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SMO 2006 Junior Round 1

Question 1

What are the last two digits of $1\times2\times3\times4\times\cdots\times2004\times2005\times2006$?
  1. $00$
  2. $20$
  3. $30$
  4. $50$
  5. $60$

Question 2

Let $x$ be a real number. What is the minimum value of $x^2-4x+3$?
  1. $-3$
  2. $-1$
  3. $0$
  4. $1$
  5. $3$

Question 3

James calculates the sum of the first $n$ positive integers and finds that the sum is $5053$. If he has counted one integer twice, which one is it?
  1. $1$
  2. $2$
  3. $3$
  4. $4$
  5. $5$

Question 4

Which of the following is a possible number of diagonals of a convex polygon?
  1. $21$
  2. $32$
  3. $45$
  4. $54$
  5. $63$

Question 5

What is the largest positive integer $n$ satisfying $n^{200}\lt 5^{300}$?
  1. $9$
  2. $10$
  3. $11$
  4. $12$
  5. $13$

Question 6

The diagram shows an equilateral triangle $ADE$ inside a square $ABCD$. What is the value of $\dfrac{\text{area of }\triangle ADE}{\text{area of }\triangle DEC}$?

Question illustration
  1. $\frac{\sqrt3}{4}$
  2. $\frac14$
  3. $\frac{\sqrt3}{2}$
  4. $\sqrt3$
  5. $2$

Question 7

What is the value of
$$(x+1)(x+2006)\left[\frac1{(x+1)(x+2)}+\frac1{(x+2)(x+3)}+\cdots+\frac1{(x+2005)(x+2006)}\right]?$$
  1. $x+2004$
  2. $2005$
  3. $x+2006$
  4. $2006$
  5. $2007$

Question 8

Suppose that only one of the following pairs $(x,y)$ yields the positive integer $\sqrt{x^2+y^2}$. Then which pair is it?
  1. $x=25530,\ y=29464$
  2. $x=37615,\ y=26855$
  3. $x=15123,\ y=32477$
  4. $x=28326,\ y=28614$
  5. $x=22536,\ y=27462$

Question 9

The value of
$$\frac1{3+1}+\frac2{3^2+1}+\frac4{3^4+1}+\frac8{3^8+1}+\cdots+\frac{2^{2006}}{3^{2^{2006}}+1}$$
is:
  1. $\frac12$
  2. $\frac12-\frac{2^{2005}}{3^{2^{2005}}-1}$
  3. $\frac12-\frac{2^{2006}}{3^{2^{2006}}-1}$
  4. $\frac12-\frac{2^{2007}}{3^{2^{2007}}-1}$
  5. None of the above

Question 10

Suppose that $p$ and $q$ are prime numbers and they are roots of the equation $x^2-99x+m=0$ for some $m$. What is the value of $\dfrac pq+\dfrac qp$?
  1. $9413$
  2. $\frac{9413}{194}$
  3. $\frac{9413}{99}$
  4. $\frac{9413}{97}$
  5. None of the above

Question 11

What is the remainder when $2006\times2005\times2004\times2003$ is divided by $7$?

Question 12

If $\dfrac{139}{22}=a+\dfrac1{b+\frac1c}$, where $a,b,c$ are positive integers, find $a+b+c$.

Question 13

Let $x$ be a positive real number. Find the minimum value of $x+\dfrac1x$.

Question 14

Find the value, in simplest form, of $\sqrt{45+20\sqrt5}+\sqrt{45-20\sqrt5}$.

Question 15

Let $n$ be the number
$$\bigl(\underbrace{999\cdots999}_{2006\text{ nines}}\bigr)^2-\bigl(\underbrace{666\cdots666}_{2006\text{ sixes}}\bigr)^2.$$
Find the remainder when $n$ is divided by $11$.

Question 16

Given that $w\gt 0$ and $w-\dfrac1w=5$, find $\left(w+\dfrac1w\right)^2$.

Question 17

$N$ pieces of candy are made and packed into boxes, with each box containing $45$ pieces. If $N$ is a non-zero perfect cube and $45$ is one of its factors, what is the least possible number of boxes that can be packed?

Question 18

Consider the following “star” figure. Given that $\angle p+\angle q+\angle r+\angle s+\angle t=500^\circ$ and $\angle A+\angle B+\angle C+\angle D+\angle E=x^\circ$, find $x$.

Question illustration

Question 19

Given that $n$ is a positive integer and $S=1+2+3+\cdots+n$, the units digit of $S$ cannot be some numbers. Find the sum of these numbers.

Question 20

Let $m=76^{2006}-76$. Find the remainder when $m$ is divided by $100$.