AMO 2027 Grades 4-5 SET 7 - Self-study

If numbers are arranged in 3 rows A, B and C according to the following table, which row will contain the number 1000?

A: 1, 6, 7, 12, 13, 18, 19, …

B: 2, 5, 8, 11, 14, 17, 20, …

C: 3, 4, 9, 10, 15, 16, 21, …

In the diagrams below, figure 1 has 5 small squares, figure 2 has 10 small squares and figure 3 has 17 small squares. Continuing the pattern, how many small squares does figure 20 have?

Diagram labels: Figure 1; Figure 2; Figure 3.

The figure below is divided into 8 congruent squares as shown. The area of the figure is 72cm². What is the length of the darkened border in centimeters?

Different letters correspond to different digits in the puzzle below. Given A = 6, T = 3 and M = 9, find the 4-digit number IJMO.

$\overline{MATH}+\overline{IJMO}=\overline{BINGO}$.



Answer entry: enter any valid four-digit value of IJMO.

In a basketball game, Max, Jackson, and Cole scored 106 points altogether. Max scored 15 more points than Jackson. Jackson scored 7 fewer than Cole. How many points did Cole score?

The amount of money a performer earns is directly proportional to the number of people attending the performance. The performer earns 120 dollars at a performance where 8 people attend. The performer uses 43% of the money earned to pay the costs involved in putting on each performance. The rest of the money earned is the performer's profit. What is the profit the performer makes at a performance where 8 people attend?

Two suspects, Adam and Ben, are on trial for shoplifting. Four witnesses are called in during the trial. Below are the witnesses’ testimonies.

• 1st witness: ”I know Adam is innocent.”

• 2nd witness: ”I know Ben is innocent.”

• 3rd witness: ”The facts presented by the first and second witnesses are true.”

• 4th witness: ”I am very certain that the third witness is lying.”

If only the 4th is telling the truth, which suspect(s) is/ are guilty of shoplifting?

In a 3-digit number, the hundreds digit is one half of the ones digit. The tens digit is one more than the ones digit. If the sum of the digits is 11, find the number.

Logan weighed 3 full bottles of milk and 2 empty bottles and got 145 grams. He then weighed 4 full bottles of milk and 5 empty bottles and got 240 grams. How many grams does one empty bottle weigh?

The bar graph shows the results when a die was thrown a number of times. How many sixes were thrown?

In the following alphabetic, all different letters stand for different digits. Find the four-digit sum PEEL.

$\overline{AM}+\overline{LAP}=\overline{PEEL}$.

James had some local and foreign stamps in the ratio 8 : 9. After he gave 88 local stamps to his brother, the ratio of the number of local stamps to the number of foreign stamps became 2 : 5. How many local stamps did he have at first?

After school, Maya and Naomi headed to the beach, 6 miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi’s and Maya’s average speeds?

Graph labels: Distance (miles); Time (minutes); Naomi; Maya.

Evaluate

a) $1+1\frac12+2+2\frac12+3+3\frac12+\cdots+9+9\frac12+10+10\frac12$

b) $\frac12+\frac14+\frac18+\frac1{16}+\frac1{32}+\frac1{64}$

c) $1-\frac13+2-\frac23+3-\frac33+4-\frac43+5-\frac53+6-\frac63$

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

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

Let n be a whole number greater than 1. It leaves a remainder of 1 when divided by any single digit whole number greater than 1. Find the smallest possible value of n.