AMO 2027 Grades 8-9 SET 25 - Self-study

56% of 57 is the same as 76% of what integer?

In the figure below, $PSR$, $RTQ$ and $PAQ$ are three semicircles of diameters 10cm, 3cm and 7cm respectively. Given that the area of the shaded region is $k\pi$. Find $k$.

The number 1000 can be written as the product of two positive integers, neither of which contains zeros. The sum of these two integers is

Thirty-six points are arranged in a unit-square array as shown. Find the sum of the interior angles, in degrees, of figure $ABCDEF$.

The digits in 5512 are rearranged to form all possible distinct 4-digit positive integers. Find the last three digits of the sum of all such integers.

Compute the sum of all the numbers found in the figure.

Square $ABCD$, shown here, has side length 8 units and is divided into four congruent squares. One of these squares contains an inscribed circle, two other squares contain diagonals and the fourth square has perpendicular line segments drawn from the midpoints of adjacent sides to form a square in the interior. In square units, what is the total area of the shaded regions? Take $\pi=22/7$.

If $2016=2^A\times3^B\times7^C$ for positive integers $A,B,C$, what is the simplified value of $A^B+C$?

Thirty-six points are arranged in a unit-square array as shown. Find the area of the 8-sided figure in square units.

Use the standard factorial definition $n!=n(n-1)\cdots2\cdot1$. Simplify
$$\frac{20!+21!}{2\times19!+4\times19!+16\times19!}.$$
Source clarification: the printed definition is $n!=n\times(n-1)!\times\cdots\times2\times1$, with an extra factorial sign after $(n-1)$. We use the standard definition stated above.

What is the smallest positive integer which cannot occur as the difference between two positive prime numbers?

Find the value of $$\sum_{j=2}^{\infty}\left(\sum_{i=1}^{\infty}\frac1{9^i}\right)^{j/3}.$$

The diagram shows 28 lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$. Find the length of segment $AE$.

$MNPQ$ is a square formed by 3 identical rectangles. The perimeter of each rectangle is 240 cm. How long is $PQ$?

In the diagram, $ABC$ is a right triangle. Given that $DE$ is perpendicular to $AC$, $AB=5$ cm, $AC=13$ cm and $DE=3$ cm, find the area of the polygon $ABED$.