2011 Israel National Olympiad

Part of Israel National Olympiad

Subcontests

(6)

1

Israel 2011 Q6 - Red and blue numbered cards

N

N

blue cards. Each card has a positive integer between

1

N

(inclusive) written on it. Prove that we can choose a (non-empty) subset of the red cards and a (non-empty) subset of the blue cards, so that the sum of the numbers on the chosen red cards equals the sum of the numbers on the chosen blue cards.

1

Israel 2011 Q5 - Process with two lists

We have two lists of numbers: One initially containing 1,6,11,...,96, and the other initially containing 4,9,14,...,99. In every turn, we erase two numbers from one of the lists, and write

\frac{1}{3}

of their sum (not necessarily an integer) in the other list. We continue this process until there are no possible moves.
[*] Prove that at the end of the process, there is exactly one number in each list. [*] Prove that those two numbers are not equal.

1

Israel 2011 Q4 - Four tangent circles

\alpha_1,\alpha_2,\alpha_3

be three congruent circles that are tangent to each other. A third circle

\beta

is tangent to them at points

A_1,A_2,A_3

respectively. Let

P

\beta

which is different from

A_1,A_2,A_3

i=1,2,3

B_i

be the second intersection point of the line

PA_i

\alpha_i

\Delta B_1B_2B_3

is equilateral.

1

Israel 2011 Q3 - Uniform triplets of ministers

In some foreign country's government, there are 12 ministers. Each minister has 5 friends and 6 enemies in the government (friendship/enemyship is a symmetric relation). A triplet of ministers is called uniform if all three of them are friends with each other, or all three of them are enemies. How many uniform triplets are there?

1

Israel 2011 Q2 - Evaluate sum of roots

Evaluate the sum

\sqrt{1-\frac{1}{2}\cdot\sqrt{1\cdot3}}+\sqrt{2-\frac{1}{2}\cdot\sqrt{3\cdot5}}+\sqrt{3-\frac{1}{2}\cdot\sqrt{5\cdot7}}+\dots+\sqrt{40-\frac{1}{2}\cdot\sqrt{79\cdot81}}

1

Israel 2011 Q1 - Equal sets of weights

We are given 5771 weights weighing 1,2,3,...,5770,5771. We partition the weights into

n

sets of equal weight. What is the maximal

n

for which this is possible?