Skip to content / Đến nội dung
ThichHocToan.com

SMO 2022 Senior Round 1

Question 1

Suppose the roots of $\dfrac{x^2}{2}+mx+n=0$ are $\dfrac m2$ and $\dfrac n3$. Find the smallest value of $mn$.
  1. $-1080$
  2. $-90$
  3. $0$
  4. $90$
  5. $1080$

Question 2

Which of the following is true?
  1. $\sqrt[6]{1/333}\lt \sqrt[3]{1/18}\lt \sqrt{1/7}$
  2. $\sqrt[3]{1/18}\lt \sqrt[6]{1/333}\lt \sqrt{1/7}$
  3. $\sqrt[3]{1/18}\lt \sqrt{1/7}\lt \sqrt[6]{1/333}$
  4. $\sqrt{1/7}\lt \sqrt[6]{1/333}\lt \sqrt[3]{1/18}$
  5. None of the above.

Question 3

Suppose $\sqrt{(\log_{377\times377}2022)(\log_{377}2022)}=\log_k2022$. Find $k$.

(A) $\sqrt{337}$

(B) $337^{\sqrt2}$

(C) $337\sqrt2$

(D) $(\sqrt{337})^{\sqrt2}$

(E) $\sqrt{337\times2}$

Question 4

Suppose $y=\cos^2x-7\cos x+25$, where $x$ is any real number. Find the range of $y$.
  1. $17\le y\le33$
  2. $18\le y\le33$
  3. $19\le y\le33$
  4. $20\le y\le33$
  5. None of the above

Question 5

Suppose $\sin(180^\circ+x)=-\dfrac79$, where $450^\circ\lt x\lt 540^\circ$. Find $\sin(2x)$.
  1. $\dfrac{49}{81}\sqrt2$
  2. $\dfrac{56}{81}\sqrt2$
  3. $-\dfrac{56}{81}$
  4. $-\dfrac{49}{81}\sqrt2$
  5. $-\dfrac{56}{81}\sqrt2$

Question 6

Find the value of $$\left(\frac{\cos10^\circ+\cos50^\circ+\cos70^\circ+\cos110^\circ}{\cos20^\circ}\right)^8.$$

Question 7

Suppose $x^{20}+\dfrac{x^{10}}2-\dfrac{3^{2x}}9+\dfrac1{16}=0$ for some positive real number $x$. Find the value of $4\cdot3^x-12x^{10}$.

Question 8

How many positive integers less than or equal to $2022$ cannot be expressed as $$\lfloor2x+1\rfloor+\lfloor5x+1\rfloor$$ for some real number $x$? Here, $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. For example, $\lfloor-2.1\rfloor=-3$, $\lfloor3.9\rfloor=3$.

Question 9

Suppose $$y=\frac{\tan^2x-\tan x+\sqrt{33}}{\tan^2x+\tan x+1},$$ where $-90^\circ\lt x\lt 90^\circ$. Find the maximum possible value of $\sqrt{33}(y-5)$.

Question 10

In the figure below, $PQRS$ is a square inscribed in a circle. Let $W$ be a point on the arc $PQ$ such that $WS=\sqrt{20}$. Find $(WP+WR)^2$.

Question illustration

Question 11

The figure below shows a quadrilateral $ABCD$ such that $AC=BD$ and $P$ and $Q$ are the midpoints of the sides $AD$ and $BC$ respectively. The lines $PQ$ and $AC$ meet at $R$ and the lines $BD$ and $AC$ meet at $S$. If $\angle PRC=130^\circ$, find the angle $\angle DSC$ (in $^\circ$).

Question illustration

Question 12

How many distinct terms are there if $(x^2+y^2)^{11}(x^{11}+y^{11})^9$ is algebraically expanded and simplified?

Question 13

If $\sqrt{x^2+7x-4}+\sqrt{x^2-x+4}=x-1$, find the value of $3x^2+14x$.

Question 14

Let $k=-1+\sqrt{2022^{1/5}-1}$, and let $f(x)=(k^2+2k+2)^{10x}$. Find the value of $\log_{2022}f(2022)$.

Question 15

Find the smallest odd integer $N$, where $N\gt 2022$, such that when $1808$, $2022$ and $N$ are each divided by a positive integer $p$, where $p\gt 1$, they all leave the same remainder.

Question 16

If $\dfrac{12}{x}+\dfrac{48}{y}=1$, where $x$ and $y$ are positive real numbers, find the smallest possible value of $x+y$.

Question 17

Find the largest value of $40x+60y$ if $x-y\le2$, $5x+y\ge5$ and $5x+3y\le15$.

Question 18

Suppose
$$\cos x-\cos y=\frac12,\qquad\sin x-\sin y=-\frac13.$$
If $\sin(x+y)=\dfrac mn$, where $\dfrac mn$ is expressed as a fraction in its lowest terms, find the value of $m+n$.

Question 19

For some positive integer $n$, the number $n^3-3n^2+3n$ has a units digit of $6$. Find the product of the last two digits of the number $7(n-1)^{12}+1$.

Question 20

Find the largest positive integer $n$ for which $\dfrac{20n+2020}{3n-6}$ is a positive integer.