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SMO 2012 Senior Round 1

Question 1

Suppose $\alpha,\beta$ are the real roots of
$$x^2+2\sqrt{\sqrt2+1}\,x+\sqrt{\sqrt2+1}=0.$$
Find $\frac1{\alpha^3}+\frac1{\beta^3}$.
  1. $3\sqrt{\sqrt2+1}(\sqrt2-1)-8$
  2. $8-6\sqrt{\sqrt2+1}(\sqrt2-1)$
  3. $3\sqrt{\sqrt2+1}(\sqrt2-1)+8$
  4. $6\sqrt{\sqrt2+1}(\sqrt2-1)-8$
  5. None of the above.

Question 2

Find
$$\frac{2011^2\cdot2012-2013}{2012!}+\frac{2013^2\cdot2014-2015}{2014!}.$$
  1. $\frac1{2009!}+\frac1{2010!}+\frac1{2011!}+\frac1{2012!}$
  2. $\frac1{2009!}+\frac1{2010!}-\frac1{2011!}-\frac1{2012!}$
  3. $\frac1{2009!}+\frac1{2010!}-\frac1{2012!}-\frac1{2013!}$
  4. $\frac1{2009!}+\frac1{2010!}+\frac1{2013!}+\frac1{2014!}$
  5. $\frac1{2009!}+\frac1{2010!}-\frac1{2013!}-\frac1{2014!}$

Question 3

The increasing sequence $T=2,3,5,6,7,8,10,11,\ldots$ consists of all positive integers that are not perfect squares. What is its $2012$th term?
  1. 2055
  2. 2056
  3. 2057
  4. 2058
  5. 2059

Question 4

Let $O$ be the centre of the inscribed circle of triangle $ABC$, and $D$ the point on $AC$ with $OD\perp AC$. If $AB=10$, $AC=9$, $BC=11$, find $CD$.

Question illustration
  1. 4
  2. 4.5
  3. 5
  4. 5.5
  5. 6

Question 5

Find
$$\frac{\cos^4 75^\circ+\sin^4 75^\circ+3\sin^2 75^\circ\cos^2 75^\circ}{\cos^6 75^\circ+\sin^6 75^\circ+4\sin^2 75^\circ\cos^2 75^\circ}.$$
  1. $\sqrt2/4$
  2. $1/2$
  3. $3/4$
  4. 1
  5. $\cos75^\circ+\sin75^\circ$

Question 6

If the roots of $x^2+3x-1=0$ are also roots of $x^4+ax^2+bx+c=0$, find $a+b+4c$.
  1. -13
  2. -7
  3. 5
  4. 7
  5. 11

Question 7

Find the sum of the digits of all numbers in the sequence $1,2,3,4,\ldots,1000$.
  1. 4501
  2. 12195
  3. 13501
  4. 499500
  5. None of the above.

Question 8

Find the number of real solutions of $x/100=\sin x$, where $x$ is in radians.
  1. 30
  2. 32
  3. 62
  4. 63
  5. 64

Question 9

In triangle $ABC$, $AB=AC$, $\angle ABC=40^\circ$, and $D\in AC$ with $BD$ bisecting $\angle ABC$. Extend $BD$ beyond $D$ to $E$ with $DE=AD$. Find $\angle ECA$.

Question illustration
  1. $20^\circ$
  2. $30^\circ$
  3. $40^\circ$
  4. $45^\circ$
  5. $50^\circ$

Question 10

Let $m,n$ be positive integers with $m>n$. If the last three digits of $2012^m$ and $2012^n$ agree, find the smallest possible $m+n$.
  1. 98
  2. 100
  3. 102
  4. 104
  5. None of the above.

Question 11

Four distinct positive real numbers $a,b,c,d$ satisfy
$$(a^{2012}-c^{2012})(a^{2012}-d^{2012})=2011,$$
$$(b^{2012}-c^{2012})(b^{2012}-d^{2012})=2011.$$
Find $(cd)^{2012}-(ab)^{2012}$.

Question 12

Determine the number of integer pairs $(x,y)$ satisfying $\frac1y-\frac1{y+2}=\frac1{3\cdot2^x}$.

Question 13

Let $S=\{1,2,\ldots,10\}$. A collection $\mathcal F$ of subsets of $S$ is intersecting if $A\cap B\ne\varnothing$ for every two subsets $A,B$ in $\mathcal F$. What is the maximum size of $\mathcal F$?

Question 14

Let $M$ contain all integers $m$ for which the polynomial
$$2(m-1)x^2-(m^2-m+12)x+6m$$
has either one repeated integer root or two distinct integer roots. Find the number of elements of $M$.

Question 15

Find the minimum of $\left|\sin x+\cos x+\frac{\cos x-\sin x}{\cos2x}\right|$.

Question 16

Find the number of arrangements in a line of the letters $A,A,B,B,C,C,D,E$ having no consecutive identical letters.

Question 17

Suppose $x=3^{\sqrt{2+\log_3x}}$ is an integer. Determine $x$.

Question 18

Let $f(x)=(x-a_1)(x-a_2)(x-a_3)(x-a_4)(x-a_5)$, where $a_1,\ldots,a_5$ are distinct integers. Given $f(104)=2012$, evaluate $a_1+a_2+a_3+a_4+a_5$.

Question 19

Positive real numbers $x,y,z,\lambda$ satisfy
$$yz=6\lambda x,\qquad xz=6\lambda y,\qquad xy=6\lambda z,\qquad x^2+y^2+z^2=1.$$
Find $(xyz\lambda)^{-1}$.

Question 20

Find the least value of $(x+y)(y+z)$ for positive real $x,y,z$ satisfying $xyz(x+y+z)=1$.