AMO 2027 Grades 8-9 SET 19 - Self-study

Whole numbers are arranged five at a time in each row, as shown. What is the sum of the row number and the column number of 2017?

Given that $f:\mathbb R\to\mathbb R$ is a quadratic polynomial with $f(1)=1$, $f(2)=\frac12$, and $f(3)=\frac13$, find $f(4)$.

How many times does the digit 0 appear in the list of integers from 1000 to 2021, including both endpoints? Count every occurrence, even if a number contains more than one zero. Enter the integer only.

Find the sum of the digits of the product $2017\times99999$.

Original printed choices: A. 201697983; B. 36; C. 201700000; D. 35; E. 10.

**Response format:** This self-study version uses numerical input. Enter the digit sum itself, not an option letter; the printed choices are retained for source comparison.

In the figure, $AB$ is a diameter of the circle, and $C,D$ lie on the same side of it, in the order shown. If $\angle ABC=75^\circ$ and $\angle DAC=33^\circ$, find $\angle DCA$ in degrees.

Find the integer $X$ satisfying all these conditions:
- $X<5000$.
- Its digits, read from left to right, are in strictly increasing order.
- $X-3$ is a perfect cube.
- $X$ is not divisible by 8 and is not prime.

**Necessary convention for this self-study version:** $X$ is positive and has at least two digits. The original source does not state a digit-length restriction. Enter the integer only.

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.

Two vectors in three-dimensional Euclidean space are $4\hat i+9\hat j-13\hat k$ and $7\hat i+6\hat k+5\hat j$. Find their dot product.

For how many whole numbers from 4002 to 4499, including both endpoints, does the product of the middle two digits exceed 6? Enter the integer only.

How many positive three-digit integers have exactly three positive factors? Enter the integer only.

Find the positive difference between the sum of all possible integers $x$ and the sum of all possible integers $y$ satisfying $x^y=81$.

**Convention used by the source solution and by this exercise:** List every integer solution pair $(x,y)$ once, and add its first and second coordinates separately. Thus a repeated coordinate is counted once for each pair in which it occurs.

What is the greatest number of regions into which seven straight lines can divide the interior of a circle?

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

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

A rhombus has side length 17 and one diagonal of length 16. Find the length of its other diagonal. (The source's phrase “has a length of 17” is interpreted as side length, as in its solution.)