Morteza Has $100$ sets. at each step Mahdi can choose two distinct sets of them and Morteza tells him the intersection and union of those two sets. Find the least steps that Mahdi can find all of the sets. Proposed by Morteza Saghafian

Prove that for all integers $k\geq 2,$ the number $k^{k-1}-1$ is divisible by $(k-1)^2.$

In the sequence of powers of $2$ (written in the decimal system, beginning with $2^1 = 2$) there are three terms of one digit, another three of two digits, another three of $3$, four out of $4$, three out of $5$, etc. Clearly reason the answers to the following questions: a) Can there be only two terms with a certain number of digits? b) Can there be five consecutive terms with the same number of digits? c) Can there be four terms of n digits, followed by four with $n + 1$ digits? d) What is the maximum number of consecutive powers of $2$ that can be found without there being four among them with the same number of digits?

Prove for $ 0 < k \leq 1$ and $ a_i \in \mathbb{R}^\plus{},$ $ i \equal{} 1,2 \ldots, n$ the following inequality holds: \[ \left( \frac{a_1}{a_2 \plus{} \ldots \plus{} a_n} \right)^k \plus{} \ldots \plus{} \left( \frac{a_n}{a_1 \plus{} \ldots \plus{} a_{n\minus{}1}} \right)^k \geq \frac{n}{(n\minus{}1)^k}.\]

Find all triplets $ (x,y,z)$, that satisfy: $ \{\begin{array}{c}\ \ x^2 - 2x - 4z = 3\ y^2 - 2y - 2x = - 14 \ z^2 - 4y - 4z = - 18 \end{array}$

\[\int_2^0 x^2+3\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Let $f_n$ be a polynomial with real coefficients for all $n \in \mathbb{Z}$. Suppose that \[f_n(k) = f_{n+k}(k) \quad n, k \in \mathbb{Z}.\] (a) Does $f_n = f_m$ necessarily hold for all $m,n \in \mathbb{Z}$? (b) If furthermore $f_n$ is a polynomial with integer coefficients for all $n \in\mathbb{Z}$, does $f_n = f_m$ necessarily hold for all $m, n \in\mathbb{Z}$? [i]Proposed by usjl[/i]

A convex quadrillateral $ABCD$ is given and the intersection point of the diagonals is denoted by $O$. Given that the perimeters of the triangles $ABO, BCO, CDO,ADO$ are equal, prove that $ABCD$ is a rhombus.

Let $ABC$ with $AB<AC<BC$ be an acute angled triangle and $c$ its circumcircle. Let $D$ be the point diametrically opposite to $A$. Point $K$ is on $BD$ such that $KB=KC$. The circle $(K, KC)$ intersects $AC$ at point $E$. Prove that the circle $(BKE)$ is tangent to $c$.

Let $O{}$ be the circumcenter of the triangle $ABC$. On the rays $AC$ and $BC$ consider the points $C_a$ and $C_b$ respectively, such that $AC_a$ and $BC_b$ are equal in length to $AB$. Let $O_c{}$ be the circumcenter of the triangle $CC_aC_b$. Define the points $O_a{}$ and $O_b{}$ similarly. Prove that $O{}$ is the orthocenter of the triangle $O_aO_bO_c$. [i]Proposed by A. Zaslavsky[/i]

Draw on the plane $(p, q)$ all points with coordinates $(p,q)$, for which the equation $\sin^2x+p\sin x+q=0$ has solutions and all its positive solutions form an arithmetic progression.

Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.

Let $\alpha>1$ and consider the function $f(x)=x^{\alpha}$ for $x \ge 0$. For $t>0$, define $M(t)$ as the largest area that a triangle with vertices $(0, 0), (s, f(s)), (t, f(t))$ could reach, for $s \in (0,t)$. Let $A(t)$ be the area of the region bounded by the segment with endpoints $(0, 0)$ ,$(t, f(t))$ and the graph of $y =f(x)$. (a) Show that $A(t)/M(t)$ does not depend on $t$. We denote this value by $c(\alpha)$. Find $c(\alpha)$. (b) Determine the range of values of $c(\alpha)$ when $\alpha$ varies in the interval $(1, +\infty)$. [hide=Note]Google translated from [url=http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales]http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales[/url][/hide]

A fraction with $1010$ squares in the numerator and $1011$ squares in the denominator serves as a game board for a two player game. $$\frac{\square + \square +...+ \square}{\square + \square +...+ \square+ \square}$$ Players take turns in moves. In each turn, the player chooses one of the numbers $1, 2,. . . , 2021$ and inserts it in any empty field. Each number can only be used once. The starting player wins if the value of the fraction after all the fields is filled differs from number $1$ by less than $10^{-6}$. Otherwise, the other player wins. Decide which of the players has a winning strategy. (Pavel Šalom)

$S$ and $S'$ are two intersecting spheres. The line $BXB'$ is parallel to the line of centers, where $B$ is a point on $S, B'$ is a point on $S'$ and $X$ lies on both spheres. $A$ is another point on $S$, and $A'$ is another point on S' such that the line $AA'$ has a point on both spheres. Show that the segments $AB$ and $A'B'$ have equal projections on the line $AA'$.

Consider the base 27 number \[ n = ABCDEFGHIJKLMNOPQRSTUVWXYZ , \] where each letter has the value of its position in the alphabet. What remainder do you get when you divide $n$ by 100? (The remainder is an integer between 0 and 99, inclusive.)

The positive real $x, y, z$ are such that $x^2+y^2+z^2 = 3$. Prove that$$\frac{x^2+yz}{x^2+yz +1}+\frac{y^2+zx}{y^2+zx+1}+\frac{z^2+xy}{z^2+xy+1}\leq 2$$

The vertices of a right triangle $ABC$ inscribed in a circle divide the circumference into three arcs. The right angle is at $A$, so that the opposite arc $BC$ is a semicircle while arc $BC$ and arc $AC$ are supplementary. To each of three arcs, we draw a tangent such that its point of tangency is the mid point of that portion of the tangent intercepted by the extended lines $AB,AC$. More precisely, the point $D$ on arc $BC$ is the midpoint of the segment joining the points $D'$ and $D''$ where tangent at $D$ intersects the extended lines $AB,AC$. Similarly for $E$ on arc $AC$ and $F$ on arc $AB$. Prove that triangle $DEF$ is equilateral.

Let $ ABC$ be an equilateral triangle inscribed in circle $ O$. $ M$ is a point on arc $ BC$. Lines $ \overline{AM}$, $ \overline{BM}$, and $ \overline{CM}$ are drawn. Then $ AM$ is: [asy]defaultpen(linewidth(.8pt)); unitsize(2cm); pair O = origin; pair B = (1,0); pair C = dir(120); pair A = dir(240); pair M = dir(90 - 18); draw(Circle(O,1)); draw(A--C--M--B--cycle); draw(B--C); draw(A--M); dot(O); label("$A$",A,SW); label("$B$",B,E); label("$M$",M,NE); label("$C$",C,NW); label("$O$",O,SE);[/asy]$ \textbf{(A)}\ \text{equal to }{BM + CM}\qquad \textbf{(B)}\ \text{less than }{BM + CM}\qquad \textbf{(C)}\ \text{greater than }{BM + CM}\qquad$ $ \textbf{(D)}\ \text{equal, less than, or greater than }{BM + CM}\text{, depending upon the position of }{ {M}\qquad}$ $ \textbf{(E)}\ \text{none of these}$

$x,y,z$ positive real numbers such that $xyz=1$ Prove that: $\frac{1}{x+y^{20}+z^{11}}+\frac{1}{y+z^{20}+x^{11}}+\frac{1}{z+x^{20}+y^{11}}\leq1$

What is the value of $$3 + \frac{1}{3+\frac{1}{3+\frac{1}{3}}}?$$ $\textbf{(A) } \frac{31}{10} \qquad \textbf{(B) } \frac{49}{15} \qquad \textbf{(C) } \frac{33}{10} \qquad \textbf{(D) } \frac{109}{33} \qquad \textbf{(E) } \frac{15}{4}$

Each cell of a $3 $ × $3$ grid is labeled with a digit in the set {$1, 2, 3, 4, 5$} Then, the maximum entry in each row and each column is recorded. Compute the number of labelings for which every digit from $1$ to $5$ is recorded at least once.

Let $P$ be a point in the interior of an equilateral triangle with height $1$, and let $x,y,z$ denote the distances from $P$ to the three sides of the triangle. Prove that \[ x^2+y^2+z^2 ~\ge~ x^3+y^3+z^3 +6xyz \]

Determine all pairs $(x, y)$ of integers such that \[1+2^{x}+2^{2x+1}= y^{2}.\]

For all integers $n \geq 9,$ the value of $$\frac{(n+2)!-(n+1)!}{n!}$$ is always which of the following? $\textbf{(A) } \text{a multiple of }4 \qquad \textbf{(B) } \text{a multiple of }10 \qquad \textbf{(C) } \text{a prime number} \\ \textbf{(D) } \text{a perfect square} \qquad \textbf{(E) } \text{a perfect cube}$