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SMO 2011 Senior Round 1

Question 1

Suppose $a,b,c$ are nonzero real numbers such that
$$\frac{a^2}{b^2+c^2}\lt\frac{b^2}{c^2+a^2}\lt\frac{c^2}{a^2+b^2}.$$
Which of the following statements is always true?
  1. $a\lt b\lt c$
  2. $|a|\lt|b|\lt|c|$
  3. $c\lt b\lt a$
  4. $|b|\lt|c|\lt|a|$
  5. $|c|\lt|b|\lt|a|$

Question 2

Suppose $\theta$ is an angle between $0$ and $\pi/2$, and $\sin2\theta=a$. Which expression equals $\sin\theta+\cos\theta$?
  1. $\sqrt{a+1}$
  2. $(\sqrt2-1)a+1$
  3. $\sqrt{a+1}-\sqrt{a^2-a}$
  4. $\sqrt{a+1}+\sqrt{a^2-a}$
  5. None of the above.

Question 3

Let $x$ be real. If $a=2011x+9997$, $b=2011x+9998$, $c=2011x+9999$, find $a^2+b^2+c^2-ab-bc-ca$.
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4

Question 4

Suppose real numbers $x,y$ satisfy $\frac1x-\frac1{2y}=\frac1{2x+y}$. Find $\frac{y^2}{x^2}+\frac{x^2}{y^2}$.
  1. $\frac23$
  2. $\frac92$
  3. $\frac94$
  4. $\frac49$
  5. $\frac29$

Question 5

In the figure, $ABC$ is a triangle, and $D,E$ are points on $AB,BC$, respectively. Given $DE\parallel AC$ and $CE:EB=1:3$, the area of $\triangle ABC$ is $1440\text{ cm}^2$ and the area of $\triangle ADE$ is $x\text{ cm}^2$. What is $x$?

Question illustration
  1. 288
  2. 240
  3. 320
  4. 384
  5. 270

Question 6

Determine the value of
$$\frac{2}{\frac1{\sqrt2+\sqrt[4]8+2}+\frac1{\sqrt2-\sqrt[4]8+2}}.$$
  1. $4-\sqrt2$
  2. $2-2\sqrt2$
  3. $4+\sqrt2$
  4. $2\sqrt2+4$
  5. $4\sqrt2-2$

Question 7

Let $x=\frac1{\log_{1/3}(1/2)}+\frac1{\log_{1/5}(1/4)}+\frac1{\log_{1/7}(1/8)}$. Which statement is true?
  1. $1.5\lt x\lt2$
  2. $2\lt x\lt2.5$
  3. $2.5\lt x\lt3$
  4. $3\lt x\lt3.5$
  5. $3.5\lt x\lt4$

Question 8

Determine the last two digits of $7^{5^6}$.
  1. 01
  2. 07
  3. 09
  4. 43
  5. 49

Question 9

Real numbers $x,y$ satisfy $x>1$, $y>1$. Find the smallest possible value of
$$\frac{\log_x2011+\log_y2011}{\log_{xy}2011}.$$
  1. 4
  2. 6
  3. 8
  4. 10
  5. 12

Question 10

Real numbers $a,b,c$ satisfy $a+b=c-1$ and $ab=c^2-7c+14$. Find the largest possible value of $a^2+b^2$.
  1. 5
  2. 6
  3. 8
  4. 9
  5. 10

Question 11

Find the value of
$$\frac{2011^2+2111^2-2\times2011\times2111}{25}.$$

Question 12

Find the largest natural number $n$ satisfying $n^{6033}\lt2011^{2011}$.

Question 13

Find the integer closest to $\frac{(1+\sqrt3)^4}{4}$.

Question 14

In the diagram, $\triangle ABC$ is isosceles with $AB=AC$, and $M,N$ are the midpoints of $AB,AC$, respectively. Given $CM\perp BN$, $BC=20\text{ cm}$, and $[ABC]=x\text{ cm}^2$, find $x$.

Question illustration

Question 15

Find the smallest positive integer $n$ such that $\sqrt{5n}-\sqrt{5n-4}\lt0.01$.

Question 16

Find the value of
$$\frac1{1+11^{-2011}}+\frac1{1+11^{-2009}}+\frac1{1+11^{-2007}}+\cdots+\frac1{1+11^{2009}}+\frac1{1+11^{2011}}.$$

Question 17

Let $x=\sin^4(\pi/8)+\cos^4(\pi/8)+\sin^4(7\pi/8)+\cos^4(7\pi/8)$. Find $36x$.

Question 18

In the diagram, the three triangle sides have lengths $a,b,c$ cm. Given $\frac{a^2+b^2}{c^2}=2011$, find $\frac{\cot C}{\cot A+\cot B}$.

Question illustration

Question 19

There are 2011 participants in a mathematics competition, at least 1000 of them female. Given any 1011 participants, at least 11 of them are male. How many male participants are there?

Question 20

Let $f:\mathbb Q\setminus\{0,1\}\to\mathbb Q$ satisfy
$$x^2f(x)+f\left(\frac{x-1}{x}\right)=2x^2$$
for every rational $x\ne0,1$. Here $\mathbb Q$ is the set of rational numbers. Find $f(1/2)$.