HKIMO 2026 THPT (Grades 10–12 shared) National Final (Heat) Set 05

A box has $2015$ balls, each red, green, blue, yellow or black. Any draw of $1800$ balls guarantees at least one of each colour. What is the minimum number to draw to guarantee at least three different colours?

Editorial working convention: Draw without replacement; seek the worst-case guarantee from the stated information.

According to the pattern, find the next number: $1024,0,512,256,384,320,352,\ldots$.

Editorial working convention: Continue by multiplying each successive difference by −1/2, equivalently taking the mean of the previous two terms.

Primes $x,y$ satisfy $x^2-y^3=1$. Find $x+y$.

Six teams $A,B,C,D,E,F$ play a tournament over five days. Each team plays once each day, and each pair meets exactly once. $A$ 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 does $E$ play on day 5?

Seven people make these statements about today. A: Yesterday was Wednesday. B: Tomorrow will not be Tuesday. C: Tomorrow will be Wednesday. D: Today is Tuesday or yesterday was 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?

Express $(\sqrt3/2+i/2)^{2015}$ in the form $a+bi$.

The polynomial equation $x^4-x^3+ax^2+bx+3=0$, where $a,b$ are constants, has a double root at $1$. How many roots have nonzero imaginary part?

Evaluate $\displaystyle\sum_{k=2}^{1000}\frac1{k^2-1}$.

Find the remainder when $x^{2015}-20x-15$ is divided by $x^2-1$.

Factorise $x^2-3xy+2y^2+x-4y-6$.

Positive real numbers $a,b,c$ satisfy $a^2+b^2+c^2\le24$. Find the maximum of $a^{2/3}+b^{2/3}+c^{2/3}$.

For real $-3\le x\le3$, find the maximum of $x+\sqrt{9-x^2}$.

Given $\log 2=0.301$, evaluate $\displaystyle\log\left(\sum_{n=0}^{2015}\binom{2015}{n}\right)$ to two decimal places.

Editorial working convention: Use the supplied approximation and ordinary half-up rounding.

Find the least positive integer $x$ satisfying $8x\equiv9\pmod{17}$ and $3x\equiv5\pmod7$.

Both $a$ and $\sqrt{23-8\sqrt7}+a\sqrt{23+8\sqrt7}$ are rational. Find $a$.

Two tangents to the circle $x^2+y^2=1$ pass through $P=(3,4)$. Find the sum of their slopes.

In acute triangle $ABC$, altitudes $AD$ and $BE$ meet at $H$. Join $D$ and $E$. How many pairs of similar triangles are there?

Triangle sides satisfy $0\le a\le6\le b\le8\le c\le12$. When the area is greatest, find the perimeter.

Find the shortest distance from $(5,-10)$ to the line $4y=3x$.

In the figure, $TA$ and $DC$ are tangent to the circle at $A$ and $C$. Points $T,B,C$ are collinear in that order. Given $\angle ATB=50^\circ$ and $\angle BCD=75^\circ$, find $\angle ABC$ in degrees.