HKIMO 2020 Grade 9 National Final (Heat) Set 01

How many integers x satisfy $-6<(8-2 x)/8\le 4$?

Find the next number in the pattern $2,5,10,17,26,37,50,\underline{\quad},\ldots$.

There are 625 balls. Peter and Mary alternately take 3 to 10 balls. The person taking the last ball wins, and Peter moves first. Before play Peter may add balls. What is the least number he must add to guarantee victory?

Given $A: B=4:3$, $B: C=5:1$, $C: D=3:2$ and $A+B+C+D=2920$, find C.

A box contains 24 red socks, 10 white socks and 20 blue socks. How many socks must be drawn blindly to guarantee 5 nonwhite pairs and 2 nonred pairs? Each pair consists of two socks of the same colour.

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

Given $x>0$ and the repeating continued fraction $x=2-\dfrac{6}{3+\dfrac{8}{2-\dfrac{6}{3+\cdots}}}$, find x.

Find the minimum of $6 x^2+60 x+49$ for real x.

Let $f(x)=ax^2+bx+c$, with $f(-5)=-24$, $f(5)=76$ and $f(-1)=27$. Find $f(1)$.

Factorize $x^2-7 y^2+6 xy+2 x+14 y$.

Find the remainder of $28^{64}$ when divided by 13.

If $6 x\equiv 11\pmod{13}$, find the smallest three-digit even integer x.

If $k^2+(n-623)k-2499=0$ and $n=1+3+5+7+\cdots+k$, find nk.

The number $\overline{20202021 ab}$ is divisible by 44. Find the maximum of a·b.

For real x, find the maximum of $36/\sqrt{x^2+8 x+97}$.

An interior angle of a regular n-gon is $(9 n+21)^\circ$. Find n.

Find the area enclosed by the coordinate axes and $x=15-\frac 94 y$.

A triangle with side lengths 13, 12, 7 has area $n\sqrt 3$. Find n.

An acute angle x satisfies $\cos x=7/13$. Find $\tan x-\sin x$ in surd form.

In the figure C is the incenter of triangle PQR, with $CE\perp PQ$, $CF\perp QR$, $CG\perp RP$. Given CP=53, PG=45 and CF=7 x, find x.