AMO 2027 Grade 8 SET 4 - Self-study

Find $1-2+3-4+5-6+\cdots+99-100+101$.

The product of two or more consecutive integers is a six-digit palindrome beginning with the digits 47. Find the sum of those integers. A palindrome reads the same backwards and forwards. Use consecutive positive integers, as intended by the source.

Stacy lies on Tuesdays, Wednesdays and Thursdays and tells the truth on all other days. Anthony lies on Saturdays, Sundays and Mondays and tells the truth on the other days. On what day can both say, “Tomorrow, I will lie”?

Enter the weekday code: Monday = 001, Tuesday = 002, Wednesday = 003, Thursday = 004, Friday = 005, Saturday = 006, Sunday = 007. Leading zeros are optional.

Find the sum of all distinct prime factors of 656656.

The largest circle lying completely inside an equilateral triangle has radius 16 cm. Find the altitude of the triangle in cm. Enter the numerical length only.

Find $(9.87)^2+19.74\times0.13+0.0169$.

In the figure, the distance between neighbouring horizontal or vertical dots is 1 unit. Find the area of the triangle. Enter the numerical area in square units.

If the product of 450 and a positive integer $N$ is a perfect cube, what is the least possible value of $N$? Choose one option.

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

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

Evaluate $\sqrt{4^3}-\sqrt[3]{2^6}$.

Determine the integer closest to $\sqrt[3]{\frac{64}{0.03}}+2$. The $+2$ is outside the cube root. Choose one option.

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.

Find the greatest positive integer that divides 377, 127 and 852 with the same remainder in each case. Enter the integer.

Some of the 12 edges of a cube are coloured red so that every face has exactly three red edges. How many edges are red?