Bginner

Since $f(x)-f(x-2)=5x+1$, a linear function, $f$ must be a quadratic function.  Let $f$ be defined by $f(x)=px^2+qx+r$ for some constants $p$, $q$, $r$.  By substituting $x$ and $x-2$ into the formula for $f(x)$ we get $f(x)-f(x-2)=4px-4p+2q=5x+1$ for all $x$.  This gives two equations $4p=5$ and $-4p+2q=1$.  Solve for $p$ and $q$ and substitute them into $p+q$.
 

The above answer works for all $x$.  If we know that $x<5$, we have
$8x+5-|x-5|= 8x+5 - (5-x) = 8x+5-5+x= 9x$.
 

Wait!  The question is not asking for how many 5-digit palindromes there are.  It wants the sum of all those palindromes instead.   For example, were I to change the 5 to a 2, the situation would be like this.   There are nine 2-digit palindromes: 11, 22, 33, 44, 55, 66, 77, 88, and 99, but the sum of all the 2-digit palindromes is $11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99 = 495$.
 

The answer is the harmonic mean of 45 and 63.

Let $a$ be the number of students in class A;
$b$ be the number of students in class B;
and $c$ be the number of students in class C.
The assumptions give you three equations in 3 unknowns $a$, $b$, and $c$.
\begin{eqnarray*}
a+b &=& 75 \\
a+c &=& 61 \\
b/c &=& 5/3
\end{eqnarray*}
The answer is $a$ after you solve this system of equations.
 

Let 
\begin{eqnarray*}
f_1 &=& 1+x+x^2+\dots+x^{68}+x^{69} \\
f_5 &=& 1+x^5+x^{10}+\dots+x^{60}+x^{65} \\
f_{10} &=& 1+x^{10}+\dots+x^{50}+x^{60} \\
f_{25} &=& 1+x^{25}+x^{50}.
\end{eqnarray*}
Find the term for $x^{69}$ in the expansion of $f_1f_5f_{10}f_{25}$.  The answer is the coefficient of the term for $x^{69}$.
 

The answer can be written out as an expression in terms of $a$ alone or $b$ alone.    You need to know either the value of $a$ or $b$ in order to know the exact value of $a^2-b^2+2a$.
 

Multiply the expression $(6x-12)(x-8)$ out to a form $ax^2+bx+c$ where $a$, $b$, and $c$ are explicit constants.  Then substitute the values of $a$ and $b$ into the expression $-b/(2a)$ to get the answer.
 

$8x+5-\sqrt{(x-5)^2}$

It's important to know if letters can be repeated.