# [Solutions] European Mathematical Cup 2021

### Junior Division

1. We say that a quadruple of nonnegative real numbers $(a,b,c,d)$ is balanced if$$a+b+c+d=a^2+b^2+c^2+d^2.$$Find all positive real numbers $x$ such that$$(x-a)(x-b)(x-c)(x-d)\geq 0$$for every balanced quadruple $(a,b,c,d)$.
2. Let $ABC$ be an acute-angled triangle such that $|AB|<|AC|$. Let $X$ and $Y$ be points on the minor arc ${BC}$ of the circumcircle of $ABC$ such that $|BX|=|XY|=|YC|$. Suppose that there exists a point $N$ on the segment $\overline{AY}$ such that $|AB|=|AN|=|NC|$. Prove that the line $NC$ passes through the midpoint of the segment $\overline{AX}$.
3. Let $\ell$ be a positive integer. We say that a positive integer $k$ is nice if $k!+\ell$ is a square of an integer. Prove that for every positive integer $n \geqslant \ell$, the set $\{1, 2, \ldots,n^2\}$ contains at most $n^2-n +\ell$ nice integers.
4. Let $n$ be a positive integer. Morgane has coloured the integers $1,2,\ldots,n$. Each of them is coloured in exactly one colour. It turned out that for all positive integers $a$ and $b$ such that $a<b$ and $a+b \leqslant n$, at least two of the integers among $a$, $b$ and $a+b$ are of the same colour. Prove that there exists a colour that has been used for at least $2n/5$ integers.

### Senior Division

1. Alice drew a regular $2021$-gon in the plane. Bob then labeled each vertex of the $2021$-gon with a real number, in such a way that the labels of consecutive vertices differ by at most $1$. Then, for every pair of non-consecutive vertices whose labels differ by at most $1$, Alice drew a diagonal connecting them. Let $d$ be the number of diagonals Alice drew. Find the least possible value that $d$ can obtain.
2. Let $ABC$ be a triangle and let $D, E$ and $F$ be the midpoints of sides $BC, CA$ and $AB$, respectively. Let $X\ne A$ be the intersection of $AD$ with the circumcircle of $ABC$. Let $\Omega$ be the circle through $D$ and $X$, tangent to the circumcircle of $ABC$. Let $Y$ and $Z$ be the intersections of the tangent to $\Omega$ at $D$ with the perpendicular bisectors of segments $DE$ and $DF$, respectively. Let $P$ be the intersection of $YE$ and $ZF$ and let $G$ be the centroid of $ABC$. Show that the tangents at $B$ and $C$ to the circumcircle of $ABC$ and the line $PG$ are concurrent.
3. Let $\mathbb{N}$ denote the set of all positive integers. Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$x^2-y^2+2y(f(x)+f(y))$$ is a square of an integer for all positive integers $x$ and $y$.
4. Find all positive integers $d$ for which there exist polynomials $P(x)$ and $Q(x)$ with real coefficients such that degree of $P$ equals $d$ and $$P(x)^2+1=(x^2+1)Q(x)^2.$$
 MOlympiad.NET là dự án thu thập và phát hành các đề thi tuyển sinh và học sinh giỏi toán. Quý bạn đọc muốn giúp chúng tôi chỉnh sửa đề thi này, xin hãy để lại bình luận facebook (có thể đính kèm hình ảnh) hoặc google (có thể sử dụng $\LaTeX$) bên dưới. BBT rất mong bạn đọc ủng hộ UPLOAD đề thi và đáp án mới hoặc liên hệbbt.molympiad@gmail.comChúng tôi nhận tất cả các định dạng của tài liệu: $\TeX$, PDF, WORD, IMG,...