HKIMO 2019 Grade 9 National Final (Heat) Set 02

How many integers x satisfy $-4<(8-2x)/9\le6$?

Find the missing number in the pattern $1,1,2,3,7,22,155,\underline{\quad},\ldots$.

There are 155 balls. Peter and Mary alternately take 3 to 10 balls; whoever takes the last ball wins. Peter moves first and may add balls before play. What is the minimum addition guaranteeing his victory?

Given $A:B=4:3$, $B:C=5:2$, $C:D=4:5$ and $A+B+C+D=2910$, find C.

A box contains 34 red socks, 24 white socks and 29 blue socks. How many socks must be drawn blindly to guarantee 3 nonwhite pairs and 2 nonred pairs? Each pair contains two socks of the same colour.

If $x^4+mx^3+5x^2+nx-6$ is divisible by $x^2-4x+3$, find mn.

Given $x>0$ and $x=4-\dfrac8{3+\dfrac2{4-\dfrac8{3+\cdots}}}$, find x.

Find the minimum of $6x^2+48x-67$ for real x.

Let $f(x)=ax^2+bx+c$, $f(-5)=-16$, $f(5)=64$, $f(-1)=-7$. Find $f(1)$.

Factorize $x^2-2y^2-xy+7x-14y$.

Find the remainder of $26^{64}$ on division by 14.

The number $\overline{20209241ab}$ is divisible by 44. Find the maximum of a·b.

For real x, find the maximum of $24/\sqrt{x^2+6x+45}$.

If $4x\equiv11\pmod{17}$, find the smallest three-digit even integer x.

If $k^2+(n-223)k-899=0$ and $n=1+3+5+7+\cdots+k$, find nk.

An interior angle of a regular n-gon is $(7n+34)^\circ$. Find n.

Find the area enclosed by the coordinate axes and $x=11-\frac74y$.

An acute angle x has $\sin x=5/13$. Find $2/\cos x-\tan^2x$.

A triangle with sides 15, 11, 6 has area $n\sqrt2$. Find n.

In the figure ABDC is a square of side 20. E, F, G are the midpoints of AC, AB, BD respectively. Find the area of triangle EHI. The figure defines H=CF∩ED and I=CF∩EG.