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1. Sum of squares divided by 365 ●○○○○ 10 ptsWhat is the value of the expression $\frac{10^{2}+11^{2}+12^{2}+13^{2}+14^{2}}{365}$?
2. Sum of squares from means ●○○○○ 10 ptsReal numbers $x$ and $y$ have an arithmetic mean of 18 and a geometric mean of $\sqrt{92}$. Find $x^2+y^2$.
3. Sum of squares after two increments ●●○○○ 20 ptsEach of 100 given numbers was increased by $1$. Then each number was increased by $1$ once more. Given that after the first increase the sum of the squares of the numbers did not change, find by how much the sum of the squares changed after the second increase.
4. Alternating block sum of squares ●●○○○ 20 ptsCompute $100^{2}+99^{2}-98^{2}-97^{2}+96^{2}+95^{2}-94^{2}-93^{2}+\ldots+4^{2}+3^{2}-2^{2}-1^{2}$.
5. Minimum sum of squares after error ●●○○○ 20 ptsAshok multiplied three real numbers. He used 73 instead of 37 for one of the numbers, and as a result the product increased by 720. Find the minimum possible value of the sum of the squares of the other two numbers.
6. Quadratic equation in two variables ●●○○○ 20 ptsSolve the equation $x^{2}-2x+y^{2}-4y+5=0$ for real numbers $x$ and $y$. Find $x$ and $y$.
7. Alternating sum of minimum values ●●●○○ 50 ptsFor a positive integer $n$, define $f_n(x)=\left(\sum_{i=1}^{n}|x-i|\right)^2-\sum_{i=1}^{n}(x-i)^2$ on the interval $[0,n+1]$. Let $a_n$ be the minimum value of $f_n(x)$. Find $\sum_{n=1}^{11}(-1)^{n+1}a_n$.
8. Sum of squared tangents in a tetrahedron section ●●●●○ 100 ptsA regular tetrahedron is intersected by a plane that is perpendicular to one of its faces and meets that face along a line $e$. The intersection lines of this plane with the other three face planes of the tetrahedron form angles $\varphi_1, \varphi_2, \varphi_3$ with $e$. Determine the value of $\tan^2 \varphi_1+\tan^2 \varphi_2+\tan^2 \varphi_3$.