IKMC 2027 - Grades 7-8 - Self-study SET 1 - Thematic Batch 3

Kateina travels to school from Otrokovice to Olomouc. Previously, she paid the full fare of 75 EUR. Now she can use a 75% student discount code. How much must Kateina now pay?

A chain is made as shown below. How many links are needed to make a chain 1.7 m long?

The side lengths of the equilateral triangles follow a pattern. Find the side length of the grey triangle.

How many unit squares are needed to form a solid with the projections shown below?

Editorial note: The source says “unit squares”; the illustrated solid is made of unit cubes. The original wording is retained. The labels mean Front, Top and Side.

Tommy’s parents have important work to do, so they must leave Tommy at home alone. His mother lovingly prepares three special dishes, French, Russian and German, for his breakfast, lunch and dinner that day. In how many orders can he enjoy those special dishes during the day?

Using the four digits 0, 1, 2 and 3 exactly once each, how many dates of the form DD/MM can be displayed? For example, 3 December is displayed as 03/12.

Find the value of $2014\times2014:2014-2014$.

There are two unusual trees: one grows only green leaves and the other only red leaves. Each tree grows 4 additional leaves in 3 days. How many additional leaves do the two trees grow altogether in 12 days, given that they grow leaves at the same rate?

John sets his computer to display the date as “DD.MM.YYYY”. For example, 1 February 2020 is displayed as shown. During 2020, on how many dates can John see a date display using only the digits 0, 1 and 2?

The sum of the digits of the year 2021 is 5. Throughout the twenty-first century, how many years have a digit sum of 5?

Two unit squares of side length 1 share a vertex, as shown. One side of one square lies along a diagonal of the other square. What is the area of the grey region?

Richard counts on the fingers of his left hand in this order: little finger first, ring finger second, middle finger third, index finger fourth, thumb fifth, index finger sixth, middle finger seventh, ring finger eighth, little finger ninth, ring finger tenth, and so on. Which finger is number 2088?

Points $K,L,M,N$ are respectively the midpoints of $AD,AB,CB,DC$ of rectangle $ABCD$. Points $O,P,R,S$ are the midpoints of the sides of quadrilateral $KLMN$, as shown. What fraction of the area of rectangle $ABCD$ is shaded?

Anna, Ben, Catherine, David, Ella and Fred stand in a queue in that order to receive sweets. The clown gives sweets first to the child at the front, then to the child at the back, then to the first child at the front who has not yet received sweets, then to the last child at the back who has not yet received sweets, and so on until everyone has received sweets. Which statement below is NOT true?

Original printed choices: A) Ben receives sweets before David, but after Ella; B) Catherine receives sweets before Ella, but after Fred; C) Anne receives sweets after Ben, but before Ella; D) Fred receives sweets before Ella and Catherine; E) Fred receives sweets after both David and Ben

Editorial task: Resolve the printed question, explain any source inconsistency, and distinguish the literal result from a corrected or intended interpretation.

Positive integers $x$ and $y$ satisfy $(x+4)^2+y^2=(x+y)^2$. How many values of $x$ are possible?

Rectangle $ABCD$ is formed from smaller rectangles. Its perimeter is 90 m. What is the area, in square metres, of each small rectangle?

Editorial note: The intended interpretation is that the small rectangles are congruent; this condition is not stated explicitly in the source.

If the average of two positive numbers is 30% less than the larger number, by what percentage is that average greater than the smaller number?

Rabbit wants to row across the river to visit Squirrel but does not know the river’s width. Clever Rabbit devises a way to measure it while staying on his own bank. First he drives a stake at a point called A, as shown.
Rabbit stands at A and looks towards the tree across the river, calling its position P. From that direction he turns 90°, parallel to the riverbank, and walks straight to any point, where he drives a second stake B. He continues straight beyond B until his distance from B equals AB, then drives stake C. From direction CB he turns 90° and walks until stake B lines up with P, where he drives stake D. Rabbit measures CD = 10 m and uses it to calculate AP, the river’s width. How wide is the river?

Martha writes a multiplication of two two-digit numbers on paper. She then colours over three digits, as shown. What is the sum of the three covered digits?

In a match between the Silver Wolves and the Brown Bears, the Silver Wolves win. The Silver Wolves score 3 goals in the first half. In the second half, the two teams score a total of 2 goals. How many different final scores are possible?