HKIMO 2026 Grade 8 National Final (Heat) Set 05

Roses, lilies and tulips cost 16, 8 and 1 dollars respectively. Amy spends 300 dollars on 30 flowers, buying at least one of each kind. How many tulips does she buy?

Find the next number in the pattern $1,2,2,4,8,32,256,\ldots$.

Find the integer part of $\displaystyle\frac1{\frac1{20}+\frac1{21}+\frac1{22}+\cdots+\frac1{29}}$.

Six teams A, B, C, D, E, F play a five-day tournament. Every team plays once each day and meets every other team exactly once, so each day has three games. 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 meets B 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 neither Tuesday nor Saturday. Exactly three statements are true. What day is today?

For real a, $|a+2|+|a+3|+|a+4|=7$. Find the sum of all possible a.

Find $1\times3\times5+2\times4\times6+3\times5\times7+\cdots+10\times12\times14$.

Express $8/91$ as a recurring decimal.

Real x, y satisfy $|x-9|+y^2-4y+4=0$. Find $x+2y$.

Nonzero real x satisfies $x^2+7x+1=0$. Find $x^2+1/x^2$.

Find the units digit of $(4+1)(4^2+1)(4^4+1)(4^8+1)\cdots(4^{2048}+1)$.

Find the remainder when $2015^{2015}$ is divided by 9.

Positive integers x, y satisfy $\frac6{11}<\frac{x}{y}<\frac9{13}$. When y is as small as possible, find x.

Positive integers x, y satisfy $1/x+1/y=1/2015$. Find the greatest possible x.

Positive integers x<y satisfy $xy+x+y=41$ and $x^2y+xy^2=330$. Find $x^2+y^2$.

Join 2015 unit squares to form a rectangle. Find the smallest perimeter.

Editorial tiling model: use every unit square without cutting, gaps or overlaps.

Find the area of the convex quadrilateral with vertices $A=(3,3)$, $B=(6,2)$, $C=(9,7)$, $D=(6,7)$.

In the diagram, ABCD is a square of side 40. If $CD:DE=EF:FA=1:2$, find AG.

AFEC is a rectangle. In the figure, $AB:BC=1:1$, $AG:GD=3:1$, $AC=8$ and $CE=12$. Find the area of triangle GDE.

In triangle ABC shown, $\angle B=60^\circ$, $\angle C=45^\circ$ and $AC=\sqrt6$. Find AB.