**A. 653.** Let \(\displaystyle n\ge2\) be an integer. Prove that there exist integers \(\displaystyle a_1,\dots,a_{n-1}\) such that \(\displaystyle a_1 \arctg 1 + a_2 \arctg 2 +\ldots+ a_{n-1}\arctg(n-1) = \arctg n\) if and only if \(\displaystyle n^2+1\) divides \(\displaystyle (1^2+1)(2^2+1)\ldots\big((n-1)^2+1\big)\).

Based on a problem of IMC 2015, Blagoevgrad

(5 points)

**A. 655.** Two circles, \(\displaystyle k_1\) and \(\displaystyle k_2\) meet at points \(\displaystyle A\) and \(\displaystyle B\). Points \(\displaystyle C\) and \(\displaystyle D\) lie on \(\displaystyle k_1\), while points \(\displaystyle E\) and \(\displaystyle F\) lie on circle \(\displaystyle k_2\) in such a way that \(\displaystyle A\), \(\displaystyle C\), \(\displaystyle E\) are collinear and \(\displaystyle B\), \(\displaystyle D\), \(\displaystyle F\) are collinear, too. Points \(\displaystyle G\) and \(\displaystyle H\) are other two points on lines \(\displaystyle ACE\) and \(\displaystyle BDF\), respectively. The line \(\displaystyle CH\) meets \(\displaystyle FG\) and \(\displaystyle k_1\) the second time at \(\displaystyle I\) and \(\displaystyle J\), respectively. The line \(\displaystyle DG\) meets \(\displaystyle EH\) and \(\displaystyle k_1\) the second time at \(\displaystyle K\) and \(\displaystyle L\), respectively. Circle \(\displaystyle k_2\) meets the lines \(\displaystyle EHK\) and \(\displaystyle FGI\) the second time at \(\displaystyle M\) and \(\displaystyle N\), respectively. The points \(\displaystyle A,B,C,\ldots,N\) are distinct. Show that \(\displaystyle I\), \(\displaystyle J\), \(\displaystyle K\), \(\displaystyle L\), \(\displaystyle M\) and \(\displaystyle N\) are either concyclic or collinear.

(5 points)

**B. 4743.** The inscribed circle of triangle \(\displaystyle ABC\) touches sides \(\displaystyle BC\), \(\displaystyle AC\) and \(\displaystyle AB\) at points \(\displaystyle A_1\), \(\displaystyle B_1\) and \(\displaystyle C_1\), respectively. Let the orthocentres of triangles \(\displaystyle AC_1B_1\), \(\displaystyle BA_1C_1\) and \(\displaystyle CB_1A_1\) be \(\displaystyle M_A\), \(\displaystyle M_B\) and \(\displaystyle M_C\), respectively. Show that triangle \(\displaystyle A_1B_1C_1\) is congruent to triangle \(\displaystyle M_AM_BM_C\).

Proposed by *Sz. Miklós,* Herceghalom

(4 points)

**B. 4747.** In a certain lottery game, 6 numbers are drawn every week, out of the numbers 1 to 45. The draw of the first week of this year produced surprising results, since five consecutive numbers appeared. The numbers drawn were 37, 38, 39, 40, 41, 45. The news spread fast in the press. The question arises whether the excitement was justified: are these numbers so special? Let a number sequence be called perfect if it consists of six consecutive numbers, and nearly perfect if exactly five numbers out of the six are consecutive. How many perfect and nearly perfect combinations are there? Considering that the lottery game has been played for 26 years, and there have been 1227 weekly draws so far, what is the probability that during a time interval of this length at least one perfect or nearly perfect sequence of numbers is drawn?

Proposed by *M. E. Gáspár,* Budapest

(3 points)

**B. 4749.** The feet of the altitudes drawn from vertices \(\displaystyle B\) and \(\displaystyle C\) of an acute-angled triangle \(\displaystyle ABC\) on the sides \(\displaystyle AC\) and \(\displaystyle AB\) are \(\displaystyle D\) and \(\displaystyle E\), respectively. The midpoint of side \(\displaystyle BC\) is \(\displaystyle F\). The intersection of line segments \(\displaystyle AF\) and \(\displaystyle DE\) is \(\displaystyle M\), and the orthogonal projection of point \(\displaystyle M\) onto the line segment \(\displaystyle BC\) is \(\displaystyle N\). Prove that line segment \(\displaystyle AN\) bisects line segment \(\displaystyle DE\).

Proposed by *B. Bíró,* Eger - In memoriam Attila Kálmán

(6 points)

**C. 1315.** In a chocolate factory, chocolate mass is poured into molds to make 100-gram bars. Owing to a malfunction of machinery, one out of 45 bars breaks in the process. A quality control inspector spots these broken bars before they would get wrapped, and returns them to the molten chocolate mass. However, the inspector misses one out of 21 broken bars and lets them go on to the wrapping machine. Out of 10 tonnes of chocolate mass, how many broken bars of chocolate will get to the market?

(5 points)

This problem is for grade 1 - 10 students only.

**C. 1317.** The interior angles lying at vertices \(\displaystyle A\), \(\displaystyle B\), \(\displaystyle C\) and \(\displaystyle D\) of a pentagon \(\displaystyle ABCDE\) are \(\displaystyle 90^\circ\), \(\displaystyle 60^\circ\), \(\displaystyle 150^\circ\) and \(\displaystyle 150^\circ\), respectively. Furthermore \(\displaystyle AB=2BC=\frac 43 AD\). Prove that the line segment joining the intersection of lines \(\displaystyle AE\) and \(\displaystyle CD\) to the intersection of lines \(\displaystyle AD\) and \(\displaystyle BC\) is parallel to \(\displaystyle AB\).

(5 points)

**K. 476.** Find all positive integers for which the number obtained by rounding to the nearest thousands is twice the number obtained by rounding to the nearest hundred. (As a rule, numbers ending in 5, 50, 500, \(\displaystyle \dots\;\) round upwards.)

(6 points)

This problem is for grade 9 students only.

**K. 478.** Farmer Thomas wanted to buy 4 metres of chain that costs 210 forints (HUF, Hungarian currency) a metre. The shop assistant tried to talk him into buying all the 10 metres they have in stock, but Farmer Thomas insisted on buying 4 metres only. However, he noticed that the assistant, on purpose, made a mistake in measuring the 4 metres, and cut off a shorter piece. Therefore he decided to ask for the other piece instead, which the assistant had to sell him for the price of 6 metres in order to avoid being caught cheating. Had he not noticed the cheating, it would have cost Farmer Thomas 14/9 as much to buy a metre of chain as it actually cost with this clever manoeuvre. How many metres of chain did Thomas get?

(6 points)

This problem is for grade 9 students only.

**K. 479.** In the expression \(\displaystyle \big({(-a^{-b})}^{-c}\big)^{-d}\) the numbers 1, 2, 3, 4 are substituted for \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\), \(\displaystyle d\) in some order. In which case will the value of the expression be a maximum, and in which case will it be a minimum?

(6 points)

This problem is for grade 9 students only.