AMO 2027 Grades 6-7 SET 27 - Self-study

The mean of four consecutive odd numbers is 24. Find the sum of the middle two numbers.

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

Each of the whole numbers 1 through 9 is written on a separate card. Three cards are given to each of three players. How many ways can the cards be given so that Player One’s cards sum to 8 and Player Two’s cards sum to 15? Cards within a player’s hand are unordered.

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

How many 4-digit odd integers can be formed from the digits 0, 1, 4, 6 and 8 if no digit can be repeated?

**Input:** Choose one option.

The place cards shown are folded along the dotted lines so that only a number or a letter is visible. Chrissy enters the room and sees all six cards, some showing numbers and some showing letters. The visible numbers sum to 11. How many different sets of numbers are possible?



**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.

Let $N$ be the greatest possible negative quotient when two distinct integers chosen from $\{-7,-5,-4,-3,-2,0,1,2,6\}$ are divided. Find $213+245N$.

**Input:** Choose one option.

How many three-digit natural numbers are divisible by 7?

**Input:** Choose one option.

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?

**Input:** Choose one option.

Suppose $\frac2N$, $\frac3N$ and $\frac5N$ are all fractions in lowest terms. How many composite whole-number values of $N$ are there from 20 to 150 inclusive?

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

Find the value of $\displaystyle\frac{\frac1{2+\frac12}+\frac1{1+\frac14}}{1-\frac25}$.

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

How many 4-digit multiples of 37 are there in total?

**Input:** Choose one option.

Compute the simplified numeric value of $(16\div8\times4)$ subtracted from $(16\times8\div4)$.

**Input:** Choose one option.

How many 4-digit numbers have a digit sum of 5?

**Input:** Choose one option.

Faye and Ingrid are fruit vendors. Faye has 21 more fruits than $1\frac12$ times Ingrid’s number of fruits. Together they have 216 fruits. How many fruits does Faye have?

**Input:** Choose one option.

Jessica has a rectangle made of construction paper. She folds it in half to form another rectangle, then folds that rectangle in half to form a 6-cm-by-6-cm square. What is the area, in $\mathrm{cm}^2$, of the original rectangle?

**Input:** Choose one option.