Found problems: 60
Find the smallest number $n$ such that there exist polynomials $f_1, f_2, \ldots , f_n$ with rational coefficients satisfying \[x^2+7 = f_1\left(x\right)^2 + f_2\left(x\right)^2 + \ldots + f_n\left(x\right)^2.\]
[i]Proposed by Mariusz Skałba, Poland[/i]
If real numbers $a, b, c, d, e$ satisfy $a + 1 = b + 2 = c + 3 = d + 4 = e + 5 = a + b + c + d + e + 3$, what is the value of $a^2 + b^2 + c^2 + d^2 + e^2$ ?
Ana and Beto play against each other. Initially, Ana chooses a non-negative integer $N$ and announces it to Beto. Next Beto writes a succession of $2016$ numbers, $1008$ of them equal to $1$ and $1008$ of them equal to $-1$. Once this is done, Ana must split the succession into several blocks of consecutive terms (each term belonging to exactly one block), and calculate the sum of the numbers of each block. Finally, add the squares of the calculated numbers. If this sum is equal to $N$, Ana wins. If not, Beto wins. Determine all values of $N$ for which Ana can ensure victory, no matter how Beto plays.
Find the smallest number $n$ such that there exist polynomials $f_1, f_2, \ldots , f_n$ with rational coefficients satisfying \[x^2+7 = f_1\left(x\right)^2 + f_2\left(x\right)^2 + \ldots + f_n\left(x\right)^2.\]
[i]Proposed by Mariusz Skałba, Poland[/i]
Prove that there exist infinitely many integers $n$ such that $n, n + 1, n + 2$ are each the sum of two squares of integers.
a.) A 7-tuple $(a_1,a_2,a_3,a_4,b_1,b_2,b_3)$ of pairwise distinct positive integers with no common factor is called a shy tuple if $$ a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2$$and for all $1 \le i<j \le 4$ and $1 \le k \le 3$, $a_i^2+a_j^2 \not= b_k^2$. Prove that there exists infinitely many shy tuples.
b.) Show that $2016$ can be written as a sum of squares of four distinct natural numbers.
Find the smallest positive integer $n$ so that $\sqrt{\frac{1^2+2^2+...+n^2}{n}}$ is an integer.
We consider the sums of the form $\pm 1 \pm 4 \pm 9\pm ... \pm n^2$. Show that every integer can be represented in this form for some $n$. (For example, $3 = -1 + 4$ and $8 = 1-4-9+16+25-36-49+64$.)
Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.
Prove that if a natural number $N$ is represented in the form as the sum of three squares of integers divisible by $3$, then it is also represented as the sum of three squares of integers not divisible by $3$.