Found problems: 85335
Evaluate $ \int_0^1 \frac{(x^2\plus{}3x)e^x\minus{}(x^2\minus{}3x)e^{\minus{}x}\plus{}2}{\sqrt{1\plus{}x(e^x\plus{}e^{\minus{}x})}}\ dx$.
We are given $n$ identical cubes, each of size $1\times 1\times 1$. We arrange all of these $n$ cubes to produce one or more congruent rectangular solids, and let $B(n)$ be the number of ways to do this.
For example, if $n=12$, then one arrangement is twelve $1\times1\times1$ cubes, another is one $3\times 2\times2$ solid, another is three $2\times 2\times1$ solids, another is three $4\times1\times1$ solids, etc. We do not consider, say, $2\times2\times1$ and $1\times2\times2$ to be different; these solids are congruent. You may wish to verify, for example, that $B(12) =11$.
Find, with proof, the integer $m$ such that $10^m<B(2015^{100})<10^{m+1}$.
What is the product of real numbers $a$ which make $x^2+ax+1$ a negative integer for only one real number $x$?
$
\textbf{(A)}\ -1
\qquad\textbf{(B)}\ -2
\qquad\textbf{(C)}\ -4
\qquad\textbf{(D)}\ -6
\qquad\textbf{(E)}\ -8
$
Find the smallest natural number $n$ which has the following properties:
a) Its decimal representation has a 6 as the last digit.
b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $n$.
Lysithea and Felix each have a take-out box, and they want to select among $42$ different types of sweets to put in their boxes. They each select an even number of sweets (possibly $0$) to put in their box. In each box, there is at most one sweet of any type, although the boxes may have sweets of the same type in common. The total number of sweets they take out is $42.$ Let $N$ be the number of ways can they select sweets to take out. Compute the remainder when $N$ is divided by $42^2 - 1.$
Let $ ABC $ be a right triangle in $ A, $ and let be a point $ D $ on $ BC. $ The bisectors of $ \angle ADB $ and $ \angle ADC $ intersect $ AB $ and $ AC $ (respectively) in $ M $ and $ N $ (respectively). Show that the small angle between $
BC $ and $ MN $ is equal to $ \frac{1}{2}\cdot\left| \angle ABC -\angle BCA \right| $ if and only if $ D $ is the feet of the perpendicular from $ A. $
[i]Bogdan Enescu[/i]
Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that
$$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$
If $M>1$, prove that the polynomial
$$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$
has no positive roots.
Eight identical cubes with of size $1 \times 1 \times 1$ each have the numbers $1$ through $6$ written on their faces with the number $1$ written on the face opposite number $2$, number $3$ written on the face opposite number $5$, and number $4$ written on the face opposite number $6$. The eight cubes are stacked into a single $2 \times 2 \times 2$ cube. Add all of the numbers appearing on the outer surface of the new cube. Let $M$ be the maximum possible value for this sum, and $N$ be the minimum possible value for this sum. Find $M - N$.
Bob plotted $2003$ green points on the plane, so all triangles with three green vertices have area less than $1$.
Prove that the $2003$ green points are contained in a triangle $T$ of area less than $4$.
A convex$N-$gon with equal sides is located inside a circle. Each side is extended in both directions up to the intersection with the circle so that it contains two new segments outside the polygon. Prove that one can paint some of these new $2N$ segments in red and the rest in blue so that the sum of lengths of all the red segments would be the same as for the blue ones.
[i]($8$ points)[/i]
For all $n\ge 1$, let $a_n=\sum_{k=1}^{n-1}\frac{\sin(\frac{(2k-1)\pi}{2n})}{\cos^2(\frac{(k-1)\pi}{2n})\cos^2(\frac{k\pi}{2n})}$. Determine $\lim_{n\rightarrow \infty}\frac{a_n}{n^3}$.
For a math tournament, each person is assigned an ID which consists of two uppercase letters followed by two digits. All IDs have the property that either the letters are the same, the digits are the same, or both the letters are the same and the digits are the same. Compute the number of possible IDs that the tournament can generate.
Let $A$ be a set of real numbers satisfying the following:
(a) $\sqrt(n^2+1) \in A$ for all positive integers $n$,
(b) if $x \in A$ and $y \in A$, then $x-y \in A$.
Prove that every integer can be written as a product of two different elements in $A$.
Consider a sequence $ x_1, x_2, x_3, ...$, defined by
\begin{align*}x_1 &= \sqrt [3]{3}\\
x_2 &= \sqrt [3]{3} ^ {\sqrt [3]{3}},\end{align*}and in general
\[ x_n \equal{} (x_{n \minus{} 1}) ^ {\sqrt [3]{3}}\,\,\text{ for }\,\,n > 1.
\]What is the smallest value of $ n$ for which $ x_n$ is an integer?
$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 27$
Let $H$ and $O$ be the orthocenter and the circumcenter of triangle $ABC$. The circumcircle of triangle $AOH$ meets the perpendicular bisector of $BC$ at point $A_1 \neq O$. Points $B_1$ and $C_1$ are defined similarly. Prove that lines $AA_1$, $BB_1$ and $CC_1$ concur.
Show that if $\sqrt{x}-\sqrt{y}=10$, then $x-2y\leq200$.
In regular tetrahedron $ABCD$, $E\in AB,F\in CD$, satisfying: $\frac{|AE|}{|EB|}=\frac{|CF|}{|FD|}=\lambda(\lambda\in R_+)$. Note that $f(\lambda)=\alpha_{\lambda}+\beta_{\lambda}$, where $\alpha_{\lambda}=<EF,AC>,\alpha_{\lambda}=<EF,BD>$.
$\text{(A)}$ $f(\lambda)$ increases in $(0,+\infty)$
$\text{(B)}$ $f(\lambda)$ decreases in $(0,+\infty)$
$\text{(C)}$ $f(\lambda)$ increases in $(0,1)$, decreases in $(1,+\infty)$
$\text{(D)}$ $f(\lambda)$ is a fixed value in $(0,+\infty)$
Prove that the number $2^9 +2^{99}$ is divisible by $100$.
For any $a,b,c>0$ satisfying $a+b+c+ab+ac+bc= 3$, prove that
\[2\leq a+b+c+abc\leq 3\]
[i]Proposed by Mohammad Ahmadi[/i]
find all functions $f,g:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $f$ is increasing and also:
$f(f(x)+2g(x)+3f(y))=g(x)+2f(x)+3g(y)$
$g(f(x)+y+g(y))=2x-g(x)+f(y)+y$
Derek rolls three $20$-sided dice. Given their sum is $23,$ find the probability one of them shows a 20.
[i]Individual #2[/i]
Let $F$ be the field of $p^2$ elements, where $p$ is an odd prime. Suppose $S$ is a set of $(p^2-1)/2$ distinct nonzero elements of $F$ with the property that for each $a\neq 0$ in $F$, exactly one of $a$ and $-a$ is in $S$. Let $N$ be the number of elements in the intersection $S \cap \{2a: a \in S\}$. Prove that $N$ is even.
The acute triangle $ABC$ satisfies $\overline {AB}<\overline {BC}<\overline {CA}$. Let $H$ a orthocenter of $ABC$, $D$ a intersection point of $AH$ and $BC$, $E$ a intersection point of $BH$ and $AC$, and $M$ a midpoint of segment $BC$.
A circle with center $E$ and radius $AE$ intersects the segment $AC$ at point $F$($\neq A$), and circumcircle of triangle $BFC$ intersects the segment $AM$ at point $S$.
Let $P$($\neq D$), $Q$($\neq F$) a intersection point of circumcircle of triangle $ASD$ and $DF$, circumcircle of triangle $ASF$ and $DF$ respectively. Also, define $R$ as a intersection point of circumcircles of triangle $AHQ$ and $AEP$. Prove that $R$ lies on line $DF$.
Prove that for $\forall$ $z\in \mathbb{C}$ the following inequality is true:
$|z|^2+2|z-1|\geq 1$,
where $"="$ is reached when $z=1$.
Let $x_1,x_2,x_3$ be three positive real roots of the equation $x^3+ax^2+bx+c=0$ $(a,b,c\in R)$ and $x_1+x_2+x_3\leq 1. $ Prove that $$a^3(1+a+b)-9c(3+3a+a^2)\leq 0$$