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

An ant stands at each vertex of a cube. Each ant either always tells the truth or always lies. More than half of the ants are liars. Two ants are neighbours when their vertices share an edge. Every ant says: “All three ants next to me are liars.” Exactly how many ants are liars?

Editorial note: The source’s Vietnamese typographical variants all refer to ants.

I draw a 3 × 3 square on paper. I have four patterned pieces: one grey 1 × 3 piece and three 2 × 1 pieces, one with stars, one with dots and one plain white, as shown. In how many ways can all four pieces cover the 3 × 3 square?

Tony’s aquarium contains two goldfish and three angelfish. He wants to give Anna some fish, including at least two angelfish. Find the number of possible selections through the following questions.

(1) Choosing $k$ elements from $n$ elements, where $k,n\in\mathbb N$, is called a combination, denoted $C_n^k$. Using $C_n^k=\frac{n(n-1)\cdots(n-k+1)}{k(k-1)\cdots1}$, how many ways can Tony choose 1 fish, 2 fish, 3 fish, 4 fish, and 5 fish from the five fish? Hence find the total number of ways to choose fish from his five fish.

(2) How many ways can Tony choose at least one fish with no angelfish included?

(3) How many ways can Tony choose goldfish, including choosing zero? How many ways can he choose one angelfish from the three? Hence find the number of selections from the five fish containing exactly one angelfish.

(4) How many ways can Tony choose fish to give Anna with at least two angelfish included?

Given $\frac dc=4$, $\frac cb=3$, and $\frac ab=\frac23$, find $\frac ad$.

How many triangles are there in the figure below?

Trapezoid $ABCD$ is divided into four triangles of areas $S_1,S_2,S_3,S_4$ as shown. Given $S_2=3S_1$, which statement is true?

Evaluate $4^{2014}+4^{2014}+4^{2014}+4^{2014}$.

Dr John’s laboratory has six types of bacteria: A, B, C, D, E and F. When placed together, A eats C; F eats C and D; C eats E; D eats B; and B eats E. Which listed combination allows the greatest number of types to be eaten?

A robot originally costs 125 EUR. I buy it after three successive discounts, each reducing the immediately preceding price by 20%. How much do I pay?

In the figure, the orange square block ABCD rests on the green triangular block PGR and the purple triangular block MNL. Initially AN = 4 cm. If the green block is pushed so that G moves to the midpoint of the original segment AG, what is the new length AN?

Original printed choices: A) 6cm; B) 8cm; C) 10cm; D) 12cm; E) 14cm

Editorial task: Solve the literal problem, identify the source discrepancy, and explain any intended correction.

Three boats set off at the same time, each at a constant speed. The speed difference between the first and second equals that between the second and third. The first boat finishes in 36 minutes and the last in 45 minutes. How many minutes does the second boat take?

Editorial note: As intended in the source solution, all three boats cover the same course.

Two regular hexagons $ABCDEF$ and $KLMNOP$ are positioned as shown, with O the midpoint of EF. If the perimeter of ABCDEF is 12 cm, what is the perimeter of KLMNOP?

A 6 × 6 square grid is shown in Figure 1. What is the minimum number of cells that must be coloured so that the 3 × 1 rectangle in Figure 2 cannot lie entirely within uncoloured cells?

Editorial note: As a grid-cell placement problem, the rectangle follows grid lines and may be horizontal or vertical.

Students who arrive late must stay at school longer: 10 minutes for the first late arrival, 20 for the second, 30 for the third, 40 for the fourth, and so on. During the winter term Simon stays a total of 9 hours 10 minutes. How many times was he late?

A four-layer tower is built from 30 unit cubes of side 1 cm as shown. The top layer contains 1 cube, the second 4, the third 9 and the bottom 16. All exterior faces, including the bottom, are painted.

(1) Find the area of each face of a unit cube.

(2) What shape do we see when looking at the tower directly from above? Find the visible area from above.

(3) What shape do we see when looking at the tower from below? Find the visible area from below.

(4) What shape do we see when looking at the tower from its four sides? Find the area visible from all four sides.

(5) What is the total painted area of the tower?

Triangle ABC has $\angle A=68^\circ$. Its three angle bisectors meet at I. Find $\angle BIC$.

Fill the empty cells in the grid with the numbers 1,3,4,5,6,7,8 and 9 so that every row, column and main diagonal has the same sum. What is the sum of the two grey cells?

Original printed choices: A) 12; B) 10; C) 9; D) 8; E) 7

Editorial task: Solve the literal problem, identify the source discrepancy, and explain any intended correction.

A cylindrical glass is 10 cm high and partly filled with water. The glass is shown in two positions. What is the height of the water when the glass stands upright?

A cube has edge length 4 cm. Painting its edges requires 0.7 g of paint. The cube is cut into small cubes with edge length 1 cm. How many grams of paint are needed to paint all edges of all the small cubes?

Editorial note: This translation follows the explicit Vietnamese edge-painting wording; the parallel English source uses the ambiguous word “sides”. Treat the small cubes separately and use a constant paint amount per unit edge length.

Three squares are placed side by side as shown. Segments AE and HC meet at P. Find $\angle CPE$.