Problems

In a square, the midpoints of its sides were marked and some segments were drawn. There is another square formed in the centre. Find its area, if the side of the square has length $10$.

In a parallelogram $ABCD$, point $E$ belongs to the side $AB$, point $F$ belongs to the side $CD$ and point $G$ belongs to the side $AD$. What is more, the marked red segments $AE$ and $CF$ have equal lengths. Prove that the total grey area is equal to the total black area.

A little Jimmy visited his four aunties today. Each of them prepared a cake for him and his parents. Auntie Martha made a carrot cake, Auntie Camilla made a sponge, Auntie Becky made a chocolate cake and Auntie Anne made a fudge. Jimmy would like to visit the aunties the next time when aunties all make the same cakes again. What day should he pick, if Auntie Martha makes a carrot cake every two days, Auntie Camilla makes a sponge every three days, Auntie Becky makes a chocolate cake every four days and Auntie Anne makes a fudge every seven days?

a) Two numbers, $a$ and $b$, are relatively prime and their product is equal to $3^5\times 7^2$. What could these numbers be? Find all the possibilities.

b) The gcd of two numbers, $c$ and $d$, is $20$ and their product is $2^4\times 5^3$. What could these numbers be? Find all the answers.

Two numbers are $a=2\times 3^5\times {31}^2\times 7$ and $b=7^2\times 2^4\times 3^2\times {29}^2$. Find their greatest common divisor and least common multiple.

A valiant adventurer enters a dragon’s cave looking for the Holy Grail. She knows that Holy Grail is a chalice that is tall, made of gold, has encrusted rubies, and has an ancient inscription written on it. Upon entering, the knight discovers a long corridor of chalices, all marked with natural numbers starting from 1. He examines the first chalices and discovers, that every 10th chalice is tall, every 15th is made of gold, every 28th has encrusted rubies and every 27th has an ancient description. Assuming that is universally true for all the chalices in the cave, which chalice should the adventurer check so she doesn’t waste too much time checking all of them?

The gcd of numbers $a$ and $b$ is $40$. What can be their smallest possible product? What could be their greatest possible product?

a) While visiting Cape Verde, Pirate Jim and Pirate Bob bought several chocolate chip cookies each. Jim paid 93 copper coins for his cookies and Bob paid 102 copper coins. What could be the price of a single cookie if it is a natural number?

b) Captain Hook and Captain Kid bought several tricorn hats each. Captain Hook paid 6 silver coins more than Captain Kid. What could be the price of a tricorn hat if it is an integer?

Show that $a-b$ is divisible by the greatest common divisor of $a$ and $b$.

a) Can you measure $6$ litres of water using two buckets of volumes $7$ and $10$ litres respectively?

b) Can you measure $7$ litres of water using buckets of volumes $9$ and $12$ litres respectively?