MathDB

3

Part of 2002 Romania Team Selection Test

Problems(5)

Isosceles triangle

Source: Romanian TST 2002

3/19/2004
Let MM and NN be the midpoints of the respective sides ABAB and ACAC of an acute-angled triangle ABCABC. Let PP be the foot of the perpendicular from NN onto BCBC and let A1A_1 be the midpoint of MPMP. Points B1B_1 and C1C_1 are obtained similarly. If AA1AA_1, BB1BB_1 and CC1CC_1 are concurrent, show that the triangle ABCABC is isosceles.
Mircea Becheanu
ratiogeometryrectanglegeometry proposed
x_n is the sum of digits of the number [an+b]

Source: Romanian TST 2002

2/5/2011
Let a,ba,b be positive real numbers. For any positive integer nn, denote by xnx_n the sum of digits of the number [an+b][an+b] in it's decimal representation. Show that the sequence (xn)n≥1(x_n)_{n\ge 1} contains a constant subsequence.
Laurentiu Panaitopol
pigeonhole principlenumber theory proposednumber theory
nice

Source:

4/7/2005
Let nn be a positive integer. SS is the set of nonnegative integers aa such that 1<a<n1<a<n and aa−1−1a^{a-1}-1 is divisible by nn. Prove that if S={n−1}S=\{ n-1 \} then n=2pn=2p where pp is a prime number.
Mihai Cipu and Nicolae Ciprian Bonciocat
modular arithmeticnumber theory proposednumber theory
PMs can move to a group if it increases his relative rating

Source: Romanian TST 2002

2/5/2011
After elections, every parliament member (PM), has his own absolute rating. When the parliament set up, he enters in a group and gets a relative rating. The relative rating is the ratio of its own absolute rating to the sum of all absolute ratings of the PMs in the group. A PM can move from one group to another only if in his new group his relative rating is greater. In a given day, only one PM can change the group. Show that only a finite number of group moves is possible.
(A rating is positive real number.)
ratioinvariantcombinatorics proposedcombinatorics
Card game with np cards in p rounds

Source: Romanian TST 2002

2/5/2011
There are nn players, n≥2n\ge 2, which are playing a card game with npnp cards in pp rounds. The cards are coloured in nn colours and each colour is labelled with the numbers 1,2,…,p1,2,\ldots ,p. The game submits to the following rules: each player receives pp cards. the player who begins the first round throws a card and each player has to discard a card of the same colour, if he has one; otherwise they can give an arbitrary card. the winner of the round is the player who has put the greatest card of the same colour as the first one. the winner of the round starts the next round with a card that he selects and the play continues with the same rules. the played cards are out of the game. Show that if all cards labelled with number 11 are winners, then p≥2np\ge 2n.
Barbu Berceanu
combinatorics proposedcombinatorics