2013 Israel National Olympiad

Part of Israel National Olympiad

Subcontests

(6)

1

Israel 2013 Q6 - Fractional inequality

x_1,...,x_n

be positive real numbers, satisfying

x_1+\dots+x_n=n

\frac{x_1}{x_2}+\frac{x_2}{x_3}+\dots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}\leq\frac{4}{x_1\cdot x_2\cdot\dots\cdot x_n}+n-4

1

Israel 2013 Q5 - Triangle in the integer lattice

A point in the plane is called integral if both its

x

y

coordinates are integers. We are given a triangle whose vertices are integral. Its sides do not contain any other integral points. Inside the triangle, there are exactly 4 integral points. Must those 4 points lie on one line?

1

Israel 2013 Q4 - Multiples of 7 made of digits 0,1,9

Determine the number of positive integers

n

satisfying:
[*]

n<10^6

n

is divisible by 7 [*]

n

does not contain any of the digits 2,3,4,5,6,7,8.

1

Israel 2013 Q3 - Sum arctans of polynomial roots

p(x)=x^4-5773x^3-46464x^2-5773x+46

. Determine the sum of

\arctan

-s of its real roots.

1

Israel 2013 Q2 - Product free subsets

A=\{n\in\mathbb{Z}\mid 0<n<2013\}

B\subseteq A

is called reduced if for any two numbers

x,y\in B

x\cdot y \notin B

. For example, any subset containing the numbers

3,5,15

cannot be reduced, and same for a subset containing

4,16

.
[*] Find the maximal size of a reduced subset of

A

. [*] How many reduced subsets are there with that maximal size?

1

Israel 2013 Q1 - Star shape in a coin

In the picture there are six coins, each with radius 1cm. Each coin is tangent to exactly two other coins next to it (as in the picture). Between the coins, there is an empty area whose boundary is a star-like shape. What is the perimeter of this shape? https://i.imgur.com/aguQRVd.png