AMO 2027 Grade 6 SET 3 - Self-study

A child gets 10 dollars in pocket money each day. He saves 1 dollar on the first day, 2 dollars on the second, 3 dollars on the third, 4 dollars on the fourth, and so on. On which day will he first have enough saved to buy his sister a 50-dollar school bag?

**Input:** Choose one option.

Four sisters paint at the same constant rate and take turns, with no overlap or gaps. Alice finishes $1/4$ of the job; then Beth finishes $2/3$ of what is left; then Cathy finishes $3/4$ of the new remainder. Dee completes the last part in 20 minutes. How many minutes does the entire job take?

**Input:** Enter the integer only, without units.

What is the value of $(115+114+113+112+111)-(101+102+103+104+105)$?

**Input:** Choose one option.

How many shortest paths along a 3 × 3 grid lead from one corner to the diagonally opposite corner?

Editorial clarification: use 3 rows and 3 columns of vertices, joined horizontally and vertically, as illustrated. This is a grid of 2 × 2 square cells. Travel from A to B along the drawn segments.



**Input:** Choose one option.

The numbers 1 through 12 are placed in the diagram, one in each circle, so that the sum of the numbers along each side of the triangle is the same. What is the largest possible value of this sum per side?



**Input:** Enter the integer only, without units.

A book uses 843 digits to number all its pages consecutively, starting with page 1. How many pages are in the book?

**Input:** Enter the integer only, without units.

Evaluate $(5\times7\times2\times17)\div(14\times34\times35)$.

Original printed options: A. $14$; B. $0.71$; C. $0.071$; D. $17$; E. $\dfrac1{17}$.

**Input:** For this self-study version, enter the exact value as a fraction a/b, without units, rather than an option letter or a rounded decimal. The original options are retained above for comparison.

A paper is folded as shown below and punched in its folded state. When the paper is opened out again, what will be the pattern of holes formed?

The original fold sequence and the four original pattern options are shown below.



**Input:** Choose one option.

How many 4-digit numbers have a digit sum of 5?

**Input:** Choose one option.

Assume all sections within each shape have equal area. Consider only the flower, circle or other shape itself, excluding its surroundings. Which shape has the smallest fraction shaded? Use the grey-filled regions in the original illustration.



**Input:** Choose one option.

How many whole numbers between 41 and 50 inclusive are multiples of either 2 or 3 or 5?

**Input:** Choose one option.

A ladder has 7 rungs. An ant climbs from A, the middle of the bottom rung, to B, the middle of the top rung, without retracing any part of its path. How many different paths are possible? Count routes along the rungs and side rails only, ignoring pauses; no segment may be traversed twice.



**Input:** Enter the integer only, without units.

Find the sum $150+195+255+210+120$.

**Input:** Choose one option.

The figure is formed by overlapping three rectangles measuring 2 cm by 6 cm. An ant completes one trip around its perimeter, moving at 1 cm/s on horizontal segments and 2 cm/s on vertical segments. How many seconds does the trip take?



**Input:** Enter the integer only, without units.

Each of the whole numbers 1 through 9 is written on a separate card. Three cards are given to each of three players. How many ways can the cards be given so that Player One’s cards sum to 8 and Player Two’s cards sum to 15? Cards within a player’s hand are unordered.

**Input:** Enter the integer only, without units.