In a group of dogs and chickens, the number of legs was 28 more than twice the number of heads. How many dogs were there? (assume that none of chickens and dogs is missing a leg.)
Let S be the set of positive integers that divide at least one of the numbers 1; 11; 111; 1111; ... For example, 3 is in S since 3 divides 111 . The number of elements in S that are less than 100 is
(a) 22 (b) 28 (c) 34 (d) 40 (e) 48
Two circles, one of radius 8 and one of radius 18, are tangent (i.e., they intersect at exactly one point). There are two lines each of which is tangent to both circles, as shown in the diagram. What is the distance from the intersection of these lines to the center of the circle with radius 8 ?
How many pairs (x, y) of integers satisfy x4 - y4 = 16 ?
Let S be the set of positive integers that divide at least one of the numbers {1, 11, 111, 1111, ...}. For example, 3 is in S since 3 divides 111 . The number of elements in S that are less than 100 is: (a) 22 (b) 28 (c) 34 (d) 40 (e) 48
In the figure below, three congruent circles are tangent to each other and to the sides of an equilateral triangle of side length \(a\) as shown. What is the radius of the circles?
Find x:
\(\sqrt{x+1-4\sqrt{x-3}}+\sqrt{x+6-6\sqrt{x-3}}=1\)
Two cylindrical candles of the same height but different diameters are lit at the same time. The first is consumed in 4 hours and the second in 3 hours. Assuming that they burn at a constant rate, how long after being lit was the first candle twice the height of the second candle?
Three circles of equal size are inscribed inside a bigger circle of radius 1, so that every circle is tangent to every other circle. What is the radius of each of the smaller circles?
Suppose a, b, and c are positive integers with a < b < c such that \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=1\). Calculate a + b + c?
Suppose a and b are two different real numbers, and the function f(x) = x2 + ax + b satisfies f(a) = f(b). What is the value of f(2)?
There are 10 sticks in a bag. The length of each stick is an integer (measured in inches). It is not possible to make a triangle out of any three sticks from that bag. What is the shortest length (in inches) the longest stick could possibly be?
Given three circles of radius 2, tangent to each other as shown in the following diagram, what is the area for the shaded region?
Find a.b given that:
\(a=\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{...}}}}\)
\(b=\sqrt{9-\sqrt{9-\sqrt{9-\sqrt{...}}}}\)
