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First Round Challenge: A 90 minute, 25 multiple choice question first round Challenge aimed at students year 13 or below. The problems on the Senior Maths Challenge are designed to make students think. Stimulating problems for both beginners and experienced problem-solvers.
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Five line segments of length 2, 2, 2, 1, and 3 connect two corners of a square as shown in the diagram. What is the shaded area?
A drinks carton is formed by arranging four congruent triangles as shown. PQ = QR = 4 cm and PR = PS = QR = PQ = 10 cm. What is the volume, in cm³, of the carton?
What is the area of the part of the xy-plane within which x3y2 − x2y2 − xy4 + xy3 ≥ 0 and 0 ≤ x ≤ y?
Saba, Rayan, and Derin are cooperating to complete a task. When all three work together, it takes 5 minutes to complete the task. How many minutes does it take for Derin to complete the task on his own?
The numbers x, y, z, and w are all integers. x and y are variable, and z and w are constant and positive. The four integers are related by the equation xy = z(x + w). When y takes its maximum possible value, which expression is equal to y - x?
A square has its vertices on the edges of a regular hexagon. Two of the edges of the square are parallel to two edges of the hexagon, as shown in the diagram. The sides of the hexagon have length 1. What is the length of the sides of the square?
Let x be a real number. What is the minimum value of (x² - 4x + 3)(x² + 4x + 3)?
Three squares ABC, DEF, and GHI have vertices which sit on the sides of triangle JKL as shown. The squares have areas of 10, 90, and 40, respectively. What is the area of triangle JKL?
The length of a rectangular piece of paper is three times its width. The paper is folded so that one vertex lies on top of the opposite vertex, thus forming a pentagonal shape. What is the area of the pentagon as a fraction of the area of the original rectangle?
A triangle with interior angles 60°, 45°, and 75° is inscribed in a circle of radius 2. What is the area of the triangle?
How many pairs of integers (x, y) satisfy the equation √x - √y + 23 = 2√2 - y?
The numbers a and b satisfy the equations 2^a + 2^b = p and 2^a − 2^b = q. What is the value of 2^(a+b) in terms of p and q?
The perimeter of a logo is created from two vertical straight edges, two small semicircles with horizontal diameters, and two large semicircles. Both the straight edges and the diameters of the small semicircles have a length of 2. The logo has rotational symmetry. What is the shaded area?
Triangle PQR is equilateral. A semicircle with centre O is drawn with its diameter on PR so that one end is at P and the curved edge touches QR at X. The radius of the semicircle is √3. What is the length of QX?
Laura and Dina have a running race. Laura runs at constant speed and Dina runs n times as fast where n > 1. Laura starts B meters in front of Dina. What distance does Dina run before overtaking Laura?
A square is inscribed inside a quadrant of a circle. The circle has a radius of 10. What is the area of the square?
Triangle LMN represents a right-angled field with LM = r, LX = p and XN = q. Jenny and Vicky walk at the same speed in opposite directions along the edge of the field, starting at X at the same time. Their first meeting is at M. Which of these expressions gives q in terms of p and r?
The diagram shows two overlapping triangles: Triangle ABC with interior angles 60°, 30°, and 90°, and Triangle DEF, which is a right-angled isosceles triangle. What is the ratio of the area of Triangle ABC to Triangle DEF?
How many pairs of real numbers (x, y) satisfy the simultaneous equations x² - y = 2022 and y² - x = 2022?
The expression (7n + 12) / (2n + 3) takes integer values for certain integer values of n. What is the sum of all such integer values of the expression?
The number 840 can be written as p!/q!, where p and q are positive integers less than 10. What is the value of p + q?
The numbers x and y are such that 3^x + 3^(y+1) = 5√3 and 3^(x+1) + 3^y = 3√3. What is the value of 3^x + 3^y?
How many solutions are there of the equation 1 + 2 sin X − 4 sin2 X − 8 sin3 X = 0 with 0° < X < 360°?
A circle of radius r and a right-angled isosceles triangle are drawn such that one of the shorter sides of the triangle is a diameter of the circle. What is the shaded area?
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