IKMC 2027 - Grades 11-12 - Self-study SET 1 - Thematic Batch 3

A convex polygon has exactly 702 diagonals. How many vertices does it have?

A school has 100 pupils, 10 of whom have previously had Covid-19. Three pupils are selected at random. What is the probability that at least one of them has not previously had Covid-19?

Original printed choices: A) 0,9; B) 0,95; C) 0,99; D) 0,995; E) 0,999

Editorial note: Explain whether the printed question and choices are consistent. Give the literal mathematical result and distinguish any intended correction.

It is now 8:25 in the morning. What is the measure of the acute angle formed by the minute hand and the hour hand?

Original printed choices: A) 102,5⁰; B) 150,5°; C) 103,5°; D) 152,5°; E) 105,5°

Editorial note: Explain whether the printed question and choices are consistent. Give the literal mathematical result and distinguish any intended correction.

Three three-digit numbers are formed using the digits 1 through 9, each exactly once. Which of the following values cannot be their sum?

The sequence ($x_{n}$) is defined by $x_{1}$=a−b, $x_{2}$=c−d and $x_{n+1}$=$x_{n}$−$x_{n−1}$ for every n≥2. What is the sum of its first 2021 terms?

A square of side 2 is divided into four triangles as shown. The three grey triangles have equal areas. What is the area of the white triangle?

The sequence ($a_{n}$) begins with $a_{1}$=123. If $a_{n}$ is even, $a_{n+1}$=$a_{n}$/2+1; if $a_{n}$ is odd, $a_{n+1}$=$a_{n}$−5. What is $a_{2022}$?

Let m be the smallest positive integer whose square is the sum of 1000 consecutive positive integers. What is m?

A real function f satisfies f (1)=2 and f (x+y)=f (x)f (y) for all real x, y. Evaluate f (2)/f (1)+f (3)/f (2)+…+f (2021)/f (2020).

Which of the following point sets represents the solutions (x; y) of cos x=cos y on coordinate axes?

The vertices of a cube are assigned the positive integers 1 through 8, each used once. The sums at the vertices of three faces are 15,18 and 19. Which triple can be the sums on the other three faces?

Editorial note: The source repeats a word in its Vietnamese statement; it is retained.

How many distinct real solutions does ||$4^{x}$−3|−2|=1 have?

A sequence of positive numbers ($a_{n}$) satisfies $a_{n}$²=(n−3)$a_{n+1}$+2 n+3 for every positive integer n. What is $a_{1}$?

A triangle is partitioned into smaller triangles as shown. The number inside each small triangle is its perimeter. What is the perimeter of the large triangle?

Consider f:(−1,∞)→(−∞,1), x↦f (x)=x/(x+1). Which graph is the reflection of its graph in the line y=x?

In how many ways can 7 identical balls be distributed among boxes A, B, C, if A contains at most 4 balls, B at least 2 balls, and C either 1 or 2 balls?

In the diagram, triangle ABC has area 8√2. What is the volume of the cube?

Alex says that every pomegranate has 613 seeds. Julia does not believe this, so she buys a pomegranate at a fruit shop and counts its seeds. She finds exactly 652 seeds and thus establishes that Alex’s statement is incorrect. They then make some other predictions: every pomegranate has 652 seeds; no pomegranate has 613 seeds; some pomegranate has exactly 652 seeds; every pomegranate has 613 or 652 seeds; some pomegranate has exactly 613 seeds. How many of these predictions are certainly true?

In isosceles triangle ABC, points D, E, F lie on BC, CA, AB and divide those sides in the ratio 1:2 as shown. M and N are respectively the intersections of AD with CF and BE. The area ratio FMNB: ABC is a/b in lowest terms. What is a×b?

Two straight rods lean against a rectangular doorway as shown. Their supporting endpoints are at heights u and v above the floor. What is the height of the rods’ intersection above the floor?