AMO 2018 Secondary 1 (Grade 7)

Find the sum of all whole numbers from 41 to 50 inclusive that are multiples of 2, 3 or 5.

Simplify to a positive integer: $\dfrac{40}{3+\dfrac{4}{1-\frac15}}$.

Fedrick is twice as old as his sister Mandy. Mandy is 3 years older than her brother Ray. Their ages sum to 33. How old is Ray?

Find the integer nearest to $10-\sqrt{\dfrac{25}{0.51}}$.

A palindrome reads the same forwards and backwards. The number 2017102 is a 7-digit palindrome. Let $A$ be the least palindrome greater than 2017102, and $B$ the greatest palindrome less than it. Find $\dfrac{A-B}{10}$.

Seven points are given, with no three on one straight line. How many different triangles have three of these points as vertices?

Different letters represent different digits in the cryptarithm shown. Neither P nor C is zero. Find the product of the digits of the smallest possible value of PINT.

Nicholas lists the positive integers from 1 to 50. On Monday he crosses out every multiple of 2; on Tuesday, of 3; on Wednesday, of 5; on Thursday, of 7; on Friday, of 11; and on Saturday, of 13. How many numbers remain?

Two non-consecutive page numbers have sum 70 and difference 36. Find their product.

The 36 points form a square grid with unit spacing. Polygon ABCDEFG has straight sides as shown. Find its area in square units.

Find the 100th term of $265,349,820,520,133,\ldots$. Each term after the first is the sum of the cubes of the digits of the preceding term. For example, $2^3+6^3+5^3=349$.

Choose a numerator and a denominator from $1,2,\ldots,10$ to form fractions. After reducing to lowest terms, how many distinct fractions are strictly between $\frac12$ and 1?

Define $n!=n(n-1)\cdots2\cdot1$; for example, $4!=24$ and $3!=6$. Simplify $\dfrac{29!\times29!}{28!\times28!}$.

Each term of a sequence is obtained by adding the same constant to the preceding term. Its first term is 12 and its seventh term is 66. Find its 50th term.

Continue the displayed pattern indefinitely. Which number is directly below 406?

Given $2016=2^5\times3^2\times7$ and $2018=2\times1009$, find the number of distinct positive divisors of $2016\times2018$.

The 36 points form a square grid with unit spacing. Polygon ABCDE has straight sides as shown. Find the sum of its interior angles, in degrees.

The product of two primes is 9991. Find the smaller prime.

The 36 points form a unit square grid. Polygon ABCDE has straight sides as shown. Its perimeter is $x+y\sqrt5+z\sqrt{13}$ units. Find $x+y+z$, with $x,y,z$ the nonnegative integer coefficients in this form. Use the two right triangles as aids. Source note: the triangle with legs 2 and 3 is incorrectly labelled $\sqrt5$ in the original; its hypotenuse is $\sqrt{13}$.

Find the sum of the interior angles of a regular 7-sided polygon, in degrees.