HKIMO 2026 Grade 7 National Final (Heat) Set 05

Roses, lilies and tulips cost 16,8 and 1 dollars each, respectively. Amy buys 30 flowers for 300 dollars, with at least one of each type. How many roses does she buy?

Following the pattern, find the next term: $1,2,3,6,11,20,37,68,\ldots$.

Find the integer part of $\displaystyle\frac 1{\frac 1{10}+\frac 1{11}+\frac 1{12}+\cdots+\frac 1{19}}$.

Six teams A,B,C,D,E,F play a round-robin tournament over five days. Every team plays once per day and every pair meets once, so there are three games daily. A beats B on day 1; D beats C on day 2; E beats A on day 3; B beats C on day 4. Which team plays A on day 5?

Seven people make these statements about today. A: Yesterday was Wednesday. B: Tomorrow will be Tuesday. C: Tomorrow will be Wednesday. D: The day after tomorrow will be Tuesday. E: Today is Tuesday. F: Today is neither Monday, Tuesday nor Sunday. G: Today is not Saturday. Exactly one statement is true. What day is today?

For real $a$, $|a+2|+|a+3|=7$. Find the sum of all possible $a$.

Evaluate $1\times 2\times 3+2\times 3\times 4+\cdots+10\times 11\times 12$.

Convert $16/37$ to a recurring decimal.

How many positive integers $a$ satisfy $3 a+7\le a+19$?

Find the value of $1+\frac 13+\frac 19+\frac 1{27}+\cdots$.

Positive integer $n$ makes $n-7,n^2+2,n-2$ all prime. Find $n$.

Find the remainder when $2015^{2015}$ is divided by 8.

Natural numbers $x,y$ satisfy $\frac 7{10}<\frac xy<\frac{11}{14}$. When $y$ is as small as possible, find $x$.

How many positive fractions less than 1, in lowest terms, have denominator 2015?

Positive integers $x<y$ satisfy $xy+x+y=19$ and $x^2 y+xy^2=84$. Find $x^2+y^2$.

Combine 2015 equilateral unit triangles to form a quadrilateral. Find the greatest possible perimeter. Editorial assembly convention: triangles have disjoint interiors, meet edge-to-edge, and form one filled quadrilateral.

The figure shows cube $ABCDEFGH$. If $AF=4\sqrt 3$, find its volume.

In the diagram, $ABCE$ is a square of side 20. Points D and F lie on CE and EA, respectively, with $CD:DE=EF:FA=1:2$. Lines BD and FC meet at G. Find $AG$.

A circle is inscribed in a square and circumscribed about a smaller square. The larger square has side 4 cm. Find the area of the smaller square in square centimetres.

The legs of a right triangle are in ratio 1:7. Its hypotenuse is 10 cm. Find its area in square centimetres.