Problems

Alice took a red marker and marked 5 points with integer coordinates on a coordinate plane. Miriam took a blue marker and marked a midpoint for each pair of red points. Prove that at least 1 of the blue points has integer coordinates.

Each point on a circle was painted red or green. Show that there is an isosceles triangle whose vertices are on the circumference of the circle, such that all three vertices are red or all three are green.

Will and Neal are writing numbers on the blackboard. Each number is only composed of digits $0$ and $1$ in its decimal representation (as in, normal base 10 numbers). Will then says “I wonder if there is such a number we could write that can be divided by $2018$". Is there?

Anna has a garden shaped like an equilateral triangle of side $8$ metres. She wants to plant $17$ plants, but they need space – they need to be at least $2$ metres apart in order for their roots to have access to all the microelements in the ground. Show that Anna’s garden is unfortunately too small.

A math circle student Emilio wrote a computer program for his house robot, Basil. Starting from 1, Basil should keep writing bigger and bigger numbers formed by 1s: 1, 11, 111, etc. The program terminates when Basil writes a number that is a multiple of 19. Prove that the program will terminate in fewer than 20 steps.

We know that the product $c\times d$ is divisible by a prime $p$. Show that either $c$ or $d$ must be divisible by $p$.

The number $b^2$ is divisible by $8$. Show that it must be divisible by $16$.

Find a number which:

a) It is divisible by $4$ and by $6$, is has a total of 3 prime factors, which may be repeated.

b) It is divisible by $6,9$ and $4$, but not divisible by $27$. It has $4$ prime factors in total, which may be repeated.

c) It is divisible by $5$ and has exactly $3$ positive divisors.

The number $a$ has a prime factorization $2^3\times 3^2\times 7^2\times 11$. Is it divisible by $54$? Is it divisible by $154$?

a) The number $a$ is even. Should $3a$ definitely also be even?

b) The number $5c$ is divisible by $3$. Is it true that $c$ is definitely divisible by $3$?

c) The product $a\times b$ is divisible by $7$. Is it true that one of these numbers is divisible by $7$?

d) The product $c\times d$ is divisible by $26$. Is it true that one of these numbers is divisible by $26$?