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The sequence (an) is given by the conditions a1=1000000, an+1=nan/n+n. Prove that an infinite subsequence can be found within it, which is an arithmetic progression.

In the infinite sequence (xn), the first term x1 is a rational number greater than 1, and xn+1=xn+1xn for all positive integers n.

Prove that there is an integer in this sequence.

Note that in this problem, square brackets represent integers and curly brackets represent non-integer values or 0.

On the plane coordinate axes with the same but not stated scale and the graph of the function y=sinx, x (0;α) are given.

How can you construct a tangent to this graph at a given point using a compass and a ruler if: a) α(π/2;π); b) α(0;π/2)?

The sequence a1,a2, is such that a1(1,2) and ak+1=ak+kak for any positive integer k. Prove that it cannot contain more than one pair of terms with an integer sum.The sequence a1,a2, is such that a1(1,2) and ak+1=ak+kak for any positive integer k. Prove that it cannot contain more than one pair of terms with an integer sum.

Prove that if the numbers x,y,z satisfy the following system of equations for some values of p and q: y=x2+px+q,z=y2+py+q,x=z2+pz+q, then the inequality x2y+y2z+z2xx2z+y2x+z2y is satisfied.

The nonzero numbers a, b, c are such that every two of the three equations ax11+bx4+c=0, bx11+cx4+a=0, cx11+ax4+b=0 have a common root. Prove that all three equations have a common root.

The teacher wrote on the board in alphabetical order all possible 2n words consisting of n letters A or B. Then he replaced each word with a product of n factors, correcting each letter A by x, and each letter B by (1x), and added several of the first of these polynomials in x. Prove that the resulting polynomial is either a constant or increasing function in x on the interval [0,1].

We are given a polynomial P(x) and numbers a1, a2, a3, b1, b2, b3 such that a1a2a30. It turned out that P(a1x+b1)+P(a2x+b2)=P(a3x+b3) for any real x. Prove that P(x) has at least one real root.