A subset $S$ of positive real numbers is called [i]powerful[/i] if for any two distinct elements $a, b$ of $S$, at least one of $a^{b}$ or $b^{a}$ is also an element of $S$. [b]a)[/b] Give an example of a four elements powerful set. [b]b)[/b] Prove that every finite powerful set has at most four elements.

Give an example of a second degree polynomial $P\in \mathbb{R}[x]$ such that \[\forall x\in \mathbb{R}\text{ with } |x|\leqslant 1: \; \left|P(x)+\frac{1}{x-4}\right| \leqslant 0.01.\] Are there linear polynomials with this property? [i]Octavian Stănășilă[/i]

Is it possible to place $1965$ points in a square with side $1$ so that any rectangle of area $1/200$ with sides parallel to the sides of the square contains at least one of these points inside?

Isosceles triangle $\triangle{ABC}$ has $\angle{ABC}=\angle{ACB}=72^\circ$ and $BC=1$. If the angle bisector of $\angle{ABC}$ meets $AC$ at $D$, what is the positive difference between the perimeters of $\triangle{ABD}$ and $\triangle{BCD}$? [i]2019 CCA Math Bonanza Tiebreaker Round #2[/i]

Raul's class has $15$ students, all with different heights. The Mathematics teacher wants to place them in a queue so that, at the beginning of the queue, they are ordered in ascending order of heights, from then on, they are ordered in descending order and Raul, who He is the tallest in the class, he cannot be at the extremes. In how many different ways is it possible to form this queue?

$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.

$z$ is a complex number that $\left|2z+\frac{1}{z}\right|=1$, then the range value of $\arg(z)$ is________.

Yang the Spinning Square Sheep is a square in the plane such that his four legs are his four vertices. Yang can do two different types of [i]tricks[/i]: (a) Yang can choose one of his sides, then reflect himself over the side. (b) Yang can choose one of his legs, then rotate $90^\circ$ counterclockwise around the leg. Yang notices that after $2016$ tricks, each leg ends up in exactly the same place the leg started out in! Let there be $N$ ways for Yang to perform his $2016$ tricks. What is the remainder when $N$ is divided by $100000$? [i] Proposed by James Lin [/i]

We will call a rectangular table filled with natural numbers [i]“good”[/i], if for each two rows, there exist a column for which its two cells that are also in these two rows, contain numbers of different parity. Prove that for $\forall$ $n>2$ we can erase a column from a [i]good[/i] $n$ x $n$ table so that the remaining $n$ x $(n-1)$ table is also [i]good[/i].

Determine all real numbers $ a$ such that there is a function $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ x\plus{}f(y)\equal{}af(y\plus{}f(x))\] for all real numbers $ x$ and $ y$. [i]Hery Susanto, Malang[/i]

In a group $G$, we have two elements $x,y$ such that $x^{n}=e,y^2=e,yxy=x^{-1}$, $n\ge 1$. Prove that for any $k\in\mathbb{N}$ holds $(x^ky)^2=e$. Note : e=group's identity .

All angles of the convex $n$-gon $A_1A_2\dots A_n$ are obtuse, where $n\ge5$. For all $1\le i\le n$, $O_i$ is the circumcenter of triangle $A_{i-1}A_iA_{i+1}$ (where $A_0=A_n$ and $A_{n+1}=A_1$). Prove that the closed path $O_1O_2\dots O_n$ doesn't form a convex $n$-gon.

The figure below shows a dotted grid $8$ cells wide and $3$ cells tall consisting of $1''\times1''$ squares. Carl places $1$-inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks? [asy] size(6cm); for (int i=0; i<9; ++i) { draw((i,0)--(i,3),dotted); } for (int i=0; i<4; ++i){ draw((0,i)--(8,i),dotted); } for (int i=0; i<8; ++i) { for (int j=0; j<3; ++j) { if (j==1) { label("1",(i+0.5,1.5)); }}} [/asy] $\textbf{(A) }130\qquad\textbf{(B) }144\qquad\textbf{(C) }146\qquad\textbf{(D) }162\qquad\textbf{(E) }196$

[b]7.1.[/b] Prove that from the sides of an arbitrary quadrilateral you can fold a trapezoid. [b]7.2 / 6.2[/b] The numbers $A$ and $B$ are relatively prime. What common divisors can have the numbers $A+B$ and $A-B$? [b]7.3. / 6.4[/b] $15$ magazines lie on the table, completely covering it. Prove that it is possible to remove eight of them so that the remaining magz cover at least $7/15$ of the table area. [b]7.4[/b] In a six-digit number that is divisible by $7$, the last digit has been moved to the beginning. Prove that the resulting number is also divisible at $7$. [url=https://artofproblemsolving.com/community/c6h3391057p32066818]7.5*[/url] (asterisk problems in separate posts) [b]7.6 [/b] On sides $AB$ and $ BC$ of triangle $ABC$ , are constructed squares $ABDE$ and $BCKL$ with centers $O_1$ and $O_2$. $M_1$ and $M_2$ are midpoints of segments $DL$ and $AC$. Prove that $O_1M_1O_2M_2$ is a square. [img]https://cdn.artofproblemsolving.com/attachments/8/1/8aa816a84c5ac9de78b396096cf718063de390.png[/img] PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here[/url].

How many values of $x\in\left[ 1,3 \right]$ are there, for which $x^2$ has the same decimal part as $x$?

Let $x,y,z $ be positive reals such that $\sqrt{a}=x(y-z)^2$, $\sqrt{b}=y(z-x)^2$ and $\sqrt{c}=z(x-y)^2$. Prove that \[a^2+b^2+c^2 \geq 2(ab+bc+ca)\]

Find all functions $ f: \mathbb {R} \rightarrow \mathbb {R} $ satisfying the equality $ f (x ^ 2-y ^ 2) = (x-y) (f (x) + f (y)) $ for any $ x, y \in \mathbb {R} $.

Let $n$ be a positive integer. Ana and Banana play a game. Banana thinks of a function $f\colon\mathbb{Z}\to\mathbb{Z}$ and a prime number $p$. He tells Ana that $f$ is nonconstant, $p<100$, and $f(x+p)=f(x)$ for all integers $x$. Ana's goal is to determine the value of $p$. She writes down $n$ integers $x_1,\dots,x_n$. After seeing this list, Banana writes down $f(x_1),\dots,f(x_n)$ in order. Ana wins if she can determine the value of $p$ from this information. Find the smallest value of $n$ for which Ana has a winning strategy. [i]Anthony Wang[/i]

Let $n$ be a positive integer, let $f_1(x),\ldots,f_n(x)$ be $n$ bounded real functions, and let $a_1,\ldots,a_n$ be $n$ distinct reals. Show that there exists a real number $x$ such that $\sum^n_{i=1}f_i(x)-\sum^n_{i=1}f_i(x-a_i)<1$.

Let $n$ be a positive integer. Beto writes a list of $n$ non-negative integers on the board. Then he performs a succession of moves (two steps) of the following type: First for each $i=1,2,...,n$, he counts how many numbers on the board are less than or equal to $i$. Let $a_i$ be the number obtained for each $i=1,2,...,n$. Next, he erases all the numbers from the board and writes the numbers $a_1,a_2,...,a_n$. For example, if $n=5$ and the initial numbers on the board are $0,7,2,6,2$, after the first move, the numbers on the board will bec$1,3,3,3,3$;after the second move they will be $1,1,5,5,5$, and so on. $a)$ Show that, for every $n$ and every initial configuration, there will come a time after which the numbers will no longer be modified when using this move. $b)$Find (as a function of $n$) the minimum value of $k$ such that, for any initial configuration, the moves made from move number $k$ will not change the numbers on the board.

Every edge of a tetrahedron is tangent to a given sphere. Prove that the three line segments joining the points of tangency of the three pairs of opposite edges of the tetrahedron are concurrent. [i](7 points)[/i]

What is the smallest integer multiple of 9997, other than 9997 itself, which contains only odd digits?

Let $I$ be a nonempty open subinterval of the set of positive real numbers. For which even $n \in \mathbb{N}$ are there injective function $f: I \to \mathbb{R}$ and positive function $p: I \to \mathbb{R}$, such that for all $x_1 , \ldots , x_n \in I$, \[ f \left( \frac{1}{2} \left( \frac{x_1+\cdots+x_n}{n}+\sqrt[n]{x_1 \cdots x_n} \right) \right)=\frac{p(x_1)f(x_1)+\cdots+p(x_n)f(x_n)}{p(x_1)+\cdots+p(x_n)} \] holds?

Let $x_0, a, b$ be reals given such that $b > 0$ and $x_0 \ne 0$. For every nonnegative integer $n$ a real value $x_{n+1}$ is chosen that satisfies $$x^2_{n+1}= ax_nx_{n+1} + bx^2_n .$$ a) Find how many different values $x_n$ can take. b) Find the sum of all possible values of $x_n$ with repetitions as a function of $n, x_0, a, b$.

A solid has a cylindrical middle with a conical cap at each end. The height of each cap equals the length of the middle. For a given surface area, what shape maximizes the volume?