AMO 2020 Secondary 1 (Grade 7)

Find the least positive integer $N$ such that $10100N$ is a perfect square.

Teams A, B, C, D and E have played the games shown: each edge represents one game. How many more games are needed for every pair of teams to have played exactly twice?

Evaluate $\dfrac{99\times9}{11}+\dfrac{99\times9}{1.1}+\dfrac{9.9\times9}{1.1}+\dfrac{9.9\times9}{11}$, rounding to the nearest integer.

Find the sum of all numbers in the figure.

Find the sum of the digits of the sum of all integers from $-77$ to $205$, inclusive.

Colour the six faces of a cube so that faces sharing an edge have different colours. What is the fewest number of colours needed?

The speed of light is approximately 186000 miles per second. If $186000=2^A\times3^B\times5^C\times31^D$, find $(A+B)(C+D)$.

Let $[x]$ be the greatest integer not exceeding $x$; for example $[12.7]=12$, $[2]=2$, $[-3.2]=-4$. Find the square of the value of $\dfrac{[1120.111222]\,[30.456]}{[1200.987654321]}$.

$A,O,F$ lie on a straight line. Rays $OB,OC,OD,OE$ are ordered as shown. Angles $BOD$ and $COE$ are right angles, $\angle COD=32^\circ$, and $\angle AOB=\angle EOF$. Find $\angle AOB$ in degrees.

Choose one number from $\{2,4,6,8,10\}$ and one from $\{1,3,5,7,9\}$. In how many ways is their sum prime?

Starting at the top A and following the arrows downwards, how many different paths spell AMO2020?

The road network shows tolls in dollars. Find the greatest possible toll x, expressed in cents and restricted to multiples of 10 cents, for which route ACDE is strictly cheaper than every other route from A to E.

Use each digit 1, 2, 3, 4, 5, 6, 7 once to fill the boxes in $M=\boxed{\phantom{000}}-(\boxed{\phantom{00}}-\boxed{\phantom{00}})$, forming one three-digit number and two two-digit numbers. Find the greatest possible value of $M$.

A $4\times6$ rectangle is divided into unit squares, then two opposite corner squares are removed as shown. How many squares with integer side lengths have their complete boundaries in the remaining diagram?

The figures are made of shaded unit squares. Continuing the displayed pattern, how many shaded unit squares are in figure 28?

$A,B,C$ are distinct nonzero digits and $\overline{ABC}-\overline{CBA}=396$. How many values of the three-digit number $\overline{ABC}$ are possible?

Evaluate $\dfrac{1979^3-1900^3-79^3}{150100\times1979}$.

A two-digit number is 6 more than 7 times the sum of its digits. Its tens digit is 2 more than twice its ones digit. How many such two-digit numbers are there?

A jacket is sold at 30% off its regular price. Its sale price is then increased by $k\%$ to restore the original price. Find $k$, rounded to the nearest integer.

Stacy divides 53 marbles into three piles, each containing an odd number of marbles. How many different divisions are possible? Pile order does not matter; for example (1,1,51) and (1,51,1) are the same division.