proof of existence While I was studying for my finals i found the following problem:
Let $f:[a,b]\rightarrow \mathbb{R}$ be a Riemann integrable real function, also let $\displaystyle \int\limits_a^bf(t)\,\mathrm{d}t=3$
Prove that there exist $\displaystyle t_1,t_2 \!\in \! (a,b):\int\limits_{t_1}^{t_2}f(t)\,\mathrm{d}t=1$
Could you give me any hint or a way to start the solution?

Hint: 
The function $F\colon [a,b]\to \mathbb{R}$ defined by $F(x) = \int_a^x f(t)dt$ is continuous, and $F(a)=0$, $F(b)=3$. Use the Intermediate Value Theorem on it.
Then:

 You can find $c\in(a,b)$ such that $F(c)=2$. Can you continue?

And then:

 Now, do a similar step with $G\colon [a,c]\to \mathbb{R}$ defined by $G(x) = \int_x^c f(t)dt$, to find $d\in(a,c)$ such that $G(d)=1$, i.e. $\int_d^c f(t)dt=1$.

