AIMO 2021 Secondary 2 — Preliminary (Open Trial 1)

The diagram shows a wooden block with a cylindrical hole drilled through it. The diameter of the hole is 2 cm. Calculate the volume of the solid, correct to 2 decimal places.

Positive integers $x,y$ satisfy $5x-6y=1$. Find the least possible $x$.

Given $3x+y+4z=16$ and $x+y+4z=18$, find $y+4z-x$.

Simplify $\frac{187}{253}$.

Mrs. Mackay forgets the four-digit code of her safe. No digit is 0 and the digit sum is 9. What is the maximum number of attempts needed to guarantee opening the safe?

Find all triples of positive integers $(x,y,z)$ such that $x+y+z=6$.

Convention: List the triples in your reasoning, then select their number.

Calculate $\left(\frac15-\frac14\right)+\left(\frac1{25}-\frac1{16}\right)+\left(\frac1{125}-\frac1{64}\right)+\left(\frac1{625}-\frac1{256}\right)$.

In right triangle $EXO$, $\angle E=90^\circ$. Point $F$ lies on $XO$ and $EF\perp XO$. If $EO=2$ and $XE=3$, find the area of triangle $EFO$.

Original printed choices: A: $\frac{54}{160}$; B: $\frac{54}{25}$; C: $\frac{25}{16}$; D: $\frac{21}{16}$.

Editorial response format: give the exact answer; the original choices are retained for reference.

Find the largest integer not greater than $\sqrt{31}+\sqrt{73}$.

In triangle $ABC$, $AB=AC=5$. Point $D$ lies on $BC$, with $BD=4$ and $DC=2$. Find $AD$.

Evaluate $12^2+35^2+53^2+24\times35+70\times53+106\times12$.

Original printed choices: A: 7200; B: 7220; C: 7244; D: 6400.

Editorial response format: give the exact answer; the original choices are retained for reference.

The positive integer ages of five people form an arithmetic sequence. Two of the ages are 20 and 30. How many different values can the age of the second eldest person take?

Given $x^2-5x-7=0$, find $x^4-2x^3-24x^2-11x+2019$.

Adjacent dots in each row and column are 1 cm apart. Find the area of the purple shaded region in square centimetres.

Nam spends 20 dollars on 20 stationery items consisting of books, pens and pencils. Each book costs 3 dollars, each pen 1.5 dollars and each pencil 0.5 dollars. How many dollars does he spend on books?

Convention: Assume Nam buys at least one item of each of the three types.

How many triangles can be formed by joining vertices of an octagon?

Convention: Use a convex octagon with no three vertices collinear.

As shown in Figure, $ABCD$ is a quadrilateral. It is known that $AD=AB$, $\angle DAB=90^\circ$ and $AE\perp BC$. Find the area of $ABCD$.

An ant moves from $A$, the south pole, to $F$, the north pole, along only the longitudes and latitude shown, including dotted lines, without visiting any point twice (for example $A\to D\to E\to F$). How many routes are possible?

The product $1\times2\times3\times\cdots\times2019$ is divisible by $175^n$, where $n$ is an integer. Find the largest possible $n$.

In quadrilateral $ABCD$, $\angle ADC=90^\circ$, $AD=DC=2$ and $AD\parallel BC$. Point $E$ lies on $CD$ and $AE\perp BD$. Find the area of quadrilateral $ABCE$.