Find the smallest real number $x$ for which exist two non-congruent triangles, whose sides have integer lengths and the numerical value of the area of each triangle is $x$.

Given a $\triangle ABC$, determine which is the circle with the smallest area that contains it.

Let $ABC$ be an acute triangle. A line, parallel to $BC$, intersects sides $AB$ and $AC$ at points $M$ and $P$, respectively. At which placement of points $M$ and $P$, is the radius of the circumcircle of the triangle $BMP$ is the smallest?

Find the smallest value of the expression $|3 \cdot 5^m - 11 \cdot 13^n|$ for all $m,n \in N$. (Folklore)

What is the smallest positive integer $k$ such that $k(3^3 + 4^3 + 5^3) = a^n$ for some positive integers $a$ and $n$, with $n > 1$?

In three houses $A,B$ and $C$, forming a right triangle with the legs $AC=30$ and $CB=40$, live three beetles $a,b$ and $c$, capable of moving at speeds of $2, 3$ and $4$, respectively. Suppose that you simultaneously release these bugs from point $M$ and mark the time after which beetles reach their homes. Find on the plane such a point $M$, where is the last time to reach the house a bug would be minimal.

Given two circles with the radiuses $R$ and $r$, touching each other from the outer side. Consider all the trapezoids, such that its lateral sides touch both circles, and its bases touch different circles. Find the shortest possible lateral side.

An ant, which is on an edge of a cube of side $8$, must travel on the surface and return to the starting point. It's path must contain interior points of the six faces of the cube and should visit only once each face of the cube. Find the length of the path that the ant can carry out and justify why it is the shortest path.

Into an angle $A$ of size $a$, a circle is inscribed tangent to its sides at points $B$ and $C$. A line tangent to this circle at a point M meets the segments $AB$ and $AC$ at points $P$ and $Q$ respectively. What is the minimum $a$ such that the inequality $S_{PAQ}<S_{BMC}$ is possible?

Find the smallest real constant $p$ for which the inequality holds $\sqrt{ab}- \frac{2ab}{a + b} \le p \left( \frac{a + b}{2} -\sqrt{ab}\right)$ with any positive real numbers $a, b$.

For an integer $n \ge 1$ we consider sequences of $2n$ numbers, each equal to $0, -1$ or $1$. The [i]sum product value[/i] of such a sequence is calculated by first multiplying each pair of numbers from the sequence, and then adding all the results together. For example, if we take $n = 2$ and the sequence $0,1, 1, -1$, then we find the products $0\cdot 1, 0\cdot 1, 0\cdot -1, 1\cdot 1, 1\cdot -1, 1\cdot -1$. Adding these six results gives the sum product value of this sequence: $0+0+0+1+(-1)+(-1) = -1$. The sum product value of this sequence is therefore smaller than the sum product value of the sequence $0, 0, 0, 0$, which equals $0$. Determine for each integer $n \ge 1$ the smallest sum product value that such a sequence of $2n$ numbers could have. [i]Attention: you are required to prove that a smaller sum product value is impossible.[/i]

If $x$ and $y$ are nonnegative real numbers such that $x+y=1$, determine minimal and maximal value of $$A=x\sqrt{1+y}+y\sqrt{1+x}$$

A plane is divided into $N$ regions by three families of parallel lines. No three lines pass through the same point. What is the smallest number of lines needed so that $N > 1999$?

Let two positive integers $x, y$ satisfy the condition $44 /( x^2 + y^2)$. Determine the smallest value of $T = x^3 + y^3$.

Let $a, b, c$ positive reals such that $a+b+c=1$. Prove that $$\min\{a(1-b),b(1-c),c(1-a)\}\leq \frac{1}{4}$$ $$\max\{a(1-b),b(1-c),c(1-a)\}\geq \frac{2}{9}$$

What is the smallest $n$ for which there is a solution to $$\begin{cases} \sin x_1 + \sin x_2 + ... + \sin x_n = 0 \\ \sin x_1 + 2 \sin x_2 + ... + n \sin x_n = 100 \end{cases}$$ ?