AMO 2027 Grade 7 SET 3 - Self-study

If 5 is added to both the numerator and denominator of $\frac6{11}$, how does the value of the fraction change?

**Input:** Choose one option.

Each number from 1 through 16 is written in exactly one box of the 4 × 4 grid. A path is formed by placing consecutive numbers in adjacent boxes horizontally or vertically, but not diagonally. Two numbers are shown. Find the smallest possible sum of the numbers in the starred (*) boxes.



**Input:** Choose one option.

A lattice point has integer coordinates. Form the set of points $(a,b)$ where both $a$ and $b$ are positive integers. How many of these points lie on the line $y=-4x+8$?

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

Which of the following Venn diagrams represents soccer players, high school Math enthusiasts, and students?

Editorial interpretation: “high school Math enthusiasts” means high school students who are enthusiastic about mathematics. Use the general membership relationships, not a particular surveyed group.

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Assume all sections within each given shape have equal area. Consider only the area of the shape itself (the flower, circle, etc.), excluding surrounding space. Which option represents the shaded fraction of at least one shape, in simplest form?



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Consider 7 points, no three of which lie on a straight line. How many different triangles can be formed by connecting three of these points as vertices?

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Choose a numerator and a denominator from $1,2,3,4,5,6,7,8,9,10$. Many resulting fractions are equal after reduction. How many distinct reduced fractions are strictly greater than $1/2$ and strictly less than $1$?

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

Nicholas writes the positive integers from 1 to 50 in an ordered list. On Monday he crosses out every multiple of 2; on Tuesday every multiple of 3; on Wednesday every multiple of 5; on Thursday every multiple of 7; on Friday every multiple of 11; and on Saturday every multiple of 13. How many of the original 50 integers are not crossed out?

**Input:** Choose one option.

Thirty-six points are arranged in a unit-square array as shown. Figure $ABCDE$ consists entirely of straight-line segments. If its perimeter is $x+y\sqrt5+z\sqrt{13}$ units, find $x+y+z$.



Here $x,y,z$ are the nonnegative integer coefficients after like radical terms have been collected.

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How many fractions in the list $\frac18,\frac19,\frac1{10},\frac1{11},\ldots,\frac1{64},\frac1{65},\frac1{66}$ have terminating decimal representations?

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The surface areas of two spheres are in the ratio 4:9. Find the ratio of their volumes, in the same order.

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Sasha wants an image to fit exactly and snugly into a box on her computer screen without squishing or stretching its proportions. The original image is $1\frac12$ inches wide and $2\frac14$ inches tall. The box is 4 inches wide. How tall must it be? All options are in inches.

**Input:** Choose one option.

Each of 10 cards displays two different symbols, one on the front and one on the back. The five symbols are $\clubsuit,\heartsuit,\spadesuit,\diamondsuit,\bullet$. The front of each card is shown. No two cards have the same unordered pair of symbols. What is the least number of cards you can turn over to ensure that $\bullet$ appears?



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

Lean cycles at constant speed along a straight street from A to B. After starting at A, he is $\frac5{17}$ of the way to B when he hears a car approaching from behind at 391 km/h. If he immediately turns back toward A, he meets the car at A; if he continues toward B, the car overtakes him at B. How fast does Lean cycle, in km/h? Assume both speeds stay constant and turning takes no time.

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

Compute the value of $\dfrac{2019^2-2000^2-19^2}{2000\times19}$.

**Input:** Choose one option.