**A. 698.** Let \(\displaystyle m\) and \(\displaystyle n\) be positive integers, and let \(\displaystyle H\) denote a subset of the set \(\displaystyle \{1,2,\ldots,m\}\times\{1,2,\ldots,n\}\). Show that if \(\displaystyle |H|>m+(m+n)\log_2n\), then there exist integers \(\displaystyle 1\le u<v\le m\) and \(\displaystyle 1\le x<y<z\le n\) such that the pairs \(\displaystyle (u,x)\), \(\displaystyle (u,y)\), \(\displaystyle (v,x)\) and \(\displaystyle (v,z)\) are elements of \(\displaystyle H\).

(5 points)

**A. 699.** A circle \(\displaystyle \omega\) lies in a circle \(\displaystyle \Omega\) such that their common center is the point \(\displaystyle O\). Fix a point \(\displaystyle A\ne O\) inside \(\displaystyle \omega\). Let \(\displaystyle X\) denote an arbitrary point on the circumference of \(\displaystyle \Omega\), and let \(\displaystyle Y\) denote the second intersection point of \(\displaystyle \Omega\) and the line \(\displaystyle AX\). Let \(\displaystyle Z\) denote the intersection of \(\displaystyle \omega\) and the line segment \(\displaystyle AX\). Let \(\displaystyle M\) denote the point on the line segment \(\displaystyle AZ\) for which \(\displaystyle MX\cdot MZ\cdot AY = MA\cdot MY\cdot XZ\). Let \(\displaystyle x\) and \(\displaystyle y\) denote the tangents of the circle \(\displaystyle \Omega\) at the points \(\displaystyle X\) and \(\displaystyle Y\), respectively. Let \(\displaystyle t\) denote the line that passes through \(\displaystyle M\) and either also passes through the intersection of \(\displaystyle x\) and \(\displaystyle y\), or is parallel to both \(\displaystyle x\) and \(\displaystyle y\). Finally, let \(\displaystyle T\) denote the intersection of \(\displaystyle t\) and the line \(\displaystyle OZ\).

Show that the locus of the points \(\displaystyle T\), as \(\displaystyle X\) is varied, is an ellipse, and the line \(\displaystyle t\) is a tangent of this ellipse.

(5 points)

**A. 700.** A positive integer \(\displaystyle n\) satisfies the following: it is possible to select some integers such that if we randomly choose two different integers from this list, say, \(\displaystyle i\) and \(\displaystyle j\), then \(\displaystyle i+j\) \(\displaystyle \mathrm{mod\ } n\) is equal to one of the numbers \(\displaystyle 0,1,\dots,n-1\) with equal probability. Find all numbers \(\displaystyle n\) with this property.

(5 points)

**B. 4877.** Points \(\displaystyle A\), \(\displaystyle B\), \(\displaystyle C\) and \(\displaystyle D\), in this order, lie on a straight line. Point \(\displaystyle E\) does not lie on the line, and

\(\displaystyle
AEB\sphericalangle=BEC\sphericalangle =CED\sphericalangle = 45^\circ.
\)

Let \(\displaystyle F\) and \(\displaystyle G\) be the midpoints of \(\displaystyle AC\) and \(\displaystyle BD\), respectively. What is the measure of angle \(\displaystyle FEG\)?

Proposed by *the class 11C of Fazekas Gimnázium, Budapest*

(3 points)

**B. 4879.** \(\displaystyle a)\) Is it true that for any irrational number \(\displaystyle a\) there exists an irrational number \(\displaystyle x\) such that \(\displaystyle a+x\) is rational and \(\displaystyle ax\) is irrational?

\(\displaystyle b)\) Is it true that for any irrational number \(\displaystyle a\) there exists an irrational number \(\displaystyle y\) such that \(\displaystyle a+y\) is irrational and \(\displaystyle ay\) is rational?

Proposed by *S. Róka,* Nyíregyháza

(4 points)

**B. 4880.** In the sequence \(\displaystyle a_1\), \(\displaystyle a_2\), \(\displaystyle a_3\), ... of positive integers, \(\displaystyle a_n\cdot a_{n+1}
= a_{n+2}\cdot a_{n+3}\) for all positive integers \(\displaystyle n\). Show that the sequence is eventually periodic.

Proposed by *M. E. Gáspár,* Budapest

(4 points)

**B. 4883.** Define the sequence \(\displaystyle a_1, a_2, \dots\) with the following recurrence relation:

\(\displaystyle
a_1=4, \quad a_2=2 \quad\text{and}\quad a_{n+1}=\frac{na_n^2}{na_n^2-(n+1)a_n+n+1},
\text{ if } n\ge 2.
\)

Prove that

\(\displaystyle
a_1+2\cdot a_2+3\cdot a_3+\cdots +n\cdot a_n=a_1\cdot a_2\cdot a_3\cdot\,\cdots\,
\cdot a_n
\)

for all \(\displaystyle n\ge 1\).

Proposed by *B. Kovács,* Szatmárnémeti

(6 points)

**C. 1421.** Prove that if \(\displaystyle n\in \mathbb{N}^+\) then there exist \(\displaystyle a,b\in \mathbb{N}^+\) such that \(\displaystyle a^2+b^2=13^n\).

Based on a problem by *F. Olosz,* Szatmárnémeti

(5 points)

This problem is for grade 1 - 10 students only.