Find all non-negative integers $m$ and $n$, such that $(2^n-1) \cdot (3^n-1)=m^2$.

Let $F$ be an increasing real function defined for all $x, 0 \le x \le 1$, satisfying the conditions (i) $F (\frac{x}{3}) = \frac{F(x)}{2}$. (ii) $F(1- x) = 1 - F(x)$. Determine $F(\frac{173}{1993})$ and $F(\frac{1}{13})$ .

Let $a, b, c, d$ be distinct positive real numbers. Prove that if one of the numbers $c, d$ lies between $a$ and $b$, or one of $a, b$ lies between $c$ and $d$, then $$\sqrt{(a+b)(c+d)} >\sqrt{ab} +\sqrt{cd}$$ and that otherwise, one can choose $a, b, c, d$ so that this inequality is false.

Let the edges of rectangular parallelepiped be $x,y$ and $z$ ($x<y<z$). Let $$p=4(x+y+z), s=2(xy+yz+zx) \,\,\, and \,\,\, d=\sqrt{x^2+y^2+z^2}$$ be its perimeter, surface area and diagonal length, respectively. Prove that $$x < \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )\,\,\, and \,\,\, z > \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )$$

Find $n>0$ such that $\sqrt[3]{\sqrt[3]{5\sqrt2+n}+\sqrt[3]{5\sqrt2-n}}=\sqrt2$.

A sequence of numbers $a_1, a_2 , \dots a_m$ is a [i]geometric sequence modulo n of length m[/i] for $n,m \in \mathbb{Z}^+$ if for every index $i$, $a_i \in \{ 0, 1, 2, \dots , m-1\}$ and there exists an integer $k$ such that $n | a_{j+1} - ka_{j}$ for $1 \leq j \leq m-1$. How many geometric sequences modulo $14$ of length $14$ are there?

The roots of the equation $x^3 - x + 1 = 0$ are $a, b, c$. Find $a^8 + b^8 + c^8$.

Find the maximum possible value of $$\left( \frac{a^3}{b^2c}+\frac{b^3}{c^2a}+\frac{c^3}{a^2b} \right)^2$$ where $a, b$, and $c$ are nonzero real numbers satisfying $$a \sqrt[3]{\frac{a}{b}}+b\sqrt[3]{\frac{b}{c}}+c\sqrt[3]{\frac{c}{a}}=0$$

Solve the system of equations $$\begin{cases} (x+y)(x^2-y^2)=32 \\ (x-y)(x^2+y^2)=20 \end{cases}$$

Prove that for any $n$ ($n \geq 2$) pairwise distinct fractions in the interval $(0,1)$, the sum of their denominators is no less than $\frac{1}{3} n^{\frac{3}{2}}$.

Six mathematician sit around a round table. Each of them has a number and they do the following transformation: Each time, two mathematician sitting next to each other is chosen, they will add $1$ to their own number. Is it possible to make all the six numbers equal if the initial numbers are a) 6,5,4,3,2,1 b) 7,5,3,2,1,4

Pentagon $ABCDE$ is given in the plane. Let the perpendicular from $A$ to line $CD$ be $F$, the perpendicular from $B$ to $DE$ be $G$, from $C$ to $EA$ be $H$, from $D$ to $AB$ be $I$,and from $E$ to $BC$ be $J$. Given that lines $AF,BG,CH$, and $DI$ concur, show that they also concur with line $EJ$.

Let $AA_0$ be the altitude of the isosceles triangle $ABC~(AB = AC)$. A circle $\gamma$ centered at the midpoint of $AA_0$ touches $AB$ and $AC$. Let $X$ be an arbitrary point of line $BC$. Prove that the tangents from $X$ to $\gamma$ cut congruent segments on lines $AB$ and $AC$

Let $n=p_1^{a_1}p_2^{a_2}\cdots p_t^{a_t}$ be the prime factorisation of $n$. Define $\omega(n)=t$ and $\Omega(n)=a_1+a_2+\ldots+a_t$. Prove or disprove: For any fixed positive integer $k$ and positive reals $\alpha,\beta$, there exists a positive integer $n>1$ such that i) $\frac{\omega(n+k)}{\omega(n)}>\alpha$ ii) $\frac{\Omega(n+k)}{\Omega(n)}<\beta$.

Given the circumference $s$ and the straight line $l$, passing through the centre $O$ of $s$. Another circumference $s'$ passes through the point $O$ and has its centre on the $l$. Describe the set of the points $M$, where the common tangent of $s$ and $s'$ touches $s'$.

We say that a quadruple $(A,B,C,D)$ is [i]dobarulho[/i] when $A,B,C$ are non-zero algorisms and $D$ is a positive integer such that: $1.$ $A \leq 8$ $2.$ $D>1$ $3.$ $D$ divides the six numbers $\overline{ABC}$, $\overline{BCA}$, $\overline{CAB}$, $\overline{(A+1)CB}$, $\overline{CB(A+1)}$, $\overline{B(A+1)C}$. Find all such quadruples.

Let $a$ and $b$ be real numbers such that $a+b=\log_2( \log_2 3)$. What is the minimum value of $2^a + 3^b$ ?

How many of the twelve pentominoes pictured below have at least one line of symmetry? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$ [asy]unitsize(5mm); defaultpen(linewidth(1pt)); draw(shift(2,0)*unitsquare); draw(shift(2,1)*unitsquare); draw(shift(2,2)*unitsquare); draw(shift(1,2)*unitsquare); draw(shift(0,2)*unitsquare); draw(shift(2,4)*unitsquare); draw(shift(2,5)*unitsquare); draw(shift(2,6)*unitsquare); draw(shift(1,5)*unitsquare); draw(shift(0,5)*unitsquare); draw(shift(4,8)*unitsquare); draw(shift(3,8)*unitsquare); draw(shift(2,8)*unitsquare); draw(shift(1,8)*unitsquare); draw(shift(0,8)*unitsquare); draw(shift(6,8)*unitsquare); draw(shift(7,8)*unitsquare); draw(shift(8,8)*unitsquare); draw(shift(9,8)*unitsquare); draw(shift(9,9)*unitsquare); draw(shift(6,5)*unitsquare); draw(shift(7,5)*unitsquare); draw(shift(8,5)*unitsquare); draw(shift(7,6)*unitsquare); draw(shift(7,4)*unitsquare); draw(shift(6,1)*unitsquare); draw(shift(7,1)*unitsquare); draw(shift(8,1)*unitsquare); draw(shift(6,0)*unitsquare); draw(shift(7,2)*unitsquare); draw(shift(11,8)*unitsquare); draw(shift(12,8)*unitsquare); draw(shift(13,8)*unitsquare); draw(shift(14,8)*unitsquare); draw(shift(13,9)*unitsquare); draw(shift(11,5)*unitsquare); draw(shift(12,5)*unitsquare); draw(shift(13,5)*unitsquare); draw(shift(11,6)*unitsquare); draw(shift(13,4)*unitsquare); draw(shift(11,1)*unitsquare); draw(shift(12,1)*unitsquare); draw(shift(13,1)*unitsquare); draw(shift(13,2)*unitsquare); draw(shift(14,2)*unitsquare); draw(shift(16,8)*unitsquare); draw(shift(17,8)*unitsquare); draw(shift(18,8)*unitsquare); draw(shift(17,9)*unitsquare); draw(shift(18,9)*unitsquare); draw(shift(16,5)*unitsquare); draw(shift(17,6)*unitsquare); draw(shift(18,5)*unitsquare); draw(shift(16,6)*unitsquare); draw(shift(18,6)*unitsquare); draw(shift(16,0)*unitsquare); draw(shift(17,0)*unitsquare); draw(shift(17,1)*unitsquare); draw(shift(18,1)*unitsquare); draw(shift(18,2)*unitsquare);[/asy]

What is the largest $n$ such that there exists a non-degenerate convex $n$-gon such that each of its angles are an integer number of degrees, and are all distinct?

Let $ n$ be a positive integer, let $ A$ be a subset of $ \{1, 2, \cdots, n\}$, satisfying for any two numbers $ x, y\in A$, the least common multiple of $ x$, $ y$ not more than $ n$. Show that $ |A|\leq 1.9\sqrt {n} \plus{} 5$.

Let $P(x,y)$ be a polynomial such that $\deg_x(P), \deg_y(P)\le 2020$ and \[P(i,j)=\binom{i+j}{i}\] over all $2021^2$ ordered pairs $(i,j)$ with $0\leq i,j\leq 2020$. Find the remainder when $P(4040, 4040)$ is divided by $2017$. Note: $\deg_x (P)$ is the highest exponent of $x$ in a nonzero term of $P(x,y)$. $\deg_y (P)$ is defined similarly. [i]Proposed by Michael Ren[/i]

Solve in the real numbers the equation $ 3^{x+1}=(x-1)(x-3). $

Let there be a square $ABCD$. Points $E$ and $F$ are on sides $(BC)$ and $(AB)$ such that $BF=CE$. LInes $AE$ and $CF$ intersect in point $G$. Prove that $EF$ and $DG$ are perpendicular.

On Halloween 31 children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order. "Are you a truth-teller?" The principal gave a piece of candy to each of the 22 children who answered yes. "Are you an alternater?" The principal gave a piece of candy to each of the 15 children who answered yes. "Are you a liar?" The principal gave a piece of candy to each of the 9 children who answered yes. How many pieces of candy in all did the principal give to the children who always tell the truth? $\textbf{(A) }7\qquad\textbf{(B) }12\qquad\textbf{(C) }21\qquad\textbf{(D) }27\qquad\textbf{(E) }31$

Let $\vartriangle ABC$ be a triangle with side lengths $AB = 13$, $BC = 14$, and $CA = 15$. The angle bisector of $\angle BAC$, the angle bisector of $\angle ABC$, and the angle bisector of $\angle ACB$ intersect the circumcircle of $\vartriangle ABC$ again at points $D$, $E$ and $F$, respectively. Compute the area of hexagon $AF BDCE$.