islands is the only one. [i]Proposed by chengbilly[/i]

[b]p1.[/b] What is $20\times 19 + 20 \div (2 - 7)$? [b]p2.[/b] Will has three spinners. The first has three equally sized sections numbered $1$, $2$, $3$; the second has four equally sized sections numbered $1$, $2$, $3$, $4$; and the third has five equally sized sections numbered $1$, $2$, $3$, $4$, $5$. When Will spins all three spinners, the probability that the same number appears on all three spinners is $p$. Compute $\frac{1}{p}$. [b]p3.[/b] Three girls and five boys are seated randomly in a row of eight desks. Let $p$ be the probability that the students at the ends of the row are both boys. If $p$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m + n$. [b]p4.[/b] Jaron either hits a home run or strikes out every time he bats. Last week, his batting average was $.300$. (Jaron's batting average is the number of home runs he has hit divided by the number of times he has batted.) After hitting $10$ home runs and striking out zero times in the last week, Jaron has now raised his batting average to $.310$. How many home runs has Jaron now hit? [b]p5.[/b] Suppose that the sum $$\frac{1}{1 \cdot 4} +\frac{1}{4 \cdot 7}+ ...+\frac{1}{97 \cdot 100}$$ is expressible as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p6.[/b] Let $ABCD$ be a unit square with center $O$, and $\vartriangle OEF$ be an equilateral triangle with center $A$. Suppose that $M$ is the area of the region inside the square but outside the triangle and $N$ is the area of the region inside the triangle but outside the square, and let $x = |M -N|$ be the positive difference between $M$ and $N$. If $$x =\frac1 8(p -\sqrt{q})$$ for positive integers $p$ and $q$, find $p + q$. [b]p7.[/b] Find the number of seven-digit numbers such that the sum of any two consecutive digits is divisible by $3$. For example, the number $1212121$ satisfies this property. [b]p8.[/b] There is a unique positive integer $x$ such that $x^x$ has $703$ positive factors. What is $x$? [b]p9.[/b] Let $x$ be the number of digits in $2^{2019}$ and let $y$ be the number of digits in $5^{2019}$. Compute $x + y$. [b]p10.[/b] Let $ABC$ be an isosceles triangle with $AB = AC = 13$ and $BC = 10$. Consider the set of all points $D$ in three-dimensional space such that $BCD$ is an equilateral triangle. This set of points forms a circle $\omega$. Let $E$ and $F$ be points on $\omega$ such that $AE$ and $AF$ are tangent to $\omega$. If $EF^2$ can be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, determine $m + n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Assume that $a_1, a_2, a_3$ are three given positive integers consider the following sequence: $a_{n+1}=\text{lcm}[a_n, a_{n-1}]-\text{lcm}[a_{n-1}, a_{n-2}]$ for $n\ge 3$ Prove that there exist a positive integer $k$ such that $k\le a_3+4$ and $a_k\le 0$. ($[a, b]$ means the least positive integer such that$ a\mid[a,b], b\mid[a, b]$ also because $\text{lcm}[a, b]$ takes only nonzero integers this sequence is defined until we find a zero number in the sequence)