IKMC 2027 - Grades 7-8 - Self-study SET 16 - Thematic Batch 3

Gina can afford to buy 4 sweets chosen from 3 different flavours. In how many ways can she buy the 4 sweets?

A group of girls stands in a circle. Xena is fourth to Yena’s left and seventh to Yena’s right. How many girls are in the group?

The edges of a polyhedron form its frame. Cyril has pieces of wire of lengths 1 cm, 2 cm, 3 cm, 4 cm, 5 cm, 6 cm and 7 cm. What is the smallest number of pieces needed to make the frame of a cube of edge length 1? The wires must not overlap or cross.

The sum of 8 consecutive odd integers is 128. What is the largest of them?

In the figure, $DP=PC$, $AQ=QB$ and $\angle(AD,CB)=90^\circ$. Find PQ.

A doctor, a farmer, a lawyer and a teacher are Andrew, Bill, Cynthia and Della, in some order. Andrew is neither the doctor, the farmer nor the teacher. Bill is neither the doctor nor the farmer. Which statement is true?

Agent Ethan enters a secret base. To open a lock made of three circular parts, he must align the correct letter or number from each part at the position marked “Code”. Each part has the letters or numbers shown. How many rotations are needed to guarantee opening the lock?

Editorial convention: one trial means testing one complete aligned three-part code, as in the source answer.

A ribbon is wrapped around a transparent plastic cube as shown. What can the cube look like after its faces are painted white, making it opaque, and the ribbon is wrapped in the same way?

Each vertex and each edge of a tetrahedron is labelled with one of 1,2,3,4,5,6,7,8,9,11, each used once. The label of any edge is the sum of its endpoint labels. Edge AB is labelled 9. What is the label of CD?

Evaluate $\sqrt{0.04^3}$.

John played 500 smartphone games last week and won 25% of them. This week his skills improved: he won half of his games and brought his overall win rate to 40%. How many games did he play this week?

Editorial clarification: the 40% rate is the cumulative rate over both weeks.

Each painter has six colours: blue, red, purple, green, yellow and orange. The table lists pairs that must not be mixed. At most how many pairs can a painter mix if each pair uses two different colours and no colour can be used twice?

Pablo says, “There are at least four Rembrandt paintings in the exhibition.” Leonardo says, “No, there are at most three.” Vincent says, “There is at least one.” Exactly one of the three is correct. How many Rembrandt paintings are there?

Farmer Tom must take a wolf, a sheep and a cabbage across a river. The boat can carry Tom and at most one of the three. Without Tom, a wolf left with the sheep eats it, and the sheep left with the cabbage eats it. Answer the following to find a route using the fewest crossings.

1. Which two of the three may safely be left together?
2. Which should Tom carry on his first outward trip?
3. Suppose on outward trip 1 Tom carries the sheep across, leaves it there and returns; on outward trip 2 he carries the wolf across. What should he carry back on return 2, and carry out on outward trip 3?
4. How many ways are there to transport all three across?

Editorial convention: part 4 counts shortest routes; distinguish the source’s outward-trip numbering from one-way crossings. Answer all four parts.

Luke glues 7 dice together into the shape shown. Faces that are glued together have the same number of pips. He then paints the assembled shape. How many pips are painted over?

Silke’s pond contains 960 fish. Because of pollution, half the surviving fish die every 3 hours. After 15 hours only 3.125% remain. After how many hours will exactly 15 fish remain?

Simon marks points dividing a thread into 9 equal pieces. Barbara marks points dividing it into 8 equal pieces. Carl cuts at every point marked by Simon or Barnara. How many pieces result?

A hiking map has scale 1:50000. Patrick’s route from start to finish measures 18 cm on the map. How long does the journey take at an average speed of 6 km/h?

Editorial convention: 18 cm represents the travelled route on the map.

Three circles are placed in a rectangle of length 6 m as shown. Adjacent circles are tangent. Find the shaded area.

The equal circles and their tangencies to the rectangle are read from the supplied figure, as in the source solution.

What is the smallest possible difference in days between Pi Day (14 March) and Kangaroo Day (the third Thursday in March)?