Prove that any convex polygon contains not more than \(35\) vertices with an angle of less than \(170^\circ\).
On the sides \(AB\), \(BC\) and \(AC\) of the triangle \(ABC\) points \(P\), \(M\) and \(K\) are chosen so that the segments \(AM\), \(BK\) and \(CP\) intersect at one point and \[\vec{AM} + \vec{BK}+\vec{CP} = 0\] Prove that \(P\), \(M\) and \(K\) are the midpoints of the sides of the triangle \(ABC\).
A field that will be used to grow wheat has a rectangular shape. This year, the farmer responsible for this field decided to increase the length of one of the sides by \(20\%\) and decrease the length of another side by \(20\%\). The field remains rectangular. Will the harvest of wheat change this year and, if so, then by how much?
Prove that, in a circle of radius 10, you cannot place 400 points so that the distance between each two points is greater than 1.
A circle is covered with several arcs. These arcs can overlap one another, but none of them cover the entire circumference. Prove that it is always possible to select several of these arcs so that together they cover the entire circumference and add up to no more than \(720^{\circ}\).
\(x_1\) is the real root of the equation \(x^2 + ax + b = 0\), \(x_2\) is the real root of the equation \(x^2 - ax - b = 0\).
Prove that the equation \(x^2 + 2ax + 2b = 0\) has a real root, enclosed between \(x_1\) and \(x_2\). (\(a\) and \(b\) are real numbers).
In the number \(a = 0.12457 \dots\) the \(n\)th digit after the decimal point is equal to the digit to the left of the decimal point in the number. Prove that \(\alpha\) is an irrational number.
The function \(f\) is such that for any positive \(x\) and \(y\) the equality \(f (xy) = f (x) + f (y)\) holds. Find \(f (2007)\) if \(f (1/2007) = 1\).
Solve the equation: \((x^3 - 2) (2^{\sin x} - 1) + (2^{x^3} - 4) \sin x = 0\).
We are given a \(100\times 100\) square grid and \(N\) counters. All of the possible arrangements of the counters on the grid which follow the following rule are considered: no two counters lie in adjacent squares.
What is the largest value of \(N\) for which, in every single possible arrangement of counters following this rule, it is possible to find at least one counter such that moving it to an adjacent square does not break the rule. Squares are considered adjacent if they share a side.