Found problems: 5
Find the number of ordered quadruplets $(a_1,a_2,a_3,a_4)$ of integers, for which $a_1\geq 1$, $a_2\geq 2$, $a_3\geq 3$, and $-10\leq a_4\leq 10$ and $a_1+a_2+a_3+a_4=2011$ .
Find out the number of real solutions of $x^2e^{\sin x}=1$
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Given positive integers $k,n$ and a real number $\ell$, where $k,n \geq 1$. Given are also pairwise different positive real numbers $a_1,a_2,\ldots, a_k$. Let $S = \{a_1,a_2,\ldots,a_k, -a_1, -a_2,\ldots, -a_k\}$.
Let $A$ be the number of solutions of the equation
$$x_1 + x_2 + \ldots + x_{2n} = 0,$$
where $x_1,x_2,\ldots, x_{2n} \in S$. Let $B$ be the number of solutions of the equation
$$x_1 + x_2 + \ldots + x_{2n} = \ell,$$
where $x_1,x_2,\ldots,x_{2n} \in S$. Prove that $A \geq B$.
Solutions of an equation with only difference in the permutation are different.
Find all real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying,$$\sqrt{x_{1}-1^{2}}+2 \sqrt{x_{2}-2^{2}}+\cdots+n \sqrt{x_{n}-n^{2}}=\frac{1}{2}\left(x_{1}+x_{2}+\cdots+x_{n}\right)$$
Given a prime $p>3$ and an odd integer $n>0$, show that the equation $$xyz=p^n(x+y+z)$$ has at least $3(n+1)$ different solutions up to symmetry. (That is, if $(x',y',z')$ is a solution and $(x'',y'',z'')$ is a permutation of the previous, they are considered to be the same solution.)