Let $P_1, P_2, \ldots, P_6$ be points in the complex plane, which are also roots of the equation $x^6+6x^3-216=0$. Given that $P_1P_2P_3P_4P_5P_6$ is a convex hexagon, determine the area of this hexagon.

Determine all pairs of positive real numbers $(a, b)$ with $a > b$ that satisfy the following equations: $a\sqrt{a}+ b\sqrt{b} = 134$ and $a\sqrt{b}+ b\sqrt{a} = 126$.

A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$. In what ratio does the fold line divide side $BC$?

If $\alpha $ is the real root of the equation \[E(x) = x^3 - 5x -50 = 0\] such that $x_{n+1} = (5x_n + 50)^{1/3}$ and $x_1 = 5$, where $n$ is a positive integer, prove that: [b](a)[/b] $x_{n+1}^3 - \alpha^3 = 5(x_n - \alpha)$ [b](b)[/b] $\alpha < x_{n+1} < x_n.$

Let be a differentiable function $ f:\mathbb{R}_{> 0}\longrightarrow\mathbb{R}_{> 0} , $ and a primitive $ F:\mathbb{R}_{> 0}\longrightarrow\mathbb{R}_{> 0} $ of it such that $ F=f+f\cdot f. $ Show that: [b]a)[/b] $ f $ is nondecreasing. [b]b)[/b] $ \lim_{x\to\infty } f(x)/x =1/2 $ [i]Vasile Solovăstru[/i]

For $x\ge 0$, define \[f(x)=\frac1{x+2\cos x}\] Find the set $\{ y \in \mathbb{R}: y=f(x), x\ge 0\}$

You have four fair $6$-sided dice, each numbered $1$ to $6$ (inclusive). If all four dice are rolled, the probability that the product of the rolled numbers is prime can be written as $\tfrac{a}{b}$, where $a$ and $b$ are relatively prime. What is $a+b$?

Let $\alpha$ be a non-zero real number. Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that \[ xf(x+y)=(x+\alpha y)f(x)+xf(y) \] for all $x,y\in\mathbb{R}$.

Find all strictly monotonous functions $f : N \to N$ that satisfy $f(f(n)) = 3n$ for all $n \in N$.

For pairwise distinct nonnegative reals $a,b,c$, prove that $$\frac{a^2}{(b-c)^2}+\frac{b^2}{(c-a)^2}+\frac{c^2}{(b-a)^2}>2$$.

Find the sum of all positive integers whose largest proper divisor is $55$. (A proper divisor of $n$ is a divisor that is strictly less than $n$.)

Let $n$ be a prime number. Show that there is a permutation $a_1,a_2,...,a_n$ of $1,2,...,n$ so that $a_1,a_1a_2,...,a_1a_2...a_n$ leave distinct remainders when divided by $n$

A tank has the shape of a regular hexagonal prism, whose bases are $1$ m on a side and its height is $10$ m. The lateral edges are placed in an oblique position and is partially filled with $9$ m$^3$ of water. The plane of the free surface of the water cuts to all lateral edges. One of them is left with a part of $2$ m under water. What part is under water on the opposite side edge of the prism?

A real number $a$ is given. The sequence $n_{1}< n_{2}< n_{3}< ...$ consists of all the positive integral $n$ such that $\{na\}< \frac{1}{10}$. Prove that there are at most three different numbers among the numbers $n_{2}-n_{1}$, $n_{3}-n_{2}$, $n_{4}-n_{3}$, $\ldots$. [i]A corollary of a theorem from ergodic theory[/i]

Given a triangle $ABC$ inscribed in circle $c(O,R)$ (with center $O$ and radius $R$) with $AB<AC<BC$ and let $BD$ be a diameter of the circle $c$. The perpendicular bisector of $BD$ intersects line $AC$ at point $M$ and line $AB$ at point $N$. Line $ND$ intersects the circle $c$ at point $T$. Let $S$ be the second intersection point of cicumcircles $c_1$ of triangle $OCM$, and $c_2$ of triangle $OAD$. Prove that lines $AD, CT$ and $OS$ pass through the same point.

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

A positive real number sequence $a_1, a_2, a_3,\dots $ and a positive integer \(s\) is given. Let $f_n(0) = \frac{a_n+\dots+a_1}{n}$ and for each $0<k<n$ \[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\] Then for every integer $n\geq s,$ the condition \[a_{n+1}=\max_{0\leq k<n}(f_n(k))\] is satisfied. Prove that this sequence must be eventually constant.

Solve the equation,$$ \sin (\pi \log x) + \cos (\pi \log x) = 1$$

Given integer $d>1,m$,prove that there exists integer $k>l>0$, such that $$(2^{2^k}+d,2^{2^l}+d)>m.$$

Prove that there exists an infinite sequence of positive integers $a_1,a_2,a_3,\dots$ such that for any positive integer $k$, $a_k^2+a_k+2023$ has at least $k$ distinct positive divisors.

A subset of $X$ of $\{1,2,3, \ldots 10000 \}$ has the following property: If $a,b$ are distinct elements of $X$, then $ab\not\in X$. What is the maximal number of elements in $X$?

[Help me] Suppose that the equation $x^3+px^2+qx+r = 0$ has 3 real roots $x_1; x_2; x_3$; where p; q; r are integer numbers. Put $S_n = x_1^n+x_2^n+x_3^n$ ; n = 1; 2; : : : Prove that $S_{2012}$ is an integer.

Integers $ a_1,a_2,\ldots a_9$ satisfy the relations $ a_{k\plus{}1}\equal{}a_k^3\plus{}a_k^2\plus{}a_k\plus{}2$ for $ k\equal{}1,2,...,8$. Prove that among these numbers there exist three with a common divisor greater than $ 1$.

Define the sequences $x_0,x_1,x_2,\ldots$ and $y_0,y_1,y_2,\ldots$ such that $x_0=1$, $y_0=2021$, and for all nonnegative integers $n$, we have $x_{n+1}=\sqrt{x_ny_n}$ and $y_{n+1}=\frac{x_n+y_n}{2}.$ There is some constant $X$ such that as $n$ grows large, $x_n-X$ and $y_n-X$ both approach $0$. Estimate $X$. An estimate of $E$ earns $\max(0,2-0.02|A-E|)$ points, where $A$ is the actual answer. [i]2021 CCA Math Bonanza Lightning Round #5.2[/i]

Given isosceles triangle $ABC$ ($AB = AC$). A straight line $\ell$ is drawn through its vertex $B$ at a right angle with $AB$ . On the straight line $AC$, an arbitrary point $D$ is taken, different from the vertices, and a straight line is drawn through it at a right angle with $AC$, intersecting $\ell$ at the point $F$. Prove that the center of the circle circumscribed around the triangle $BCD$ lies on the circumscribed circle of triangle $ABD$.