Problem 1 :
Let α and β be the roots of the quadratic equation
x^{2} + 2bx + b = 1
The smallest possible value of (α - β)^{2}
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
Problem 2 :
It is known that n^{2012} + n^{2010} is divisible by 10 for some positive integer n. Which of the following numbers is not a possible value for n?
(A) 2 (B) 13 (C) 35 (D) 47 (E) 59
Problem 3 :
Using the vertices of a cube as vertices , how many triangular pyramid can you form?
(A) 54 (B) 58 (C) 60 (D) 64 (E) 70
Problem 4 :
AB is a chord of a circle with center 0. CD is the diameter perpendicular to the chord AB, with AB closer to C than to D. Given that ∠.AOB = 90° , then the quotient
area of △ABC/area of △AOD = ?
(A) √2 - 1 (B) 2 - √2 (C) √2/2 (D) (1+√2)/2 (E) 1/2
Problem 5 :
The diagram below shows that ABCD is a parallelogram and both AE and BE are straight lines. Let F and G be the intersections of BE with CD and AC respectively. Given that BG = EF, find the quotient DE/ AE.
Problem 6 :
Four circles each of radius x and a square are arranged within a circle of radius 8 as shown in the following figure .
Problem 7 :
Adam has a triangular field ABC with AB = 5, BC = 8 and CA = 11. He intends to separate the field into two parts by building a straight fence from A to a point D on side BC such that AD bisects ∠.BAC. Find the area of the part of the field ABD.
Problem 8 :
For any real number x, let ⌊x⌋ be the largest integer less than or equal to x and x = x - ⌊x⌋. Let a and b be real numbers with b ≠ 0 such that
a = b⌊a/b⌋ - b{a/b}
which of the following is incorrect ?
(A) If b is an integer then a is an integer;
(B) If a is a non-zero integer then b is an integer;
(C) If b is a rational number then a is a rational number;
(D) If a is a non-zero rational number then b is a rational number;
(E) If b is an even number then a is an even number.
Problem 9 :
Given that
is an integer . Which of the following is incorrect?
(A) x can admit the value of any non-zero integer;
(B) x can be any positive number;
(C) x can be any negative number;
(D) y can take the value 2;
(E) y can take the value -2.
Problem 10 :
Suppose that A, B, C are three teachers working in three different schools X , Y, Z and specializing in three different subjects: Mathematics , Latin and Music . It is known that
(i) A does not teach Mathematics and B does not work in school Z;
(ii) The teacher in school Z teaches Music;
(iii) The teacher in school X does not teach Latin;
(iv) B does not teach Mathematics.
Which of the following statement is correct?
(A) B works in school X and C works in school Y;
(B) A teaches Latin and works in school Z;
(C) B teaches Latin and works in school Y;
(D) A teaches Music and C teaches Latin;
(E) None of the above.
Problem 11 :
Let a and b be real numbers such that a > b, 2^{a} + 2^{b} = 75 and 2^{-a} + 2^{-b} = 12^{-1}. Find the value of 2^{a-b}
Problem 12 :
Find the sum of all positive integers x such that
is an integer.
Problem 13 :
Consider the equation
√(3x^{2}-8x+1) + √(9x^{2}-24x-8) = 3
It is known that the largest root of the equation is - k times the smallest root . Find k.
Problem 14 :
Find the four digit number
Problem 15 :
Suppose x and y are real number satisfying
x^{2} + y^{2} - 22x - 20y + 221 = 0. Find xy.
Problem 16 :
Let m and n be positive integers satisfying
mn^{2} + 876 = 4mn + 217 n
Find the sum of all possible values of m.
Problem 17 :
For any real number x, let ⌊x⌋ denote the largest integer less than or equal to x. Find the value of ⌊x⌋ of the smallest x satisfying ⌊x^{2}⌋ - ⌊x⌋^{2 }= 100.
Problem 18 :
Suppose x_{1}, x_{2}, .............x_{49} are real numbers such that
x_{1}^{2}+ 2x_{2}^{2}, .............49x_{49}^{2} = 1
Find the maximum value of x_{1}+ 2x_{2}, .............49x_{49}
Problem 19 :
Find the minimum value of
√x^{2} + (20 -y)^{2} + √y^{2} + (21 - z)^{2} + √z^{2} + (20 -w)^{2} + √w^{2} + (21 - x)^{2}
Problem 20 :
Let A be a 4 digit number. When both the first digit(left most) and the third digit are increased by n, and the second digit and fourth digit are decreased by n, the new number is n times A. Find the value of A.
Problem 21 :
Find the remainder when 1021^{I022} is divided by 1023.
Problem 31 :
How many triples of non-negative integers (x, y, z) satisfying the equation
xyz + xy + yz + zx + x + y + z = 2012?
Problem 35 :
There are k people and n chairs in a row, where 2≤ k < n. There is a couple among k people. The number of ways in which all k people can be seated such that the couple is seated together is equal to the number of ways in which the (k - 2) people without the couple present, can be seated. Find the smallest value of n.
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