IKMC 2020 - Grades 11-12 - Brazil Second Application - Self-study

What is the last digit of $1\times3\times5\times7\times9^2\times7\times5\times3\times1$?

An ant normally walks 6 m along the straight segment AB. Johnny puts an upright cylinder of height 1 m on its path. The ant now follows the same line in horizontal projection, climbing vertically up the cylinder, crossing its top, and climbing down as shown. How far does it walk?

How many integers lie strictly between $2020.9^2$ and $2018.9\times2022.9$? (The source uses decimal commas.)

Evaluate $\dfrac{1010^3-2020^3+3030^3}{1010^3}$.

The pie chart gives the populations of five city zones. The central zone has as many inhabitants as the north, west and east together. The south has half as many as the west. What is the difference between the north and east percentages of the total population?

Integers satisfy $1\le a=b\le c$ and $abc=2020^2$. What is the largest possible value of $a$?

Which number is divisible by 3 for every integer n?

A dozen nanica bananas cost the same as a dozen prata bananas. If y nanica bananas cost x reais, how many reais do z prata bananas cost?

Source defect: no printed option is generally correct. Give the correct price formula.

Printed choices: A. $xyz$; B. $\frac56xyz$; C. $\frac{6xz}{5y}$; D. $\frac{5yz}{6x}$; E. $\frac{6yz}{5x}$

Each of two identical fair dice has two red, two blue and two green faces. What is the probability of obtaining different colours when both are rolled?

In the shown addition CAN+GUR=UUU, different letters stand for distinct decimal digits and leading digits are nonzero. What is the largest possible C+A+N?



Source defect: the true maximum is absent from the printed options. Give the numerical maximum.

Printed choices: A. 15; B. 18; C. 19; D. 21; E. 24

A gray square of area 36 cm² and a black square of area 25 cm² overlap as shown. A vertex of the black square lies on the gray lower side,2 cm from the lower-left corner. The black top side meets the gray right side 1 cm below the top. What is the perimeter of the white overlap?



Editorial note: the source tilt conflicts with the dimensions. Use the stated lengths and incidences; the solution shows valid parallel squares.

2020 coins start heads up. Each move flips exactly three distinct coins. What is the fewest moves needed to make every coin tails up?

Zilda builds the shown solid with eight outer faces from six equal cubes and two different rectangular blocks. Painting every piece completely before gluing takes 18 litres. How many litres would painting only the outside of the glued solid take, at the same rate per area?

Nonzero real numbers satisfy $(a-a^{-1})^2+(b-b^{-1})^2+(c-c^{-1})^2=0$. Which cannot equal $a+b+c$?

The last two digits of a 2020-digit positive integer are 99. What is the greatest possible number of digits in its square? Editorial clarification: the source phrase “a 2020 number” is interpreted as “a 2020-digit number”.

A sequence satisfies $f_1=1$, $f_2=2$ and $f_n=f_{n-1}f_{n+1}$ for $n\ge2$. How many of its first 2020 terms are even integers?

Matias writes 15 numbers around the wheel; the top number is 10. Every seven consecutive positions have the same sum. Which sum is possible?

The large tilted square touches two smaller squares standing on one line, as shown. Their areas are 9 and 4. Find the area of the tilted square.

A circle is tangent to the lower side of a rectangle and passes through its two upper vertices, as shown. A square of area 20 cm² has its lower side on the upper rectangle side and its upper vertices on the circle. Find the rectangle area. The depicted upper side of the rectangle is a diameter of the circle.

Two rectangular blocks and a cube form the larger rectangular block shown, of volume 280 cm³. The dark cube has volume 125 cm³; the smaller other block has volume 75 cm³. Find the area of the face marked?.