BMO Round 2 Practice Paper

  1. The smallest even number greater than $2$ that cannot be expressed as the sum of two prime numbers is ____.
  2. The maximum prime number which cannot be expressed in the form $n^2+1 (n\in\mathbb{Z})$ is ____.
  3. Give any $n (n>2)$ points $A_1, A_2, …, A_n$ on the plane, the distance between any two points is less than 1, than $\max{\min_{1\leq i< j \leq n}{A_iA_j}}=$____.
  4. The equation $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ has no positive integer solutions for $x,y,z$, then the minimum value for the positive integer $n$ is ____.
  5. Let $\sigma(n)=\sum_{d|n, d\in\mathbb{Z}^+}d$, then the minimum odd number satisfies $\sigma(n)=2n$ is ____.
  6. Given that $f(x)\in Z\left[x\right]$ and $f(e+\pi)=0$, then $\deg f(x)=$ ____.
  7. Let $p_n$ be the $n^{th}$ prime number from small to large, than the sum of all positive integer $n$ satisfies $p_{n+1}-p_{n}=2$ is ____.
  8. Given a positive integer $m$, if any $n$ points on the plane must contain $m$ points which can form $m$ vertices of a convex $m$ polygon, than $\min{n}=$ ____.
  9. Prove: $\int_{0} ^{1} (\Sigma_{x=1} ^N e^{2\pi ix^k \alpha}) ^2(\Sigma_{x=1} ^N e^{-2\pi ix^k \alpha})d\alpha=0$.
  10. $\forall n\in \mathbb{N}^*, f(n)=\lbrace^{3n+1, n\ is\ odd} _{\frac{n}{2}, n\ is\ even} $. For any given positive integer $n$, calculate the smallest possible integer $k$, makes $f^{(k)}(n)=1$.
  11. Let$\zeta(s)=\Sigma_{n=1} ^{+\infty} \frac{1}{n^2} (s\in C)$, prove: except for negative real numbers, all zeros of $\zeta(s)$ contain real part $\frac{1}{2}$.