Skip to content / Đến nội dung
ThichHocToan.com

AIMO 2021 Senior secondary — Preliminary (Open Trial 1)

Question 1

How many positive divisors does 2020 have?
  1. 10
  2. 12
  3. 14
  4. 16

Question 2

Using $\tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}$, express $\tan15^\circ$ in surd form.
  1. $2-\sqrt3$
  2. $2+\sqrt3$
  3. $\sqrt3$
  4. $1+\sqrt3$

Question 3

With $\sum_{i=a}^{b}f(i)=f(a)+f(a+1)+\cdots+f(b)$, evaluate $\sum_{i=1}^{10}\frac1{(2i-1)(2i+1)}$.
  1. $1/3$
  2. $9/21$
  3. $10/21$
  4. $4/5$

Question 4

ABCD is a square. E lies on AD and F lies on AB, with $DE=2AE$ and $AF=2BF$. Find the ratio of the area of triangle CEF to the area of square ABCD.

Question illustration
  1. $1/5$
  2. $1/3$
  3. $1/4$
  4. $7/18$

Question 5

How many committees consisting of one man and three women can be formed from three men and four women?
  1. 12
  2. 7
  3. 9
  4. 10

Question 6

The four consecutive side lengths of a rectangle are $2x-3,3x+2,y-1,2y+3$, respectively. Find its area.
  1. 30
  2. 33
  3. 24
  4. 26

Question 7

It is given that 1812 can be written as the difference of the cubes of two consecutive positive integers. Find their sum. The source prints “1812” on one baseline.
  1. 200
  2. 218
  3. 209
  4. 208

Question 8

Positive integers $x,y$ satisfy $11x-29y=1$. Find the least value of $x+y$.
  1. 10
  2. 11
  3. 12
  4. 13

Question 9

Let $x=\arctan(1/2)+\arctan(1/3)$, where $0<x<\pi$. Express x in radians.
  1. $2\pi/3$
  2. $\pi/2$
  3. $\pi/3$
  4. $\pi/4$

Question 10

In the cryptarithm TEN+TEN+FORTY=SIXTY, find Y. Use the standard cryptarithm convention: different letters denote different decimal digits, and leading digits are nonzero.
  1. 6
  2. 7
  3. 8
  4. 24

Question 11

E is the midpoint of CD in square ABCD. BE meets diagonal AC at F. If quadrilateral AFED has area 45, find the area of the square.

Question illustration
  1. 100
  2. 108
  3. 120
  4. 135

Question 12

For real $x,y$, find the minimum of $2x^2+4xy+3y^2+6y+4$.
  1. -3
  2. -5
  3. 5
  4. 3

Question 13

Let $[x]$ denote the greatest integer not exceeding x. Evaluate $[1^2/2]+[2^2/2]+[3^2/2]+\cdots+[20^2/2]$.

Original printed choices: A: 20; B: 1435; C: 140; D: 1260.

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

Question 14

A bag contains three red and four blue marbles. Two are drawn without replacement. Given that the first is blue, what is the probability that the second is also blue?
  1. $1/2$
  2. $3/7$
  3. $4/7$
  4. $2/3$

Question 15

Given $a_1=2,a_2=-1$ and $a_{n+2}=a_n+2a_{n-1}$ for natural numbers $n\ge1$, find $a_7$.
  1. 23
  2. 22
  3. 20
  4. 18

Question 16

The parabola $y=(x-20)^2-16$ meets the x-axis at A and B and the line $x+y=10$ at C and D. Find the area of the quadrilateral with these four vertices.
  1. 46
  2. 52
  3. 60
  4. 63

Question 17

An integer n satisfies $0\le n<9$ and $9\mid n^2-3n+5$. Find the sum of all positive possible values of n.

Question 18

Given $\cos(2\pi/7)+\cos(4\pi/7)+\cos(6\pi/7)=-0.5$, find $\sum_{k=1}^{6}\cos^2(k\pi/7)$.

Question 19

Evaluate the infinite continued fraction $5+\frac{15}{5+\frac{15}{5+\frac{15}{5+\cdots}}}$.

Question 20

Find the sum of all four-digit numbers formed by the digits 1,2,5,6. Editorial convention: use each of the four listed digits exactly once.