HKIMO 2026 Grade 9 National Final (Heat) Set 05

A box contains 100 balls: 30 blue, 30 red, 20 green and 20 yellow. Sherry draws without looking. What is the least number she must draw to guarantee 25 balls of one colour?

According to the pattern, what is the next number in $2,9,28,65,126,217,344,\ldots$?

Find the integer part of $\dfrac1{\frac2{50}+\frac2{51}+\frac2{52}+\cdots+\frac2{69}}$.

Six teams A,B,C,D,E,F play a five-day round-robin tournament. Each team plays once each day and meets every other team exactly once, giving three games per day. A beats B on day 1; D beats C on day 2; E beats A on day 3; B beats C on day 4. Whom does B play on day 5?

Seven people make statements about today. A: Yesterday was Wednesday. B: Tomorrow will not 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 one statement is true. What day is today?

For real a, find the minimum of $|a+2|+|a+3|+|a+4|+|a+5|$.

Find the value of $\sqrt{13-12\sqrt{13-12\sqrt{\cdots}}}$.

Solve $4^x-2^{x+2}-32=0$ for real x.

Positive integers x,y satisfy $x+2y^2=54$, $2y+xy=30$, and $x+2y=14$. Find 2x+y.

For nonzero real x satisfying $x^2+7x+1=0$, find $x^3+1/x^3$.

For x>0, find the minimum of $2x^2-4x+16/x^2+11$.

If $\sqrt{11+\sqrt{112}}=a+\sqrt b$ for positive integers a,b, find $a\sqrt b$.

Find the real x minimizing $\sqrt{x^2-6x+58}+\sqrt{x^2+6x+205}$.

Find the least positive integer x satisfying $x\equiv9\pmod{17}$ and $2x\equiv5\pmod7$.

Distinct primes a,b,c satisfy $(a-b)(a-c)=75$. Find b+c.

The figure shows a circle with tangent lines AB, BC and CD. The source text names the tangency points A, F and C respectively; the figure labels the lower tangency point D. Given AB ∥ DC and AD=18, find BF·FC.

ABCD is a square, and P lies inside it. Given AP=a, BP=2a and CP=3a, find the area of the square in terms of a.

In the figure, BC=2015. Points D,E,F lie on BC,CA,AB respectively, and DCEF is a parallelogram. If $\angle AFE=\angle BFD$, find the perimeter of DCEF.

In the cyclic figure, $AM=MC$, $\angle ABC=74^\circ$ and $\angle BCA=42^\circ$. Find $\angle CAM$.

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