AMO 2021 – Junior College (Grades 11–12)

Find $\sin^2 10^\circ+\sin^2 20^\circ+\sin^2 30^\circ+\sin^2 40^\circ+\sin^2 50^\circ+\sin^2 60^\circ+\sin^2 70^\circ+\sin^2 80^\circ$. The five choices below correspond, in order, to choices 1–5 in the original.

If $f(x)=x^3(a\cdot2^x-2^{-x})$ is even, find $a$.

Acute angles $\alpha,\beta$ satisfy $2\tan\alpha+3\sin\beta=7$ and $\tan\alpha-6\sin\beta=1$. Find $\sin\alpha$.

A geometric sequence $(a_n)$ satisfies $a_5a_6=4$. Find $\log_2a_1+\cdots+\log_2a_{10}$. As required by the displayed logarithms, all ten terms are positive.

The polynomial $x^4-3x^3+5x^2+ax+b$ is divisible by $x^2-2x+1$. Find $ab$.

In the expansion of $(\sqrt[3]x+\sqrt x)^{50}$, how many terms have integer exponents of $x$?

In an arithmetic sequence $(A_n)$, the sum $S_{2m-1}$ of its first $2m-1$ terms equals $19$, and $A_{m-1}+A_{m+1}-A_m^2=1$. Find $m$.

How many ordered pairs of integers $(x,y)$ satisfy $x^{2022}+y^2=2y$?

Given $\log_a b+\log_b a=3$ and $a>b>1$, find $\log_a b-\log_b a$.

$A(1,3)$ and $B(3,m)$ lie on $y=k/x$. Find the $x$-coordinate of a point $P$ on the $x$-axis minimizing $PA+PB$.

If $x^2/2+y^2/3=4$, find the greatest value of $x+y$.

Given $\sqrt[3]{5+x}+\sqrt[3]{5-x}=1$, find $x^2$.

A circular cone has vertex $P$, base radius $\sqrt3$ inches, and height $1$ inch. Points $E,F$ lie on the base circle. Find the largest possible area of $\triangle PEF$, in square inches.

Continue the hexagonal dot pattern shown. How many dots are in the 99th shape?

All logarithms are defined and $2\log(a-b)=\log(2a)+\log(3b)$. Find $a/b$.

Find $80M^2$, where $M=\dfrac{\sqrt{3+\sqrt5}+\sqrt{3-\sqrt5}}{\sqrt{5-2\sqrt6}-\sqrt{5+2\sqrt6}}$.

For every nonzero real $x$, $2f(x)+f(1/x)=2$. Find $30f(x)$.

In trapezoid $ABCD$, $CD\parallel AB$, $AD=15$ cm, $AB=22$ cm, $BC=13$ cm, and $CD=8$ cm. Find its area in square centimetres.

The parabola $y=-x^2+2x+3$ meets the axes at $A$ and $B$ as shown. Point $E$ is on the positive $x$-axis. The vertical line through the midpoint of $OE$ meets $BE$ at $N$ and the parabola at $M$, with $MN=5/2$. The sum of all possible positive $x$-coordinates of $E$ is $a+b\sqrt c$, in simplified form with integers $a,b$ and square-free positive integer $c$. Find $abc$.

A square has seven nonidentity symmetries: four reflections and rotations through $90^\circ,180^\circ,270^\circ$. Starting at vertex $P$, perform four independent, uniformly random choices from these seven symmetries, successively. The probability that the four resulting positions are all four vertices is $k/7^4$. Find $k$. Clarification: this version counts the positions after moves 1, 2, 3, 4, not the initial position. The original wording does not specify this distinction; the solution explains both readings.