IKMC 2027 - Grades 7-8 - Self-study SET 12 - Thematic Batch 3

A lotus flower rises 25 cm above the surface of a lake. A strong wind bends it until its tip just touches the water, 55 cm horizontally from the point where the stem originally emerged, as shown. How deep is the lake at the lotus?

Editorial model: The stem is straight and keeps its length as it pivots about its fixed root.

The picture shows a network of traffic lights. All lights switch simultaneously from red (grey circles) to green (white circles), and vice versa. Juan is stopped at the red entrance light and wants to leave through the exit. What is the least number of times he must stop at a red light after passing the entrance light? Assume changes are so slow that he could traverse an all-green network without stopping.

Editorial interpretation: Count stops after the entrance light has been passed; the initial wait at the entrance is excluded. Including it would give three waits.

Choose two different numbers from {−3,−2,−1,0,1,2}, then multiply them. What is the probability that their product is 0?

Editorial assumption: Every pair of different numbers is equally likely.

Which of the following fractions is the smallest?

Amy and three friends study mathematics at one square table. Two more friends arrive, so Amy adds a second table to seat six people together. Later, two more join and she adds a third table to seat eight people, as shown. Following this arrangement, how many tables are needed for 24 people?

What is the units digit of $2^{2015}-2015^2$?

Triangle ABC is divided into five smaller triangles of equal area. Find $FD/AC$.

The three pictures show the same drum from three different angles. It is decorated with 12 straps. How many straps are white?

A cube is cut three times to form eight smaller solids. Find the ratio of the total surface area of the eight solids to the surface area of the original cube.

Editorial interpretation: Use the three full face-parallel cutting planes shown in the source figure.

Two identical dice have faces numbered 0,2,3,5,7,11. Roll both and add the numbers. What is the probability that the sum is prime?

Editorial assumption: Both dice are fair and the rolls are independent.

The picture shows the top-left corner of a 100 × 100 table. Amr cuts out a 2 × 2 block and replaces its entries by A,B,C,D as shown. B is 78 greater than A and C is 45 greater than A. What is D?

A rectangular sheet of paper measuring 6 cm × 12 cm is folded along its diagonal, as shown. The excess grey regions are cut off; unfolding produces a rhombus. How long is its side?

This year Ginny is 11 and has been admitted to Hogwarts. She and her brother Ron wait for the train to school at King's Cross station. She finds that three out of every ten children arriving at the station wait at Platform $9\frac34$.

1. If 50 children arrive, how many wait at Platform $9\frac34$?

2. Hogwarts holds a feast in the Great Hall to welcome new pupils. Ginny sees Headmaster Dumbledore wearing half-moon glasses and a long silver-white beard. “I wonder how old he is?” Ginny asks. Hermione replies playfully, “Age is inversely proportional to the number of years needed to complete all the charms in the magic book. I am 12, and if I need 55 years to complete them, at his age I would need only 6 years.” Ginny works out the respected headmaster's age. How old is Dumbledore?

Triangle ABC is divided into five triangles of equal area. Calculate $FD/AC$.

A cube of side 1 cm is placed on a cube of side 3 cm, forming the solid shown. Find its surface area.

A giraffe is inside a triangular enclosure with side lengths 20 m, 16 m and 12 m. It can reach 2 m beyond the fence to eat the grass in the shaded region. Find the area of this region.

Editorial numerical convention: The answer choices use π ≈ 3.14; the exact area is expressed using π in the solution.

The picture gives two equations using a smiling face and a sad face. Find the sum of the two face values.

A class has 16 boys and 10 girls. 75% of the boys and 50% of the girls sign up for skiing. How many pupils sign up altogether?

A test has 30 questions. Ruth answers 50% more questions correctly than incorrectly. Each answer is either correct or incorrect. How many does she answer correctly?

Two equilateral triangles of side 6 cm overlap with their sides pairwise parallel, as shown. What is the perimeter of the shaded hexagon?