Found problems: 85335
Let $ABC$ be triangle with $A$-excenter $J$. The reflection of $J$ in $BC$ is $K$. The points $E$ and $F$ are on $BJ, CJ$ such that $\angle EAB=\angle CAF=90^{\circ}$. Prove that $\angle FKE+\angle FJE=180^{\circ}$.
In triangle $ ABC$, $ AB \equal{} 10$, $ BC \equal{} 14$, and $ CA \equal{} 16$. Let $ D$ be a point in the interior of $ \overline{BC}$. Let $ I_B$ and $ I_C$ denote the incenters of triangles $ ABD$ and $ ACD$, respectively. The circumcircles of triangles $ BI_BD$ and $ CI_CD$ meet at distinct points $ P$ and $ D$. The maximum possible area of $ \triangle BPC$ can be expressed in the form $ a\minus{}b\sqrt{c}$, where $ a$, $ b$, and $ c$ are positive integers and $ c$ is not divisible by the square of any prime. Find $ a\plus{}b\plus{}c$.
If the product $\dfrac{3}{2}\cdot \dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 9$, what is the sum of $a$ and $b$?
$\textbf{(A)}\ 11 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 17 \qquad \textbf{(D)}\ 35 \qquad \textbf{(E)}\ 37$
Let be a nonnegative integer $ n. $ Find all continuous functions $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ for which the following equation holds:
$$ (1+n)\int_0^x f(t) dt =nxf(x) ,\quad\forall x>0. $$
Show that there are no positive real numbers $x, y, z$ such $(12x^2+yz)(12y^2+xz)(12z^2+xy)= 2014x^2y^2z^2$ .
Determine all pairs of real numbers $a$ and $b$, $b > 0$, such that the solutions to the two equations
$$x^2 + ax + a = b \qquad \text{and} \qquad x^2 + ax + a = -b$$
are four consecutive integers.
Given triangle $ABC$ and point $P$. Points $A', B', C'$ are the projections of $P$ to $BC, CA, AB$. A line passing through $P$ and parallel to $AB$ meets the circumcircle of triangle $PA'B'$ for the second time in point $C_{1}$. Points $A_{1}, B_{1}$ are defined similarly. Prove that
a) lines $AA_{1}, BB_{1}, CC_{1}$ concur;
b) triangles $ABC$ and $A_{1}B_{1}C_{1}$ are similar.
Let $a,b,c$ real numbers. Show that there are non-negative $x,y,z,xyz\neq0$ such that
\begin{align*}
cy-bz &\ge 0, \\
az-cx &\ge 0, \\
bx-ay &\ge 0.
\end{align*}
A $5 \times 5$ grid is given, called $f_1$:
\[
\begin{array}{ccccc}
1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 \\
-1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 \\
\end{array}
\]
A new grid $f_{n+1}$ is constructed where each cell is equal to the product of its neighboring cells in grid $f_n$.
(a) Find the grids $f_6$ and $f_7$.
(b) Find the grids $f_{2008}$ and $f_{2009}$.
(c) Find $f_{2n}$ and $f_{2n+1}$ for any $n \in \mathbb{N}$.
*Note: Neighboring cells are those that share an edge, not just a vertex.*
In cube $ABCD-A_1B_1C_1D_1$, $E$ is midpoint of $BC$, $F\in AA_1$, and $A_1F:FA=1:2$. Calculate the dihedral angle between plane $B_1EF$ and plane $A_1B_1C_1D_1$.
Solve in the real numbers the equation $ 2^{1+x} =2^{[x]} +2^{\{x\}} , $ where $ [],\{\} $ deonotes the ineger and fractional part, respectively.
[i]Aurel Bârsan[/i]
In rectangle $ABCD$, $AD=1$, $P$ is on $AB$ and $DB$ and $DP$ trisect $\angle ADC$. What is the perimeter $\triangle BDP$
[list=1]
[*] $3+\frac{\sqrt{3}}{3}$
[*] $2+\frac{4\sqrt{3}}{3}$
[*] $2+2\sqrt{2}$
[*] $\frac{3+3\sqrt{5}}{2}$
[/list]
Let $ I$ be the incenter of triangle $ ABC.$ Let $ M,N$ be the midpoints of $ AB,AC,$ respectively. Points $ D,E$ lie on $ AB,AC$ respectively such that $ BD\equal{}CE\equal{}BC.$ The line perpendicular to $ IM$ through $ D$ intersects the line perpendicular to $ IN$ through $ E$ at $ P.$ Prove that $ AP\perp BC.$
A positive integer $k > 1$ is called nice if for any pair $(m, n)$ of positive integers satisfying the condition $kn + m | km + n$ we have $n | m$.
1. Prove that $5$ is a nice number.
2. Find all the nice numbers.
Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls?
$\textbf{(A) }\dfrac1{64}\qquad\textbf{(B) }\dfrac16\qquad\textbf{(C) }\dfrac14\qquad\textbf{(D) }\dfrac5{16}\qquad\textbf{(E) }\dfrac12$
Increasing the radius of a cylinder by $ 6$ units increased the volume by $ y$ cubic units. Increasing the altitude of the cylinder by $ 6$ units also increases the volume by $ y$ cubic units. If the original altitude is $ 2$, then the original radius is:
$ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 6\pi \qquad\textbf{(E)}\ 8$
Find a factor of $2^{33}-2^{19}-2^{17}-1$ that lies between $1000$ and $5000$.
Lines $l$ and $m$ are perpendicular. Line $l$ partitions a convex polygon into two parts of equal area, and partitions the projection of the polygon onto $m$ into two line segments of length $a$ and $b$ respectively. Determine the maximum value of $\left\lfloor \frac{1000a}{b} \right\rfloor$. (The floor notation $\lfloor x \rfloor$ denotes largest integer not exceeding $x$)
Calculate the integral of the Gaussian curvature of the surface
\[z^4+(x^2+y^2-1)(2x^2+3y^2-1)=0\]
A prism is called [i]binary [/i] if it can be assigned to each of its vertices a number from the set $\{-1, 1\}$, such that the product of the numbers assigned to the vertices of each face is equal to $-1$.
a) Prove that the number of vertices of the binary prisms is divisible for $8$.
b) Prove that a prism with $2000$ vertices is binary.
Let $n$ be a positive integer. Prove that for each factor $m$ of the number $1+2+\cdots+n$ such that $m\ge n$, the set $\{1,2,\ldots,n\}$ can be partitioned into disjoint subsets, the sum of the elements of each being equal to $m$.
[b]Edit[/b]:Typographical error fixed.
Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that
\[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\]
If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$
Let $f:\mathbb N\to (0,\infty)$ satisfy $\prod_{d\mid n} f(d) = 1$ for every $n$ which is not prime. Determine the maximum possible number of $n$ with $1\le n \le 100$ and $f(n)\ne 1$.
There are $n$ blocks placed on the unit squares of a $n \times n$ chessboard such that there is exactly one block in each row and each column. Find the maximum value $k$, in terms of $n$, such that however the blocks are arranged, we can place $k$ rooks on the board without any two of them threatening each other.
(Two rooks are not threatening each other if there is a block lying between them.)
Let $a, b, c$ be the roots of the equation $x^3-9x^2+11x-1 = 0$, and define $s =\sqrt{a}+\sqrt{b}+\sqrt{c}$.
Compute $s^4 -18s^2 - 8s$ .