**A. 660.** The circle \(\displaystyle \omega\) is inscribed in the quadrilateral \(\displaystyle ABCD\). The incenters of the triangles \(\displaystyle ABC\) and \(\displaystyle ACD\) are \(\displaystyle I\) and \(\displaystyle J\), respectively. Let \(\displaystyle T\) and \(\displaystyle U\) be those two points on the arcs of \(\displaystyle \omega\), lying in the triangles \(\displaystyle ABC\) and \(\displaystyle ACD\), respectively, for which the circles \(\displaystyle ATC\) and \(\displaystyle AUC\) are tangent to \(\displaystyle \omega\). Show that the line segments \(\displaystyle AC\), \(\displaystyle IU\) and \(\displaystyle JT\) are concurrent.

(5 points)

**A. 661.** Let \(\displaystyle K\) be a fixed positive integer. Let \(\displaystyle (a_0,a_1,\dots)\) be the sequence of real numbers that satisfies \(\displaystyle a_0=-1\) and

\(\displaystyle
\sum_{\substack{i_0,i_1,\dots,i_K\ge0 \\ i_0+i_1+\dots+i_K=n}}
\frac{a_{i_1}\cdot\dots\cdot a_{i_K}}{i_0+1} =0
\)

for every positive integer \(\displaystyle n\). Show that \(\displaystyle a_n>0\) for \(\displaystyle n\ge1\).

(5 points)

**B. 4763.** Let \(\displaystyle G\) be a simple undirected graph of \(\displaystyle n\) vertices. Prove that it is possible to assign infinite sets \(\displaystyle \mathcal{H}_1,\mathcal{H}_2, \dots, \mathcal{H}_n\) of natural numbers to the graph such that the intersection of two sets is infinite if the corresponding vertices are joined by an edge, and empty if there is no edge joining the vertices.

Proposed by *G. Mészáros,* Budapest

(4 points)

**B. 4765.** In a cyclic quadrilateral \(\displaystyle ABCD\), the angle bisectors of angles \(\displaystyle ADB\sphericalangle\) and \(\displaystyle ACB\sphericalangle\) intersect side \(\displaystyle AB\) at points \(\displaystyle E\) and \(\displaystyle F\), and the bisectors of \(\displaystyle CBD\sphericalangle\) and \(\displaystyle CAD\sphericalangle\) intersect side \(\displaystyle CD\) at \(\displaystyle G\) and \(\displaystyle H\), respectively. Prove that the points \(\displaystyle E\), \(\displaystyle F\), \(\displaystyle G\), \(\displaystyle H\) are concyclic.

Proposed by *B. Bíró,* Eger

(6 points)

**B. 4766.** The sequence \(\displaystyle a_{1}, a_{2}, \dots\) is defined by the following recursion: \(\displaystyle a_{1}=1\), \(\displaystyle a_{2}=5\), \(\displaystyle a_{3}=15\), and in the case of \(\displaystyle n\ge 4\), \(\displaystyle a_{n}=n^{2}+a_{n-1}+a_{n-2}-a_{n-3}\).

\(\displaystyle a)\) Calculate the sum \(\displaystyle a_{1}+a_{2}+a_{3}+\dots +a_{2015}\).

\(\displaystyle b)\) Prove that \(\displaystyle \frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+ \dots
+\frac{1}{a_{2015}}<\frac{4}{3}\).

Proposed by *B. Kovács, *Szatmárnémeti

(5 points)

**C. 1329.** King Sanislaus sends an envoy to a distant empire. In three days, a courier is sent back to the King's court with a message. It takes the courier two days to cover the same distance as the envoy takes three days to cover. With the answer, the courier returns to the envoy, and catches up with them exactly as they arrive at the destination. However, the emperor, who is a great friend of king Sanislaus, has been exiled in the meantime, so the entire envoy returns immediately, and this time they all travel at the speed of the courier. How many days elapse between the envoy starting their journey and returning to the court?

(5 points)

This problem is for grade 1 - 10 students only.

**C. 1334.** Given that the cyclic quadrilateral \(\displaystyle ABCD\) is either a trapezium (where \(\displaystyle AB\parallel
CD\)) or its side \(\displaystyle AB\) is a diameter of the circumscribed circle, then the equations

\(\displaystyle BD\cdot x+AD\cdot \sqrt{1-x^2} = AC;\)

\(\displaystyle AC\cdot x+BC\cdot \sqrt{1-x^2} = BD\)

are held for some \(\displaystyle x\) of \(\displaystyle 0<x<1\).

(5 points)

This problem is for grade 11 - 12 students only.

**K. 487.** Find the largest and the smallest eight-digit numbers in which the digits are 0, 1, 2, 3, 4, 5, 6, 7 in some order, and the sum of every pair of adjacent digits is a prime.

(6 points)

This problem is for grade 9 students only.

**K. 490.** Anthony is training ants. The performance starts with 99 ants sleeping on a 1-metre long straight rod. When a whistle is blown, they all wake up at the same time, and start crawling towards an end of the rod, at a speed of 1 cm/s. If an ant reaches the end of the rod, it will leave the rod. If two ants meet, they will turn back immediately, and start crawling in the opposite direction. The performance ends when the last ant has left the rod. What is the maximum possible duration of the performance?

(6 points)

This problem is for grade 9 students only.

**K. 491.** Andrew has chosen five numbers. He wrote them on a sheet of paper, and formed every possible sum of three terms out of them. He got the following results: 3, 4, 6, 7, 9, 10, 11, 14, 15 and 17. Which were the five original numbers that Andrew had in mind?

(6 points)

This problem is for grade 9 students only.