AMO 2027 Grades 4-5 SET 10 - Self-study

Ashley had a length of string. She gave $\frac13$ of it to Daisu and $\frac14$ of the remainder to Jamie. What fraction of the string did Jamie get?

15 identical rectangles of perimeter 20cm each are arranged as shown below. Find the perimeter of the figure.

Find the number represented by $\square$ in:

$10+9+8\times7\div\square+7-5\times5-3\times2=3$.

The source uses a colon for division; the division sign above has the same meaning.

What is the largest number of pieces you can obtain by cutting a round cake with only 4 straight cut?

A. 15

B. 18

C. 21

D. 24

E. None of the above

Answer entry: enter a number of pieces, not an option letter. The source does not specify flat versus spatial cutting; see the model notes in the solution.

There is a total of 33 triangles and rectangles, and they are pairwise disjoint. They have a total of 113 vertices. How many triangles are there?

On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widgets, and on each succeeding day, she sold two more widgets than she had sold on the previous day. How many widgets in total had Janabel sold after working 20 days?

Max had 140 more stamps than Donna. If Max gave 15 stamps to Donna, he would have thrice as many stamps as Donna. How many stamps did Max have at first?

Charlie has 160 pieces of 50¢ and 20¢ stamps. The total value of all his stamps is 50 dollars. Find the number of 50¢ and 20¢ stamps Charlie has.

Answer entry: enter 50-cent count;20-cent count, separated by semicolons.

**Editorial input note:** Use no spaces before or after semicolons.

A few years ago, there were 4 Saturdays and 4 Wednesdays in the month of March. On which day in that year was 10th of March?

The mass of a container was 45kg when it was half-filled with sand. When 20kg of sand was added, it was $\frac56$ full. Find the mass of the empty container.

A book has 100 pages. How many digits are used to number all pages in the book?

Ages of the First 12 United States Presidents at the Beginning of Their Terms in Office

| President | Age (years) | President | Age (years) |
|---|---:|---|---:|
| Washington | 57 | Jackson | 62 |
| Adams | 62 | Van Buren | 55 |
| Jefferson | 58 | Harrison | 68 |
| Madison | 58 | Tyler | 51 |
| Monroe | 59 | Polk | 50 |
| Rooney | 58 | Taylor | 65 |

The table above lists the ages of the first 12 United States presidents when they began their first term in office. According to the table, what was the mean age, in years, of these presidents at the beginning of their terms? Round your answer to the nearest tenth.

The label “Rooney” is retained exactly as printed; this question uses the supplied table rather than verifying historical information.



Answer entry: use a decimal point, for example 12.3.

Mr. Lee left Town X in a van at 8.30 a.m. and traveled at a constant speed of 50 km/h. Three hours later, Mr. Wee took the same route by car and traveled at a constant speed. After traveling 300 km, Mr. Wee caught up with Mr. Lee.

a) What is Mr Wee’s speed?

b) How far apart would they be at 1 : 30 p.m.?

Answer entry: enter (a) speed in km/h;(b) distance in km, separated by semicolons.

**Editorial input note:** Use no spaces before or after semicolons.

Find the missing number in each of (a), (b) and (c).

Readable figure data: (a) each square lists top, left, right, bottom; centre. First: 12,18,3,12;45. Middle: 19,?,13,12;55. Last: 4,8,9,3;24. (b) Starting just clockwise from the top, the eight sectors read 11,14,16,17,22,?,8,34. (c) Each diamond lists top,left,right,bottom: (7,4,2,6), (9,8,1,9), (5,1,5,3), (6,1,?,2).



Answer entry: enter (a);(b);(c), separated by semicolons.

**Editorial input note:** Use no spaces before or after semicolons.

How many leap years are there from year 1899 to 2001?