A numeric set \(M\) containing 2003 distinct numbers is such that for every two distinct elements \(a, b\) in \(M\), the number \(a^2+ b\sqrt 2\) is rational. Prove that for any \(a\) in \(M\) the number \(q\sqrt 2\) is rational.
All of the points with whole number co-ordinates in a plane are plotted in one of three colours; all three colours are present. Prove that there will always be possible to form a right-angle triangle from these points so that its vertices are of three different colours.
Is there a bounded function \(f\colon \mathbb{R} \rightarrow \mathbb{R}\) such that \(f (1)> 0\) and \(f (x)\) satisfies the inequality \(f^2 (x + y) \geq f^2 (x) + 2f (xy) + f^2 (y)\) for all \(x, y \in \mathbb{R}\)?
Ten pairwise distinct non-zero numbers are such that for each two of them either the sum of these numbers or their product is a rational number.
Prove that the squares of all numbers are rational.
The polynomial \(P (x)\) of degree \(n\) has \(n\) distinct real roots.
What is the largest number of its coefficients that can be equal to zero?
We are given a table of size \(n \times n\). \(n-1\) of the cells in the table contain the number \(1\). The remainder contain the number \(0\). We are allowed to carry out the following operation on the table:
1. Pick a cell.
2. Subtract 1 from the number in that cell.
3. Add 1 to every other cell in the same row or column as the chosen cell.
Is it possible, using only this operation, to create a table in which all the cells contain the same number?
Solve the equation \((x + 1)^3 = x^3\).
We call a number \(x\) rational if it can be represented as \(x=\frac{p}{q}\) for coprime integers \(p\) and \(q\). Otherwise we call the number irrational.
Non-zero numbers \(a\) and \(b\) satisfy the equality \(a^2b^2 (a^2b^2 + 4) = 2(a^6 + b^6)\). Prove that at least one of them is irrational.
Prove that in any set of 117 unique three-digit numbers it is possible to pick 4 non-overlapping subsets, so that the sum of the numbers in each subset is the same.
A target consists of a triangle divided by three families of parallel lines into 100 equilateral unit triangles. A sniper shoots at the target. He aims at a particular equilateral triangle and either hits it or hits one of the adjacent triangles that share a side with the one he was aiming for. He can see the results of his shots and can choose when to stop shooting. What is the largest number of triangles that the sniper can guarantee he can hit exactly 5 times?