Reference no: EM13583384
1. The sum of some consecutive integers is 2012. Find the smallest of these integers.
2. How many 5-digit numbers have at least one digit of 5?
3. Find all integers n for which n2 - n + 1 divides n2012 + n + 2001.
4. Let a and b be real numbers such that 2a2 + 3ab + 2b2 ≤ 7. Prove that max(2a + b, a + 2b) ≤ 4.
5. Find all pairs (m, n) of integers such that m3 + n3 = 2015.
6. Let P (x) = 3x3 - 9x2 + 9x. Prove that P (a2 + b2 + c2) ≥ P (ab + bc + ca) for all real numbers a, b, c.
7. For a positive integer N , let r(N ) be the number obtained by reversing the digits of N. For example, r(2013) = 3102. Find all 3-digit numbers N for which r2(N ) - N 2 is the cube of a positive integer.
8. Solve the system of equations
log xy = 5/log z, , log yz =8/log x, , log zx =9/log y
9. In circle C chords AB and XY intersect at P .
Prove that the pro-jections of P onto AX, BX, AY , BY are concyclic if and only if the midpoints of AX, BX, AY, BY are the vertices of a rectangle.
10.(a) Give example of triple (a, b, c) of even positive integers such that ab + 1, bc + 1, ca + 1 are all perfect squares.
(b) Are there triples (a, b, c) of odd positive integers such that ab + 1, bc + 1, ca + 1 are all perfect squares?