SMO 2008 Senior Round 1

Question 1

Find the value of $\dfrac{1+3+5+7+\cdots+99}{2+4+6+8+\cdots+100}$.

$48/49$
$49/50$
$50/51$
$98/99$
$100/101$

Question 2

Suppose that $x,y$ are real numbers satisfying all three conditions $3x-2y=4-p$, $4x-3y=2+p$, and $x\gt y$. What are the possible values of $p$?

$p\gt -1$
$p\lt 1$
$p\lt -1$
$p\gt 1$
$p$ can be any real number.

Question 3

If $f(x)=x^2+\sqrt{1-x^2}$, where $-1\le x\le1$, find the range of $f(x)$.

$\frac12\le f(x)\le1$
$1\le f(x)\le\frac54$
$1\le f(x)\le\frac{1+2\sqrt3}{4}$
$\frac{\sqrt3}{2}\le f(x)\le1$
$\frac12\le f(x)\le\frac{\sqrt3}{2}$

Question 4

If $a,b$ are integers and $\sqrt{7-4\sqrt3}$ is one of the roots of $x^2+ax+b=0$, find $a+b$.

$-3$
$-2$
$0$
$2$
$3$

Question 5

A bag contains 30 balls numbered 1, 2, ..., 30. Two balls are randomly chosen from the bag. Find the probability that the sum of their numbers is divisible by 3.

$1/2$
$1/3$
$7/29$
$9/29$
$11/87$

Question 6

$ABCD$ is a square with $AB=a$, and $AEFG$ is a rectangle such that $E$ lies on side $BC$ and $D$ lies on side $FG$. If $AE=b$, what is the length of $EF$?

Question illustration

$b/a$
$3a^2/(2b)$
$4a^2/(3b)$
$\sqrt2a^2/b$
$a^2/b$

Question 7

Find $\sin^4\frac\pi8+\sin^4\frac{3\pi}8+\sin^4\frac{5\pi}8+\sin^4\frac{7\pi}8$.

$1$
$3/2$
$7/4$
$2$
$5/2$

A circle with radius $x$ cm is inscribed inside a triangle $ABC$, where $\angle ACB$ is a right angle. If $AB=9$ cm and the area of triangle $ABC$ is $36\text{ cm}^2$, find $x$.

Question illustration

$2.2$
$2.6$
$3$
$3.4$
$3.8$

Question 9

How many positive integers $n$ satisfy that $7n+1$ is a perfect square and $3n+1\lt 2008$?

$6$
$9$
$12$
$15$
$18$

Question 10

Find the minimum of $(a+b)\left(\frac1a+\frac4b\right)$, where $a,b$ range over all positive real numbers.

$3$
$6$
$9$
$12$
$18$

Question 11

Find the smallest integer $n$ such that $n(\sqrt{101}-10)\gt 1$.

Question 12

Given positive real numbers $x,y$ such that $(x+y)^2=2500$ and $xy=500$, find the exact value of $x^3+y^3$.

Question 13

Find the smallest positive integer $N$ such that $2^n\gt n^2$ for every integer $n$ in $\{N,N+1,N+2,N+3,N+4\}$.

Question 14

The side lengths of a quadrilateral are 2006 cm, 2007 cm, 2008 cm and x cm. If x is an integer, find its largest possible value.

Question 15

Find the number of positive integers $x$ satisfying $\left|3+\log_x\frac13\right|\lt \frac83$.

Question 16

Two bullets are placed in two consecutive chambers of a 6-chamber pistol. The cylinder is then spun. The pistol is fired but the first shot is a blank. Let $p$ be the probability that the second shot is also a blank if the cylinder is spun after the first shot, and $q$ the probability that the second shot is also a blank if the cylinder is not spun after the first shot. Find the smallest integer $N$ such that $N\ge100p/q$.