ASMO Math 2022 — Grade 8 — School Round

Diagram below shows a parallelogram ABCD. N is the midpoint of AB and F is the midpoint of AD. Determine the ratio of the area of triangle AFN to the area of quadrilateral CDFN.

Let a, b and c be three real numbers such that $ab+bc+ac=4$. Determine the least possible value of $a^2+b^2+c^2$.

A lorry charges a rate of x cents per kg per mile. How much dollars (in cents) it cost 3 kg of goods to travel y miles, if the lorry driver decides to give his customer a 50% discount?

Determine the number of 7-digit positive integers such that the digits from left to right are non-increasing. (Examples of 7-digit non-increasing numbers are 5555555 and 9998766; An example of a number that is NOT non-increasing is 7776556)

Determine the number of ways of arranging 5 identical red balls and 13 identical blue balls on a straight line such that between any 2 red balls there is at least 1 blue ball.

Let $a_1,a_2,\ldots,a_n$ be positive integers, not necessarily distinct, with at least five distinct values. Suppose that for every pair of indices $1\leq i\lt j\leq n$, there exist indices $1\leq k\lt l\leq n$, both different from $i$ and $j$, such that

$$a_i+a_j=a_k+a_l.$$

Thus, the four indices $i,j,k,l$ are pairwise distinct. Find the smallest possible value of $n$.

Charly has 3 more 20 cents coins than 10 cents coins. The total value of her 20 cents coins is \$4.50 more than that of her 10 cents coins. How many 20 cents coins does Charley have?

The area of a rectangle remains unchanged when either its length is increased by 6 cm and width decreased by 2 cm, or its length decreased by 12 cm and its width increased by 6 cm. If the perimeter of the original rectangle is k cm, determine the value of k.

Lydia was born on 1 January 1995. Her father was born on 1 January 1963. In what year her father will be exactly 2 times as old as Lydia?

Two congruent isosceles triangles, ABC and EFG, overlap so that their bases are parallel and the vertex of each is the midpoint of the base of the other as shown. If the area of the overlap is 24cm2, what is the area of the triangle ABC?

It is given that $\sqrt{x}-\sqrt{y}=20$, where x and y are real numbers. Determine the maximum possible value of $x-5y$.

Find the largest prime number less than 5000 of the form $a^n-1$, where a and n are positive integers, and n is greater than 1.

A square piece of graph paper of side length 30 mm contains 900 smallest squares each of side length 1 mm each. Its four rectangular corners, denoted by A, B, C, D in clockwise order, are cut away from the square piece of graph paper. The resultant graph paper, which has the shape of a cross, is shown in the figure below. Let N denote the total number of rectangles, excluding all the squares which are contained in the resultant graph paper. Determine the value of $N/10$.

The numbers $1/1,1/2,\ldots,1/2010$ are written on a blackboard. A student chooses any two of the numbers, say p, q, erases them and then writes down $p+q+pq$. She continues to do this until only one number is left on the blackboard. What is this number?

Determine the remainder when $2^{55}+1$ is divided by 33.

Determine the sum of all the 5-digit integers which are not multiples of 11 and whose digits are 1, 3, 4, 7, 9.

Let a, b and c be integers with $a>b>c>0$. If b and c are relatively prime, $b+c$ is a multiple of a, and $a+c$ is a multiple of b, find the value of $abc$.

Determine the sum of all positive integers p such that the expression $(x-p)(x-13)+4$ can be expressed in the form $(x+q)(x+r)$ for distinct integers q and r.

It is given that $x=1/(2-\sqrt{3})$. Determine the value of $x^6-2\sqrt{3}x^5-x^4+x^3-4x^2+2x-\sqrt{3}$.

Figure below shows an equilateral triangle PRS. The area of triangle PRS is half that of the triangle PQR. Determine the angle PRQ in degrees.