AMO 2027 Grade 8 SET 5 - Self-study

A semiprime is a number $p_1p_2$, where $p_1,p_2$ are primes; they may be equal. Three independent fair standard six-sided dice are rolled. The probability that their sum is semiprime is $a/b$ in lowest terms, with $a,b>0$. Find $a+b$.

Determine the number of non-negative integers $M$ satisfying exactly three of the following four statements:
1. $M$ is prime.
2. $M+3$ is prime.
3. $1\lt \sqrt M\lt 8$.
4. $M+5$ has an odd number of factors.

**Answer format:** Enter the count of admissible integers, not one example of $M$. Factors mean positive integer divisors; 1 is not prime.

The diagonals of kite $KITE$ are perpendicular. Its perimeter is $2(\sqrt{x}+\sqrt{y})$ units. Find $x+y$.



**Convention:** $x$ and $y$ are the squares of the two different adjacent side lengths obtained from the right triangles in the figure; do not arbitrarily rewrite the radical sum. The diagonal intersection splits $KT$ into lengths 2 and 2, and $IE$ into lengths 3 and 4.

Compute $(75\times5\div3)+(75\div5\times3)$.

Andy and Brett start at opposite points on a circular track of circumference 600 feet and jog in the directions shown. Andy runs at 4.3 feet per second and Brett at 5.5 feet per second. How many seconds until they meet for the first time? The arrows indicate that both run clockwise. Choose one option.

How many points $(x,y)$ on the segment joining $X(-4,4)$ and $Y(12,-8)$ have both coordinates integers? Include both endpoints.

In the diagram, $ABC$ is right-angled at $C$, $AD=4$ and $CD=6$. Find $DB$ if $CD\perp AB$ and $D$ lies on $AB$.

What is the greatest three-digit number divisible by 5, 6, and 7? Choose one option.

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.

A metal cylinder is recast into a cone with twice the cylinder's radius. If the cylinder's height was 12 m, what is the new cone's height? Assume no metal is lost.

For a positive integer $n$, $n!=n(n-1)\cdots1$. For example, $4!=4\times3\times2\times1$. Find the greatest prime factor of $14!+16!$. Enter the integer.

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

Study the dot figures below. Continue the same pattern of adding a complete outer diamond layer. Let $N$ be the total number of dots from Figure 1 through Figure 31. Find $N/31$.



Enter the numerical value.

One straight line can divide the inside of a circle into two regions, as shown. What is the least number of straight lines needed to divide it into exactly 407 regions?



Enter the integer.

Betsy collects sports posters. Exactly one of the following statements is false:
- Betsy has at least 11 sports posters.
- Betsy has at least 20 sports posters.
- Betsy has at least 24 sports posters.
- Betsy has at least 28 sports posters.

What is the greatest possible number of posters in Betsy's collection? Enter the integer only.