4

Part of 2015 Caucasus Mathematical Olympiad

Problems(5)

26 students, all either always tell the truth either always tells lies

Source: Caucasus 2015 7.4

26

students in the class. They agreed that each of them would either be a liar (liars always lie) or a knight (knights always tell the truth). When they came to the class and sat down for desks, each of them said: “I am sitting next to a liar.” Then some students moved for other desks. After that, everyone says: “ I am sitting next to a knight .” Is this possible? Every time exactly two students sat at any desk.

combinatoricsTrue or False

\frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}>0 if abc<0, a+b+c=0

Source: Caucasus 2015 8.4

The sum of the numbers

a,b

c

is zero, and their product is negative. Prove that the number

\frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}

algebrainequalitiesthree variable inequality

9-digit number with different remaider divided by each of it's digits

Source: Caucasus 2015 9.4

Is there a nine-digit number without zero digits, the remainder of dividing which on each of its digits is different?

number theoryDigitsremainder

max no of consecutive naturals >25 that are sums of different primes

Source: Caucasus 2015 10.4

We call a number greater than

25

, semi-prime if it is the sum of some two different prime numbers. What is the greatest number of consecutive natural numbers that can be semi-prime?

number theorySumprime numbersconsecutive

midpoint of edge equidistant from vertices in triangular pyramid,Caucasus 11th

Source: I Caucasus 2015 11.4

The midpoint of the edge

SA

of the triangular pyramid of

SABC

has equal distances from all the vertices of the pyramid. Let

SH

be the height of the pyramid. Prove that

BA^2 + BH^2 = C A^2 + CH^2

geometry3D geometrypyramid3-Dimensional Geometry