Q1:
Find the number of ordered pairs of integers (x, y) such that x² + y² = 2013
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a,b and c are positive integers such that the simultaneous equations (a−2b)x=1, (b−2c)x=1and x+25=c have a positive solution for x. What is the minimum value of a?
f(x) is a polynomial satisfying f(x+2)−f(x)=(6x+4)^2 and f(0)=−16. Determine f(5).
An equilateral triangle ABC has AB=20root (3). P is a point placed in triangle ABC and D,E and F are the foot of the perpendiculars from P to AB, BC and AC, respectively. If PD=9 and PE=10, what is the value of the length of PF?
In a triangular base pyramid with base ABC and vertex S, all plane angles with vertex S are 90 degree. The areas of lateral faces SAB, SAC and SBC are 3,4 and 6 respectively. Find the volume of SABC.
a) 4
b) 5
c) 6
d) 12
5 people attend a party. At the end of the night, they each randomly grab a left shoe and a right shoe. The probability that each person leaves with exactly one of their own shoes can be expressed as a/b, where a and b are coprime positive integers. What is the value of a+b?
Three fair 6-sided dice each have their sides labeled 0,1,e,pi,i,sqrt(2). If these dice are rolled, the probability that the product of all the numbers on the top face is real can be expressed as a/b, where a and b are coprime positive integers. What is the value of a+b?
A straight road passes by a building. Three points A, B andC are on the road such that |AB|=|BC|=200 meters. The angle of elevation from points A, B and C to the top of the building are, 30∘, 45∘ and 60∘, respectively. The height of the building can be expressed as a√b, where a and b are positive integers, and b is not divisible by the square of any prime. What is a+b?
How many ways can the numbers 1,2,3,4,5,6,7 be arranged in a row such that the numbers in the 2nd, 4th, and 6th positions are each larger than both of their neighbours?
Triangle ABC has integer side lengths. Rectangles BCDE,ACFG,ABHJ are constructed so that CD=AC+AB, CF=AB+BC, and BH=(AC+BC)^2. If [ABHJ]=[BCDE]+[ACFG], how many different values can [ABC] have?
Triangle ABC has side lengths a,b and c. If these lengths satisfy a^2=a+2b+2c and −3=a+2b−2c, what is the measure (in degrees) of the largest angle?
An equilateral triangle ABC has AB=20√3. P is a point placed in triangle ABC and D,E and F are the foot of the perpendiculars from P to AB, BC and AC, respectively. If PD=9 and PE=10, what is the value of the length ofPF?
P is a point in triangle ABC. The lines AP,BP, and CP intersect the sides BC,CA, and AB at points D,E, and F, respectively. If [BDP]=10, [DPC]=16, [APB]=210, what is [APC]?
1. n and m are positive integers that satisfy n^3+2n^2=m^2. If 1≤n≤1000, how many possible pairs of (n,m) are there?
Let N=abc and M=cba, where a≠0 and c≠0. If N−M=297, how many possibilities are there for N?
abc does not mean a*b*c it means a 3 digit no abc
Consider the function f(x)=(9^x)/(9^x+3). What is the value of
f (1/401)+f (2/401)+............+f (399/401)+f (400/401 )?
5 balls are to be put in 3 boxes. In how many ways can this be done if:-
a)Balls are identical and boxes are different
b)balls are different and boxes are identical
c)both boxes and balls are different
d)both boxes and balls are identical
in which of the above cases the formulae (n+r-1)C(r-1) be applied?
Please explain WITH EXPLANATIONS. I am not very good at this topic. Thanks
Suppose f(x) is a degree 8 polynomial such that f(2^i)=1/2^i for all integers 0≤i≤8. If f(0)=a/b,where a and b are coprime positive integers, what is the value of a+b?
If A is the sum of the squares of the first n natural numbers (where n
(a) 40 (b) 60 (c) 59 (d) 39