1A. Marin, Maksim, Matej, and Marjan are four brothers. Their mother caught them eating all the cookies she had baked. To her question, "Who ate the cookies?", they gave the following statements:

Marin: "We all ate the cookies."

Maksim: "Marin and I together ate as many cookies as Matej and Marjan together."

Matej: "Marjan ate more cookies than me."

Marjan: "But, Marin and I together ate fewer cookies than Maksim and Matej together."

Mother knows that all four of them are telling the truth. Who ate the most cookies?
Solution. First method. Let Marin, Maxim, Matej, and Marjan take $a, b, c$, $d$ buns respectively. Then, the statements they made are:

Marin: ,, $a, b, c, d>0$ “,

Maxim: ,, $a+b=c+d$ ",

Matej: ,, $d>c$ ",

Marjan: ,, $a+dc$ we have that $a<c<d<b$, which means Maxim took the most buns.

Second method. From Matej's statement, we conclude that Matej is not the one who took the most buns, because Marjan took more than him. Let's assume that Marjan took the most buns. This means Marjan took more or the same number of buns as Maxim, so Maxim took fewer or the same number of buns as Marjan. From Maxim's statement, we conclude that Matej took fewer or the same number of buns as Marin, so that Marin and Maxim together have an equal number of buns as Matej and Marjan together. Then, Maxim and Matej together have fewer or the same number of buns as Marjan and Marin together, which
contradicts Marjan's statement, so Marjan cannot have taken the most buns.

Let's assume that Marin took the most buns. Then, Marin took more or the same number of buns as Matej. From Maxim's statement, we conclude that Marjan must have taken more or the same number of buns as Maxim. However, then Marjan and Marin together will have more or the same number of buns as Matej and Maxim together, which contradicts Marjan's statement, so Marin cannot have taken the most buns.

We conclude that Maxim is the one who took the most buns.