SMO 2010 Junior Round 1

Among the five real numbers below, which one is the smallest?

Among the five integers below, which one is the largest?

Among the four statements on real numbers below, how many are correct?
1. If $a\lt b$ and $a,b\ne0$, then $1/b\lt 1/a$.
2. If $a\lt b$, then $ac\lt bc$.
3. If $a\lt b$, then $a+c\lt b+c$.
4. If $a^2\lt b^2$, then $a\lt b$.

What is the largest integer less than or equal to $\sqrt[3]{2010^3+3\times2010^2+4\times2010+1}$?

The conditions of the road between Town $A$ and Town $B$ can be classified as up slope, horizontal or down slope, and the total length of each type of road is the same. A cyclist travels from Town $A$ to Town $B$ with uniform speeds $8$ km/h, $12$ km/h and $24$ km/h on the up slope, horizontal and down slope road respectively. What is the average speed of his journey?

In the diagram, $ABC$ and $CDE$ are equilateral triangles. Given $\angle EBD=62^\circ$ and $\angle AEB=x^\circ$, what is $x$?

A carpenter wishes to cut a wooden $3\times3\times3$ cube into twenty-seven $1\times1\times1$ cubes. He can do this easily by making $6$ cuts through the cube, keeping the pieces together in the cube shape as shown.



What is the minimum number of cuts needed if he is allowed to rearrange the pieces after each cut?

What is the last digit of $7^{7^7}$?

Given that $n$ is an odd integer less than $1000$ and the product of all its digits is $252$, how many such integers are there?

What is the value of $(\sqrt{11}+\sqrt5)^8+(\sqrt{11}-\sqrt5)^8$?

Let $x,y$ be real numbers satisfying
$$y=\sqrt{\frac{2008x+2009}{2010x-2011}}+\sqrt{\frac{2008x+2009}{2011-2010x}}+2010.$$
Find $y$.

For integers $a_1,\ldots,a_n\in\{1,2,3,\ldots,9\}$, the notation $\overline{a_1a_2\cdots a_n}$ denotes $10^{n-1}a_1+10^{n-2}a_2+\cdots+10a_{n-1}+a_n$. For example, when $a=2,b=0$, $\overline{ab}$ denotes $20$. Given $\overline{ab}=b^2$ and $\overline{acbc}=(\overline{ba})^2$, find $\overline{abc}$.

Given that $m-2$ is a positive integer and is also a factor of $3m^2-2m+10$, find the sum of all such values of $m$.

In triangle $ABC$, $AB=32$ cm, $AC=36$ cm and $BC=44$ cm. If $M$ is the midpoint of $BC$, find $AM$ in cm.

Evaluate $\dfrac{678+690+702+714+\cdots+1998+2010}{3+9+15+21+\cdots+327+333}$.

Esther and Frida are supposed to fill a rectangular array of $16$ columns and $10$ rows with the numbers $1$ to $160$. Esther does it row-wise: the first row is $1,2,\ldots,16$, the second is $17,18,\ldots,32$, and so on. Frida does it column-wise: the first column is $1,2,\ldots,10$, the second is $11,12,\ldots,20$, and so on. Some numbers occupy the same position in both arrays. Find the sum of the numbers in these positions.

The sum of two integers $A,B$ is $2010$. If their lowest common multiple is $14807$, write down the larger integer.

A sequence of polynomials $a_n(x)$ is defined recursively by
$$a_0(x)=1,\qquad a_1(x)=x^2+x+1,$$
$$a_n(x)=(x^n+1)a_{n-1}(x)-a_{n-2}(x)\quad(n\ge2).$$
For example,
$$a_2(x)=(x^2+1)(x^2+x+1)-1=x^4+x^3+2x^2+x,$$
$$a_3(x)=(x^3+1)(x^4+x^3+2x^2+x)-(x^2+x+1)$$
$$=x^7+x^6+2x^5+2x^4+x^3+x^2-1.$$
Evaluate $a_{2010}(1)$.

A triangle $ABC$ is inscribed in a semicircle of radius $5$. If $AB=10$, find the maximum value of $s^2$, where $s=AC+BC$.

Find the last two digits of $2011^{2010^{2009}}$.