A triangle has an angle of measure $\theta$. It is dissected into several triangles. Is it possible that all angles of the resulting triangles are less than $\theta$, if (a) $\theta = 70^o$ ? (b) $\theta = 80^o$ ?

A car travels $120$ miles from $A$ to $B$ at $30$ miles per hour but returns the same distance at $40$ miles per hour. The average speed for the round trip is closest to: $\textbf{(A)}\ 33\text{ mph} \qquad \textbf{(B)}\ 34\text{ mph} \qquad \textbf{(C)}\ 35\text{ mph} \qquad \textbf{(D)}\ 36\text{ mph} \qquad \textbf{(E)}\ 37\text{ mph}$

Find the largest positive integer $n <30$ such that $\frac{1}{2}(n^8 + 3n^4 -4)$ is not divisible by the square of any prime number.

Let $f(x) = x^n$ where $n$ is a fixed positive integer and $x =1, 2, \cdots .$ Is the decimal expansion $a = 0.f (1)f(2)f(3) . . .$ rational for any value of $n$ ? The decimal expansion of a is defined as follows: If $f(x) = d_1(x)d_2(x) \cdots d_{r(x)}(x)$ is the decimal expansion of $f(x)$, then $a = 0.1d_1(2)d_2(2) \cdots d_{r(2)}(2)d_1(3) . . . d_{r(3)}(3)d_1(4) \cdots .$

Phoenix is counting positive integers starting from $1$. When he counts a perfect square greater than $1$, he restarts at $1$, skipping that square the next time. For example, the first $10$ numbers Phoenix counts are $1$, $2$, $3$, $4$, $1$, $2$, $3$, $5$, $6$, $7$, $...$ How many numbers will Phoenix have counted after counting 1$00$ for the first time?

Call admissible a set $A$ of integers that has the following property: If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$. Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. [i]Proposed by Warut Suksompong, Thailand[/i]

Trapezoid $ABCD$ with bases $AB$ and $CD$ is inscribed in a circle. Prove that the quadrilateral formed by orthogonal projections of any point of this circle onto lines $AC, BC, AD$ and $BD$ is inscribed.

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

Given convex $ n$-gon $ A_1\ldots A_n$. Let $ P_i$ ($ i \equal{} 1,\ldots , n$) be such points on its boundary that $ A_iP_i$ bisects the area of polygon. All points $ P_i$ don't coincide with any vertex and lie on $ k$ sides of $ n$-gon. What is the maximal and the minimal value of $ k$ for each given $ n$?

In the plane $xOy$, a lot of points are considered $$X = \{P (a, b) | (a, b) \in \{1, 2,..., 10\} \times \{1, 2,..., 10 \}\}$$ Determine the number of different lines that can be obtained by joining two of them between the points of the set $X$; so that any two lines are not parallel.

For $p>1,\frac1p+\frac1q=1$ and $r>1$. If $x_{00},y_{00}>0$, and reals $x_{ij},y_{ij},i=1,2,\ldots,n$, $j=1,2,\ldots,m$, then prove that $$\left(\frac{\left(\displaystyle\sum_{j=1}^m\displaystyle\sum_{i=1}^n(x_{ij}+y_{ij})^r\right)^{1/r}}{(x_{00}+y_{00})^{1/q}}\right)^p\le\left(\frac{\left(\displaystyle\sum_{j=1}^m\displaystyle\sum_{i=1}^nx_{ij}^r\right)^{1/r}}{x_{00}^{1/q}}\right)^p+\left(\frac{\left(\displaystyle\sum_{j=1}^m\displaystyle\sum_{i=1}^ny_{ij}^r\right)^{1/r}}{y_{00}^{1/q}}\right)^p$$ with equality if and only if either $x_{ij}=y_{ij}=0$ for $i=1,\ldots,n,j=1,\ldots,m$ or $x_{ij}=\alpha y_{ij}$ for $i=0,1,\ldots,n,j=0,1,\ldots,m$, and some $\alpha>0$. [i]Proposed by Chang-Jian Zhao[/i]

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function with the property that for any $a,b \in \mathbb{R},$ $a<b,$ there are $c_1,c_2 \in [a,b],$ $c_1 \le c_2$ such that $f(c_1)= \min_{x \in [a,b]} f(x)$ and $f(c_2)= \max_{x \in [a,b]} f(x).$ Prove that $f$ is increasing.

Find the minimum value of $\sqrt{58-42x}+\sqrt{149-140\sqrt{1-x^2}}$ where $-1 \le x \le 1$.

How many whole numbers between 1 and 1000 do [b]not[/b] contain the digit 1? $ \textbf{(A)}\ 512 \qquad \textbf{(B)}\ 648 \qquad \textbf{(C)}\ 720 \qquad \textbf{(D)}\ 728 \qquad \textbf{(E)}\ 800$

Are there different integers $a,b,c,d,e,f$ such that they are the $6$ roots of $$(x+a)(x^2+bx+c)(x^3+dx^2+ex+f)=0?$$

Find all integer solutions to the equation $y^2=2x^4+17$.