A. 627. Let \(\displaystyle n\ge1\) be a fixed integer. Calculate the distance \(\displaystyle \inf_{p,f}
\max_{0\le x\le 1} \bigf(x)p(x)\big\), where \(\displaystyle p\) runs over polynomials of degree less than \(\displaystyle n\) with real coefficients and \(\displaystyle f\) runs over functions \(\displaystyle f(x) = \sum_{k=n}^\infty c_k x^k\) defined on the closed interval \(\displaystyle [0,1]\), where \(\displaystyle c_k\ge0\) and \(\displaystyle \sum_{k=n}^\infty c_k=1\).
Miklós Schweitzer competition, 2014
(5 points)
A. 628. Is it true that for every infinite sequence \(\displaystyle x_1,x_2,\ldots\) of integers satisfying \(\displaystyle x_{k+1}x_k=1\) for every positive integer \(\displaystyle k\), there exists a sequence \(\displaystyle k_1<k_2<\ldots<k_{2014}\) of positive integers such that as well the indices \(\displaystyle k_1,k_2,\ldots,k_{2014}\) as the numbers \(\displaystyle x_{k_1},x_{k_2},\ldots,x_{k_{2014}}\) (in this order) form arithmethic progressions?
Proposed by: E. Csóka, Warwick and Ben Green, Oxford
(5 points)
B. 4660. In a championship, every team plays every other team exactly once. 3 points are awarded for winning, 0 for losing, and 1 for a draw. In the case of equal scores, the order of the teams is determined at random. The championship is in progress at the moment. Team \(\displaystyle A\) is leading the points table. If team \(\displaystyle A\) scores exactly \(\displaystyle x\) points in the remaining rounds then they will win the championship. However, if \(\displaystyle A\) scores more than \(\displaystyle x\) points, they will not necessarily win. (It is possible for \(\displaystyle A\) to score more than \(\displaystyle x\).) How many rounds remain to be played in the championship?
Suggested by V. Vígh, Szeged
(5 points)
B. 4662. A regular triangle is drawn over each side of a triangle \(\displaystyle ABC\) on the outside. The third vertices of these triangles are \(\displaystyle D\), \(\displaystyle E\) and \(\displaystyle F\). Given the points \(\displaystyle D\), \(\displaystyle E\) and \(\displaystyle F\), construct the triangle \(\displaystyle ABC\).
Suggested by Sz. Miklós, Herceghalom
(4 points)
B. 4664. A rectangle \(\displaystyle ABDE\) is drawn to side \(\displaystyle AB\) of an acute triangle \(\displaystyle ABC\) on the inside, such that point \(\displaystyle C\) should lie on the side \(\displaystyle DE\). The rectangles \(\displaystyle BCFG\) and \(\displaystyle CAHI\) are defined in a similar way. (\(\displaystyle A\) lies on line segment \(\displaystyle FG\), and \(\displaystyle B\) lies on line segment \(\displaystyle HI\).) The midpoints of sides \(\displaystyle AB\), \(\displaystyle BC\), and \(\displaystyle CA\) are \(\displaystyle J\), \(\displaystyle K\), and \(\displaystyle L\), respectively. Prove that the sum of the angles \(\displaystyle GJH\sphericalangle\), \(\displaystyle IKD\sphericalangle\) and \(\displaystyle ELF\sphericalangle\) is \(\displaystyle 180^{\circ}\).
Suggested by Sz. Miklós, Herceghalom
(4 points)
C. 1256. A king wants to give a convict another chance to escape prison. The convict is blindfolded, and instructed to select one ball from each of three urns. The guards place the three balls in a fourth urn. Then the blindfolded man draws a ball from this fourth urn. If the ball is white, he is let go free. What is the probability of escaping prison if the number of balls of various colours in the three urns is as follows:

white 
red 
black 
urn 1 
2 
5 
3 
urn 2 
5 
2 
3 
urn 3 
3 
3 
4 

Suggested by J. Czinki, Budapest
(5 points)
K. 433. We have four objects that look all alike, but weigh 3, 5, 8 and 11 kg. We also have an equalarm balance without extra weights.
\(\displaystyle a)\) By using the balance only twice, how can we identify the 11kg object?
\(\displaystyle b)\) With two further measurements, how can we determine the masses of the other objects, too?
(6 points)
This problem is for grade 9 students only.
K. 437. Every digit in the product below is one of the digits 2, 3, 5, 7. Determine which digit each letter represents. (Different letters may stand for the same digit, but each digit must be one of the four listed.)
(6 points)
This problem is for grade 9 students only.