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)

There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold: [i]i.)[/i] Each pair of students are in exactly one club. [i]ii.)[/i] For each student and each society, the student is in exactly one club of the society. [i]iii.)[/i] Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is in exactly $m$ societies. Find all possible values of $k$. [i]Proposed by Guihua Gong, Puerto Rico[/i]

The vertices of a $33$-gon are labelled with the integers from $1$ to $33$. Each edge is then labelled with the sum of the labels of its two vertices. Is it possible for the edge labels to consist of $33$ consecutive numbers?

Find all positive integers $n \ge 2$ such that equality $i+j \equiv C_{n}^{i} + C_{n}^{j}$ (mod $2$) is true for arbitrary $0 \le i \le j \le n$.

[b]8.[/b] Let the function $f(x)$ be periodic with the period $1$, non-negative, concave in the interval $(0,1)$ and continuous at the point $0$. Prove that $f(nx)\leq nf(x)$ for every real $x$ and positive integer $n$. [b](R. 6)[/b]

Let $f: \mathbb{R}\to\mathbb{R}$ be a continuous function, and $a<b$ be two points in the image of $f$ (that is, there exists $x,y$ such that $f(x)=a$ and $f(y)=b$). Show that there is an interval $I$ such that $f(I)=[a,b]$.

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

In rectangle $ ABCD$, we have $ AB\equal{}8$, $ BC\equal{}9$, $ H$ is on $ \overline{BC}$ with $ BH\equal{}6$, $ E$ is on $ \overline{AD}$ with $ DE\equal{}4$, line $ EC$ intersects line $ AH$ at $ G$, and $ F$ is on line $ AD$ with $ \overline{GF}\perp\overline{AF}$. Find the length $ GF$. [asy]unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair D=(0,0), Ep=(4,0), A=(9,0), B=(9,8), H=(3,8), C=(0,8), G=(-6,20), F=(-6,0); draw(D--A--B--C--D--F--G--Ep); draw(A--G); label("$F$",F,W); label("$G$",G,W); label("$C$",C,WSW); label("$H$",H,NNE); label("$6$",(6,8),N); label("$B$",B,NE); label("$A$",A,SW); label("$E$",Ep,S); label("$4$",(2,0),S); label("$D$",D,S);[/asy]$ \textbf{(A)}\ 16 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 30$

Do there exist points $A, B, C, D$ in space, such that $AB = CD = 8, AC = BD = 10$, and $AD = BC = 13$?

1400 real numbers are given. Prove that one can choose three of them like $x,y,z$ such that : $$\left|\frac{(x-y)(y-z)(z-x)}{x^4+y^4+z^4+1}\right| < 0.009$$