MathDB

p1. Determine all pairs of non -negative integers

(a, b, x, y)

that satisfy the system of equations:

a + b = xy

x + y = ab

p2. Find all pairs of real numbers

(x, y, z)

that satisfy the system of equations

x=1+\sqrt{y-z^2}

y=1+\sqrt{z-x^2}

z=1+\sqrt{x-y^2}

p3. A man has

6

friends. One night at a restaurant, he met with each of them

11

times, every

2

of them

6

times, every

3

of them

4

times, every

4

of them

3

times, every

5

of them

3

times, and all of them

10

times. He eats out

9

times without meeting them. How many times did he eat at the restaurant in total?
[url=https://artofproblemsolving.com/community/c6h2371607p19389327]p4. Given an acute triangle

ABC

. Point

H

denotes the foot of the altitude drawn from

A

. Prove that

AB + AC \ge BC cos \angle BAC + 2AH sin \angle BAC

p5. It is known that

p_0 = 1

and

p_i

is the i-th prime number, for

i = 1, 2, ...

namely

p_1 = 2

p_2 = 3

...

. The prime number of

p_i

is said to be moderate if

p_i^{(n^2)} >p_{i-1} (n!)^4

for all positive integers

n

. Determine all moderate prime numbers.

Problem Statement