Skip to content / Đến nội dung
ThichHocToan.com

SMO 2014 Senior Round 1

Question 1

If $\alpha,\beta$ are the roots of the equation $3x^2+x-1=0$, where $\alpha\gt \beta$, find the value of $\dfrac\alpha\beta+\dfrac\beta\alpha$.
  1. $\frac79$
  2. $-\frac79$
  3. $\frac73$
  4. $-\frac73$
  5. $-\frac19$

Question 2

Find the value of $$\frac{2014^3-2013^3-1}{2013\times2014}.$$
  1. 3
  2. 5
  3. 7
  4. 9
  5. 11

Question 3

Find the value of $$\frac{\log_5 9\log_7 5\log_3 7}{\log_2\sqrt6}+\frac1{\log_9\sqrt6}.$$
  1. 2
  2. 3
  3. 4
  4. 6
  5. 7

Question 4

Find the smallest number among the following numbers:
  1. $\sqrt{55}-\sqrt{52}$
  2. $\sqrt{56}-\sqrt{53}$
  3. $\sqrt{77}-\sqrt{74}$
  4. $\sqrt{88}-\sqrt{85}$
  5. $\sqrt{70}-\sqrt{67}$

Question 5

Find the largest number among the following numbers:
  1. $30^{30}$
  2. $50^{10}$
  3. $40^{20}$
  4. $45^{15}$
  5. $5^{60}$

Question 6

Given that $\tan A=\dfrac{12}5$, $\cos B=-\dfrac35$ and that $A$ and $B$ are in the same quadrant, find the value of $\cos(A-B)$.
  1. $-\frac{63}{65}$
  2. $-\frac{64}{65}$
  3. $\frac{63}{65}$
  4. $\frac{64}{65}$
  5. $\frac{65}{63}$

Question 7

Find the largest number among the following numbers:
  1. $\tan47^\circ+\cos47^\circ$
  2. $\cot47^\circ+\sqrt2\sin47^\circ$
  3. $\sqrt2\cos47^\circ+\sin47^\circ$
  4. $\tan47^\circ+\cot47^\circ$
  5. $\cos47^\circ+\sqrt2\sin47^\circ$

Question 8

In the diagram below, $\triangle ABC$ is a triangle and $D,E,F$ are points on $BC,CA,AB$ respectively. It is given that $BF=BD$, $CD=CE$ and $\angle BAC=48^\circ$. Find the angle $\angle EDF$.

Question illustration
  1. $64^\circ$
  2. $66^\circ$
  3. $68^\circ$
  4. $70^\circ$
  5. $72^\circ$

Question 9

Find the number of real numbers which satisfy the equation $$x|x-1|-4|x|+3=0.$$
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4

Question 10

If $f(x)=\dfrac1x-\dfrac4{\sqrt x}+3$ where $\dfrac1{16}\le x\le1$, find the range of $f(x)$.
  1. $-2\le f(x)\le4$
  2. $-1\le f(x)\le3$
  3. $0\le f(x)\le3$
  4. $-1\le f(x)\le4$
  5. None of the above

Question 11

Suppose $x$ is real number such that $\dfrac{27\cdot9^x}{4^x}=\dfrac{3^x}{8^x}$. Find the value of $2^{-(1+\log_2 3)x}$.

Question 12

Evaluate $50(\cos39^\circ\cos21^\circ+\cos129^\circ\cos69^\circ)$.

Question 13

Suppose $a$ and $b$ are real numbers such that the polynomial $x^3+ax^2+bx+15$ has a factor of $x^2-2$. Find the value of $a^2b^2$.

Question 14

In the triangle $\triangle ABC$ below, $AC=3AD$, $BC=4EC$, $BD=5GF=5FD$. Suppose the area of $\triangle ABC$ is 900 meter$^2$. Find the area of the triangle $\triangle EFG$ in meter$^2$.

Question illustration

Question 15

Let $x,y$ be real numbers such that $y=|x-1|$. What is the smallest value of $(x-1)^2+(y-2)^2$?

Question 16

Evaluate the sum $$\frac{3!+4!}{2(1!+2!)}+\frac{4!+5!}{3(2!+3!)}+\cdots+\frac{12!+13!}{11(10!+11!)}.$$

Question 17

Let $n$ be a positive integer such that $12n^2+12n+11$ is a 4-digit number with all 4 digits equal. Determine the value of $n$.

Question 18

Given that in the expansion of $(2+3x)^n$, the coefficients of $x^3$ and $x^4$ are in the ratio $8:15$. Find the value of $n$.

Question 19

In a triangle $\triangle ABC$, it is given that $$(\sin A+\sin B):(\sin B+\sin C):(\sin C+\sin A)=9:10:11.$$ Find the value of $480\cos A$.

Question 20

Let $x=\sqrt{37-20\sqrt3}$. Find the value of $$\frac{x^4-9x^3+5x^2-7x+68}{x^2-10x+19}.$$