BY-MO 2019

Is it true that for any nonzero rational numbers $a$ and $b$ one can find integers $m$ and $n$ such that the number $(am+b)^2+(a+nb)^2$ is an integer?

(M. Karpuk)

The rhombus $ABCD$ is given. Let $E$ be one of the points of intersection of the circles $\Gamma_B$ and $\Gamma_C$ , where $\Gamma_B$ is the circle centered at $B$ and passing through $C$ , and $\Gamma_C$ is the circle centered at $C$ and passing through $B$ . The line $ED$ intersects $\Gamma_B$ at point $F$ . Find the value of angle $\angle AFB$ .

(S. Mazanik)

Positive real numbers $a$ and $b$ satisfy the following conditions: the function $f(x)=x^3+ax^2+2bx-1$ has three different real roots, while the function $g(x)=2x^2+2bx+a$ doesn't have real roots. Prove that $a-b>1$ .

(V. Karamzin)

The sum of several (not necessarily different) positive integers not exceeding $10$ is equal to $S$ . Find all possible values of $S$ such that these numbers can always be partitioned into two groups with the sum of the numbers in each group not exceeding $70$ .

(I. Voronovich)

For a positive integer $n$ write down all its positive divisors in increasing order: $1=d_1<d_2<\ldots<d_k=n$ . Find all positive integers $n$ divisible by $2019$ such that $n=d_{19}\cdot d_{20}$ .

(I. Gorodnin)

The point $M$ is the midpoint of the side $BC$ of triangle $ABC$ . A circle is passing through $B$ , is tangent to the line $AM$ at $M$ , and intersects the segment $AB$ secondary at the point $P$ . Prove that the circle, passing through $A$ , $P$ , and the midpoint of the segment $AM$ , is tangent to the line $AC$ .

(A. Voidelevich)

Find all non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients such that $P(Q(x)^2)=P(x)\cdot Q(x)^2$ .

(I. Voronovich)

Andrey and Sasha play the game, making moves alternate. On his turn, Andrey marks on the plane an arbitrary point that has not yet been marked. After that, Sasha colors this point in one of two colors: white and black. Sasha wins if after his move it is impossible to draw a line such that all white points lie in one half-plane, while all black points lie in another half-plane with respect to this line.a) Prove that Andrey can make moves in such a way that Sasha will never win.b) Suppose that Andrey can mark only integer points on the Cartesian plane. Can Sasha guarantee himself a win regardless of Andrey's moves?

(N. Naradzetski)