3. a, b are roots of x2+ax+b=0, g, d are roots of x2-ax+b-2=0. Given 1/a + 1/b + 1/d + 1/g
=5/12 and abdg = 24, find the value of the coefficient ?a?.
4. x + y + z = 15 and xy + yz + zx = 72, prove that 3 £ x £ 7.
5. Let l, a Î R, find all the set of all values of l for which the set of linear equations has a non-trivial solution.
lx + (sin a) y + (cos a) z = 0
x + (cos a) y + (sin a) z = 0
-x + (sin a) y - (cos a) z = 0
If l = 1, find all values of a.
6. Prove that for each posive integer 'm' the smallest integer which exceeds (Ö3 + 1)2m is divisible by 2m+1.
7. Prove that, for every natural k, the number (k3)! is divisible by (k!)k2+k+1.
8. Prove that the inequality: n=1år { m=1år aman/(m+n)} ³ 0. ai is any real number.
9. Prove the following inequality: k=1ån Ö[nCk] £ Ö[n(2n-1)]
10. A sequence {Un, n ³ 0} is defined by U0=U1=1 and Un+2=Un+1+Un.Let A and B be natural numbers such that A19 divides B93 and B19 divides A93.Prove by mathematical induction, or otherwise, that the number (A4+B8)Un+1 is divisible by (AB)Un for n ³ 1.
11. The real numbers a, b satisfy the equations: a3 + 3a2 + 5a - 17 = 0, b3 - 3b2 + 5b + 11 = 0. Find a+b.
12. Given 6 numbers which satisfy the relations:
y2 + yz + z2 = a2
z2 + zx + x2 = b2
x2 + xy + y2 = c2
Determine the sum x+y+z in terms of a, b, c. Give geometrical interpretation if the numbers are all positive.
13. Solve: 4x2/{1-Ö(1+2x2)}2 < 2x+9
14. Find all real roots of: Ö(x2-p) + 2Ö(x2-1) = x
15. The solutions a, b, g of the equation x3+ax+a=0, where 'a' is real and a¹0, satisfy a2/b + b2/g + g2/a = -8. Find a, b, g.
16. If a, b, c are real numbers such that a2+b2+c2=1, prove the inequalities: -1/2 £ ab+bc+ca £ 1.
17. Show that, if the real numbers a, b, c, A, B, C satisfy: aC-2bB+cA=0 and ac-b2>0 then AC-B2£0.
18. When 0<x<1, find the sum of the infinite series: 1/(1-x)(1-x3) + x2/(1-x3)(1-x5) + x4/(1-x5)(1-x7) + ....
19. Solve for x, y, z:
yz = a(y+z) + r
zx = a(z+x) + s
xy = a(x+y) + t
20. Solve for x, n, r > 1
| xCr | n-1Cr | n-1Cr-1 | | x+1Cr | nCr | nCr-1 | | x+2Cr | n+2Cr | n+2Cr-1 | | = 0 |
21. Let p be a prime and m a positive integer. By mathematical induction on m, or otherwise, prove that whenever r is an integer such that p does not divide r, p divides mpCr.
22. Let a and b be real numbers for which the equation x4 + ax3 + bx2 + ax + 1 = 0 has at least 1 real solution. For all such pairs (a,b), find the minimum value of a2+b2.
23. Prove that:
2/(x2 - 1) + 4/(x2 - 4) + 6/(x2 - 9) + ... + 20/(x2 - 100) =
11/((x - 1)(x + 10)) + 11/((x - 2)(x + 9)) + ... + 11/((x - 10)(x + 1))
24. Find all real p, q, a, b such that we have (2x-1)20 - (ax+b)20 = (x2+px+q)10 for all x.
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