If p and q are primes and has \(x^2-px+q=0\) distinct positive integral roots, find p and q .

Let a,b be the roots of equation \(x^2-px+q=0\) and let c,d be the roots of the equation \(x^2-rx+s=0\), where p,q,r,s are some positive real numbers. Suppose that :

\(M=\frac{2\left(abc+bcd+cda+dab\right)}{p^2+q^2+r^2+s^2}\)

is an integer. Determine a,b,c,d .

The rectangle ABCD is divided into 4 regions whose perimeters are indicated in the figure below,where X,Y,Z Are Distinct positive integers and X>Y .It is known that Z=\(\frac{X+Y}{3}\)and W<6.Find X

The rectangle ABCD is divided into 4 regions whose perimeters are indicated in the figure below,where  X,Y,Z are distinct positive integers and X>Y .It is known that Z=\(\frac{Z+Y}{3}\)and W<6.Find X

Let a,b be the roots of equation \(x^2-px+q=0\) and let c,d be the roots of the equation \(x^2-rx+s=0\), where p,q,r,s are some positive real numbers. Suppose that : 

                \(M=\frac{2\left(abc+bcd+cda+dab\right)}{p^2+q^2+r^2+s^2}\)

 is an integer. Determine a,b,c,d .

1) The rectangle has length p and breath q (cm), where p and q are intergers. If p and q satisfy the equation pq+q=13 + q²

then the maxnium area of the rectangle

2) Let a,b and c be positive intergers such that ab + bc=518 and ab-ac=360. Find the largest value of the product abc.

P/s: As you may now, These are some questions from the 8 round of Math Violympic. Plz help me as much as you can! Thanks for all!

Given the polynomial P= x^2 + ax +b. Find the values of a and b such that b has 2 roots that are 2 and 3.

a= -5 and  b = -1

a= -1 and  b = -6

a= 1 and  b = 1

a= -5 and  b = 6

(giải giùm mình nha các bạn)

every positive integer can be expressed as a sum of distinct of 2. note that 1 and 2 are power of 2. How many three - digit numbers are sums of exactly 9 distinct power of 2

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Find all primes p such that the number p²+11 has exactly six different divisors (including 1 and the number itself)

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