IKMC 2027 - Grades 11-12 - Self-study SET 8 - Thematic Batch 3

Numbers a and b satisfy ab=a−b. What is the value of a/b+b/a−ab?

Domain note: The requested expression is defined only for nonzero a and b.

Alex folds two edges of a unit square onto its diagonal as shown, producing the quadrilateral on the right. What is the area of that quadrilateral?

The Fibonacci sequence satisfies $F_{1}$=$F_{2}$=1 and $F_{n+2}$=$F_{n+1}$+$F_{n}$ for every positive integer n. How many of its first 2020 terms are even?

Alex has a solid assembled from four rectangular boxes as shown, each sharing a whole face with another box. Three marked faces have areas 6 cm²,15 cm²,35 cm². What is the area of the face marked x?

An equilateral triangle has side length 2. An arbitrary point is taken inside it. What is the probability that its distance from every vertex is greater than 1?

Original printed choices: A) 1 − 2π/3; B) 1 − π$\sqrt{3}$/6; C) $\sqrt{3}$/2 − π/4; D) $\sqrt{3}$/2 − π/2; E) $\sqrt{3}$ − π/2

Editorial task: Explain what information is missing or inconsistent, then derive the source’s intended answer under clearly stated additional assumptions.

Sam has four square cards, two grey and two white, with areas 1,8,16,32 cm². They are stacked in decreasing size as shown. What is the visible grey area?

Steve starts with 2018. On each move he may subtract 20 from the current number or divide it by 18. His goal is to reach 1. What is the least number of moves?

Let f : R → R be f(x)=|x+1|+|x+3|. Consider: (i) f is increasing on all positive real numbers; (ii) f is decreasing on all negative real numbers; (iii) f is constant on some interval longer than 1; (iv) f is positive for every x. How many statements are true?

Find the area of the region defined by y²$\lt$4−x² and y$\lt$−x.

Maria begins weaving on January 1 and makes 25 metres each day. At the end of every day she gives her sister one tenth of all the cloth she then has. After a gift, she first notices that more than 100 metres remain. What date is it?

How many positive integers below $10^{10}$ have digit sum 9?

Three parallel lines a, b, c pass respectively through vertices A, B, C of square ABCD. The distance between a and b is 7, and between b and c is 9. What is the square’s area?

For a real number k, let M(k) be the maximum of |4x²−4x+k| over −1≤x≤1. What is the smallest possible M(k)?

Points D, F, G lie respectively on sides AB, BC, CA of triangle ABC, with FG parallel to AB. Triangles GFC and GFD have areas 4 and 2. What is the area of ABC?

Alice, Bob, Chris, David, Emily stand in a row and recite a poem of 2022 lines. They read lines 1–5 in that order, then reverse: David reads 6, Chris 7, Bob 8, Alice 9. They reverse again: Bob reads 10, Chris 11, David 12, Emily 13, and continue this back-and-forth pattern. Who reads the final line?

Points A, B, C are marked at surface grid nodes of a 3×3×3 cube in each of the four pictures. In how many pictures is triangle ABC right-angled?

The pictured right triangles ABC and EDC overlap. Given MD=2020, DC=2021, MB=2019, what is area(ABC)/area(EDC)?

An island has 2017 people, each either a truthful knight or a liar. Seated around a circular table, each says: “Of my two neighbours, one is a knight and one is a liar.” How many knights are there?

Five predictions are made about a football match between A and B: (1) no draw; (2) B scores at least once; (3) B wins; (4) B does not lose; (5) exactly three goals in total. Exactly three predictions are true. What is the final score in the order B:A?

Two pet shops sell Krefft and Flatback turtles. The first has 50 Krefft and 10 Flatback turtles; the second has 60 Krefft and 20 Flatback turtles. Mandy randomly chooses one of the shops. What is the probability that she buys a Krefft turtle?

Original printed choices: A) 19/24; B) 11/14; C) 7/12; D) 5/8; E) 23/24

Editorial task: Explain what information is missing or inconsistent, then derive the source’s intended answer under clearly stated additional assumptions.