Question 1
What is the units digit of $7^5$?
SMO 2005 Junior Round 1
Question 2
What is the value of $(0.5)^3+3\times(0.5)^2\times(-1.5)+3\times(0.5)\times(-1.5)^2+(-1.5)^3$?
Question 3
What is the last two digits of $1+2+3+4+\cdots+2003+2004+2005$?
Question 4
How many zeroes does the number $50\times49\times48\times\cdots\times3\times2\times1$ end with?
Question 5
In a square $ABCD$, let $P$ be a point on the side $BC$ such that $BP=3PC$ and $Q$ be the mid-point of $CD$. If the area of the triangle $PCQ$ is $5$, what is the area of triangle $QDA$?
Question 6
The values of $p$ which satisfy the equations
$$p^2+9q^2+3p-pq=30,\qquad p-5q-8=0$$
may be found by solving which equation?
$$p^2+9q^2+3p-pq=30,\qquad p-5q-8=0$$
may be found by solving which equation?
Question 7
Using the digits $1,2,3,4$ only once to form a $4$-digit number, how many of them are divisible by $11$?
Question 8
If a polygon has its sum of interior angles smaller than $2005^\circ$, what is the maximum number of sides of the polygon?
Question 9
The last $2$ digits of
$$2005+2005^2+2005^3+\cdots+2005^{2005}$$
are:
$$2005+2005^2+2005^3+\cdots+2005^{2005}$$
are:
Question 10
Suppose $3$ distinct numbers are chosen from $1,2,\ldots,3n$ with their sum equal to $3n$. What is the largest possible product of those $3$ numbers?
Question 11
Suppose $a\ne0$, $b\ne0$ and $\frac ba=\frac cb=2005$. Find the value of $\frac{b+c}{a+b}$.
Question 12
Find the exact value of $\sqrt{\frac{x-2}{6}}$ when $x=2006^3-2004^3$.
Question 13
Suppose $x-y=1$. Find the value of
$$x^4-xy^3-x^3y-3x^2y+3xy^2+y^4.$$
$$x^4-xy^3-x^3y-3x^2y+3xy^2+y^4.$$
Question 14
Simplify
$$\frac{(1-\frac1{2^2})(1-\frac1{3^2})(1-\frac1{4^2})\cdots(1-\frac1{2005^2})}{(1-\frac12)(1-\frac13)(1-\frac14)\cdots(1-\frac1{2005})}.$$
$$\frac{(1-\frac1{2^2})(1-\frac1{3^2})(1-\frac1{4^2})\cdots(1-\frac1{2005^2})}{(1-\frac12)(1-\frac13)(1-\frac14)\cdots(1-\frac1{2005})}.$$
Question 15
The figure $ABCDEF$ is a regular hexagon. Evaluate the quotient
$$\frac{\text{Area of hexagon }ABCDEF}{\text{Area of triangle }ACD}.$$
$$\frac{\text{Area of hexagon }ABCDEF}{\text{Area of triangle }ACD}.$$
Question 16
Suppose $x$ and $y$ are two real numbers such that $x+y=10$ and $x^2+y^2=167$. Find the value of $x^3+y^3$.
Question 17
Find the value (in the simplest form) of $\sqrt{9+4\sqrt5}-\sqrt{9-4\sqrt5}$.
Question 18
If $x\gt 0$ and $(x+\frac1x)^2=25$, find the value of $x^3+\frac1{x^3}$.
Question 19
The diagram shows a segment of a circle such that $CD$ is perpendicular bisector of the chord $AB$. Given that $AB=16$ and $CD=4$, find the diameter of the circle.
Question 20
A palindrome number is the same either read from left to right or right to left, for example, $121$ is a palindrome number. How many $5$-digit palindrome numbers are there together?