Problems

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Found: 1936

The real numbers a,b,c are non-zero and satisfy the following equations: {a2+a=b2b2+b=c2c2+c=a2. Show that (ab)(bc)(ca)=1.

Find the representation of (a+b)n as the sum of Xn,kakbnk for general n. Here by Xn,k we denote coefficients that depend only on k and n.

The positive real numbers a,b,c,x,y satisfy the following system of equations: {x2+xy+y2=a2y2+yz+z2=b2x2+xz+z2=c2

Find the value of xy+yz+xz in terms of a,b, and c.

A five-digit number is called indecomposable if it is not decomposed into the product of two three-digit numbers. What is the largest number of indecomposable five-digit numbers that can come in a row?

Let a and b be real numbers. Find a representation of a3+b3 as a product.

  • Find a representation of the number 117=1214 as a product.

  • Let a and b be real numbers. Find a representation of a2b2 as a product.

Find all solutions of the equation: x2+y2+z2+t2=x(y+z+t).

Solve the system of equations in real numbers: {x+y=2xyz2=1

Find all solutions of the equation: xy+1=x+y.

Find all solutions of the system of equations: {x+y+z=ax2+y2+z2=a2x3+y3+z3=a3