AMO 2022 - Secondary 4 (Grade 10)

Question 1

Find $\sin^2 10^\circ+\sin^2 20^\circ+\sin^2 30^\circ+\sin^2 40^\circ+\sin^2 50^\circ+\sin^2 60^\circ+\sin^2 70^\circ+\sin^2 80^\circ$.

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Question 2

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

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Question 3

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

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$\frac13$
$\frac{3\sqrt{10}}{10}$
$\frac{3\sqrt7}{7}$
$\frac{5\sqrt5}{5}$
$\frac{10}{9}$

Question 4

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.

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$2$
$4$
$5$
$8$
$10$

Question 5

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

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$-10$
$-5$
$-2$
$5$
$10$

Question 6

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

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$2$
$4$
$6$
$9$
$10$

Question 7

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 $m$.

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