CEMC Diagnostic

Consider the quadratic equation \(x^2 - (r+7)x + r + 87 = 0\) where \(r\) is a real number. This equation has two distinct real solutions \(x\) which are both negative exactly when \(p < r < q\), for some real numbers \(p\) and \(q\). The value of \(p^2 + q^2\) is

The sum of four different positive integers is 100. The largest of these four integers is \(n\). The smallest possible value of \(n\) is

If \(x\) is 20% of \(y\) and \(x\) is 50% of \(z\), then what percentage is \(z\) of \(y\)?

For how many integers \(m\) does the line with the equation \(y = mx\) intersect the line segment with endpoints \((20, 24)\) and \((4, 202)\)?

If \(4x + 12 = 48\), the value of \(x\) is

How many positive integers \(n\) with \(n \le 100\) can be expressed as the sum of four or more consecutive positive integers?

In the diagram, point \(Q\) lies on \(PR\) and point \(S\) lies on \(QT\). What is the value of \(x\)?

In the diagram, each of the circles with centres \(X\), \(Y\) and \(Z\) is tangent to the two other circles. Also, the circle with centre \(X\) touches three sides of rectangle \(PQRS\) and the circle with centre \(Z\) touches two sides of rectangle \(PQRS\), as shown. If \(XY = 30\), \(YZ = 20\) and \(XZ = 40\), the area of rectangle \(PQRS\) is closest to

An unpainted cone has radius 3 cm and slant height 5 cm. The cone is placed in a container of paint. With the cone's circular base resting flat on the bottom of the container, the depth of the paint in the container is 2 cm. When the cone is removed, its circular base and the lower portion of its lateral surface are covered in paint. The fraction of the total surface area of the cone that is covered in paint can be written as \(\dfrac{p}{q}\) where \(p\) and \(q\) are positive integers with no common divisor larger than 1. What is the value of \(p + q\)? (The lateral surface of a cone is its external surface not including the circular base. A cone with radius \(r\), height \(h\), and slant height \(s\) has lateral surface area equal to \(\\pi rs\).)

The value of \(\dfrac{20-20}{20+20}\) is