MYTS 2016 - Grade 8 - Round 1

If $a-b=1$ and $a^3-b^3=1$, what is $a^4-b^4$?

How many digits does $201520162017\times 201820192020$ have?

Real numbers $a, b, c$ satisfy $a+b+c=2$. Find the maximum of $2 ab+bc+ca$.

Let $N=1\underbrace{00\cdots 00}_{2015}1\underbrace{00\cdots 00}_{2016}38$. By how much is N greater than the largest perfect square strictly below it?

A taxi company charges a 10000-dong opening fee and 7000 dong per kilometer, then 5000 dong per kilometer beyond the first 20 km. An pays 230000 dong. How many kilometers does she travel?

The polynomial $x^3+ax+b$ leaves remainder 4 on division by $x-3$ and remainder 6 on division by $x-4$. Find a+b.

Cut the figure along the grid lines into two congruent pieces. Sum the numbers on each piece. What is the larger sum?

A natural number N is generous if divisible by every digit and by the digit sum. For example 12 is generous, but 102 is not. Find the smallest generous number divisible by 11.

Chau cuts equal squares from the four corners of a 40 cm square metal sheet, then folds along the dashed lines to make a square-based tray as shown. What cut-square side length in cm maximizes its volume?

A bag contains 25 marbles in three colors: blue, red, yellow. Drawing 21 without looking is necessary to guarantee all three colors. What is the greatest possible number of yellow marbles?

Real numbers $a, b, c, d$ satisfy $a/2=b/4=c/6=d/(8+b)$. Find the minimum of a+b+c+d.

Four equal circles are mutually tangent in a row and tangent to the sides of a square as shown. A larger circle touches the middle two small circles and the opposite square side. If each small radius is 2 cm, find the large radius in cm.

Numbers $a, b, c, d, e, f, g, h$ are the eight distinct elements of $\{-7,-5,-3,-2,2,4,6,13\}$. Find the minimum of $(a+b+c+d)^2+(e+f+g+h)^2$.

In how many ways can 8 sweets be distributed to Lam, Hung, Vinh so each receives at least one and no sweet is broken?

Find the last two digits of $7^{107}$.

Four friends are playing and one breaks a vase. Lan says,“It was not me!” Van says,“Yen did it!” Hang says,“No, Van did it!” Yen says,“Van is lying.” Exactly one tells the truth. Who broke the vase?

Three face areas of a rectangular cuboid are 30,40,75 cm². What is its volume in cm³?

Real numbers $x, y, z$ satisfy $x+y+z=x^2-yz=18$. The maximum x is written as $2\sqrt a-b$; find a+b.

Each edge of a cube is colored blue, red or yellow so that every face contains all three colors. What is the minimum number of blue edges?

At 9 o’clock, the angle between the hour and minute hands is 90°. What is the angle between them at 10:42, in degrees?