A. 572. Two circles k_{1} and k_{2} with centres O_{1} and O_{2}, respectively, intersect perpendicularly at P and Q. Their external homothety center is H. The line t is tangent to k_{1} at T_{1} and tangent to k_{2} at T_{2}. Let X be a point in the interior of the two circles such that HX=HP=HQ, and let X' be the reflection of X about t. Let the circle XX'T_{2} and the shorter arc PQ of k_{1} meet at U_{1}, and let the circle XX'T_{1} and the shorter arc PQ of k_{2} meet at U_{2}. Finally, let the lines O_{1}U_{1} and O_{2}U_{2} meet at V. Show that VU_{1}=VU_{2}.
(5 points)
A. 573. Let D={0,1,2,...,9} be the set of decimal digits, and let RD×D be a set of ordered pairs of digits. An infinite sequence (a_{1},a_{2},a_{3},...) of digits is said to be compatible with R if (a_{j},a_{j+1})R for all positive integer j. Determine the smallest positive integer K with the property that if an arbitrary set RD×D is compatible with at least K distinct digit sequences then R is compatible with infinitely many digit sequences.
Based on the 5th problem of CIIM 2012, Guanajuato, Mexico
(5 points)
B. 4482. In Elastica, only natural numbers less than ten thousand are known. These numbers are written down in a flexible way. The rule for Elastican numbers is as follows: each number is represented in the lowest possible base such that the resulting number have at most four digits. Unfortunately, the numbers represented in this way are not always possible to decode uniquely: For example, the fourdigit Elastican number 1101 may mean 13 as well as 37. Is there a threedigit Elastican number that represents more than one natural number?
Based on the idea of T. Lakatos, Balassagyarmat
(3 points)
B. 4490. P and Q are interior points on side AC of a nonisosceles triangle ABC such that . The interior angle bisectors drawn from A and C intersect the line segment BP at points K and L, and the line segment BQ at points M and N, respectively. Prove that the lines AC, KN and LM are concurrent.
Suggested by Sz. Miklós, Herceghalom
(6 points)
B. 4491. The names of 100 convicts are placed in 100 numbered drawers in some order. Then the convicts are brought one by one from their cells in a random order, and each of them is allowed to pull out 50 drawers one by one. If he finds his own name in a drawer then he is led to a separate room. Otherwise all 100 convicts are executed immediately. Finally, if each of them succeeds, they are all set free. Show that, knowing the rules, the prisoners can follow a strategy that results in a larger than 30% probability of going free.
(6 points)
C. 1140. Ann, Beth, Connie and Dora shared a room in a class trip. The girls were given 1 bottle of orange juice, 2 bottles of apple juice, 2 bottles of peach juice and 3 bottles of mineral water to take with them on the forest walk. In how many different ways can they distribute the drinks among themselves if everyone is to get two bottles?
Suggested by A. Balga, Budapest
(5 points)
C. 1142. The points P, A, B and Q, in this order, lie on the same line such that PA=BQ, and a triangle ABC with interior angles of 36^{o}, 72^{o}, 72^{o} can be constructed out of the three line segments. Let R be the intersection of the circle of radius CP centred at C and the extension of the line AC beyond C. Find the angles subtended by the line segment PQ at the points C and R.
(5 points)
K. 350. Steve lives far away from his new workplace, so he decided to go to work by car. On the first day, he travelled at an average speed of 70 km/h (practically uniformly), and arrived 1 minute late. On the second day, he started out at the same time but travelled at an average speed of 75 km/h (practically uniformly) and arrived 1 minute early. How far does he live from his workplace?
(6 points)
This problem is for grade 9 students only.
K. 352. Make a 5×5 table, and fill it out with the numbers 1 to 25, left to right, top to bottom. Interchange the rows an arbitrary number of times, and then interchange the columns an arbitrary number of times. Add 7 to each entry of the ``mixed'' table obtained in this way. Finally, add the numbers along one of the diagonals. Prove that the result is always 100.
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 

(6 points)
This problem is for grade 9 students only.