AIMO 2021 Secondary 3 — Preliminary (Open Trial 1)

Find a 4-digit number whose digits are all the same and which has only two distinct prime divisors.

Under a point reflection, $(-5,8)$ is mapped to $(3,-4)$. Find the centre of symmetry.

If $\sin x+\cos x=\frac12$, find $\sin x\cos x$.

Colour the five regions in the figure using red, yellow, blue and green. All four colours must occur, and regions sharing an edge must have different colours. How many colourings are possible?

Find the sum of the prime factors of $3^8+3^7+1$.

Original printed choices: A: 13; B: 637; C: 690; D: 689.

Editorial format: answer directly; the printed choices are retained for source fidelity. See the solution for the source issue.

Find the remainder when $2019^{2020}$ is divided by 2020. Option B is blank in the source.

The length and width of a rectangle are each increased by 50%. How many times its original area is the new area?

Original printed choices: A: 2.5; B: 5; C: 1.25; D: 12.5.

Editorial format: answer directly; the printed choices are retained for source fidelity. See the solution for the source issue.

Evaluate $2^2+4^2+6^2+\cdots+98^2+100^2-(1^2+2^2+\cdots+49^2+50^2)$.

Original printed choices: A: 42925; B: 131325; C: 42920; D: 130050.

Editorial format: answer directly; the printed choices are retained for source fidelity. See the solution for the source issue.

Evaluate the infinite nested radical $\sqrt{10+3\sqrt{10+3\sqrt{10+\cdots}}}$.

Solve $|x^2-4x-13|<8$.

Define $C_r^n=\frac{n!}{r!}$, where $n!=n(n-1)\cdots2\cdot1$. Find $C_1^{10}+C_2^{10}+\cdots+C_8^{10}$.

Original printed choices: A: 1012; B: 1020; C: 1010; D: 1008.

Editorial format: answer directly; the printed choices are retained for source fidelity. See the solution for the source issue.

Six people are selected uniformly at random from 10 boys and 8 girls. What is the probability of choosing three boys and three girls?

Simplify $P=\frac{2x+1}{x+2}+\frac{x^2-4}{x^2+x-2}+\frac1{1-x}$.

Original printed choices: A: $\frac{x+1}{x+2}$; B: $\frac{x-1}{x+2}$; C: $\frac{x-1}{x-2}$; D: $\frac{x+1}{x-2}$.

Editorial format: answer directly; the printed choices are retained for source fidelity. See the solution for the source issue.

Nine identical white balls and three identical black balls are arranged in a row. No two black balls may be adjacent. How many arrangements are possible?

Original printed choices: A: 6; B: 4; C: 3; D: 5.

Editorial format: answer directly; the printed choices are retained for source fidelity. See the solution for the source issue.

Positive integers $a,b$ satisfy $a^3=3060b^2$. Find the smallest possible $b$.

Original printed choices: A: 6; B: 4; C: 3; D: 5.

Editorial format: answer directly; the printed choices are retained for source fidelity. See the solution for the source issue.

A triangle has vertices $A(-1,3),B(4,-1),C(3,1)$. Find its area.

Positive integers $a,b$ satisfy $(6a-1)(3b^2-11)(2b^2)=1440$. Find $b$.

Circle BCD passes through the centre of circle ABC, as shown. If $\angle BDC=40^\circ$ and $\angle BAC=a^\circ$, find $a$.

ABCD is a square and E lies on BC. Folding along MN makes A coincide with E. If $\tan\angle AEN=\frac12$, find $\sin\angle ENB$.

Find the least common multiple of 9, 2019 and 20194038. Editorial reading: the last entry is the integer 20,194,038, exactly as printed on one baseline.