Found problems: 85335
Find all triplets of positive integers $(x, y, z)$ such that $x^2 + y^2 + x + y + z = xyz + 1$.
[i]Proposed by Viktor Simjanoski[/i]
If $ a$, $ b$, $ x$, $ y$ are integers greater than 1 such that $ a$ and $ b$ have no common factor except 1 and $ x^a \equal{} y^b$ show that $ x \equal{} n^b$, $ y \equal{} n^a$ for some integer $ n$ greater than 1.
The value of $x^2-6x+13$ can never be less than:
$ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 4.5 \qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 13 $
Let $\Gamma $ be the circumcircle of triangle $ABC$. A circle $\Omega$ is tangent to the line segment $AB$ and is tangent to $\Gamma$ at a point lying on the same side of the line $AB$ as $C$. The angle bisector of $\angle BCA$ intersects $\Omega$ at two different points $P$ and $Q$.
Prove that $\angle ABP = \angle QBC$.
Let skew lines $PM, QN$ be given such that $PM\perp PQ\perp QN$. Let a plane $\sigma\perp PQ$ containing the midpoint $O$ of segment $PQ$ be given and in it a circle $k$ with center $O$ and given radius $r$. Consider all segments $XY$ with endpoint $X, Y$ on lines $PM, QN$, respectively, which contain a point of $k$. Show that segments $XY$ have the same length. Find the locus of all such points $X$.
If $x$, $y$, $k$ are positive reals such that \[3=k^2\left(\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}\right)+k\left(\dfrac{x}{y}+\dfrac{y}{x}\right),\] find the maximum possible value of $k$.
Solve the inequality
$$\sqrt{(x-2)^2(x-x^2)}<\sqrt{4x-1-(x^2-3x)^2}$$
An eight-digit number is said to be 'robust' if it meets both of the following conditions:
(i) None of its digits is $0$.
(ii) The difference between two consecutive digits is $4$ or $5$.
Answer the following questions:
(a) How many are robust numbers?
(b) A robust number is said to be 'super robust' if all of its digits are distinct. Calculate the sum of all
the super robust numbers.
In the convex quadrilateral $ABCD$, $M$ is the midpoint of side $AD$, $AD = BD$, lines $CM$ and $AB$ are parallel, and $3\angle LBAC = \angle LACD$. Find the measure of angle $\angle ACB$.
The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.
$ \textbf{(A)}\ 68\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 169\qquad\textbf{(D)}\ 208\qquad\textbf{(E)}\ 289 $
Find all rational numbers $x,y$ such that \[\left(x+y\sqrt5\right)^2=7+3\sqrt5.\]
For each positive integer $n$ and non-negative integer $k$, define $W(n,k)$ recursively by
\[
W(n,k) =
\begin{cases}
n^n & k = 0 \\
W(W(n,k-1), k-1) & k > 0.
\end{cases}
\]
Find the last three digits in the decimal representation of $W(555,2)$.
There are 6 distinct values of $x$ strictly between $0$ and $\frac{\pi}{2}$ that satisfy the equation
\[
\tan(15 x) = 15 \tan(x) .
\]
Call these 6 values $r_1$, $r_2$, $r_3$, $r_4$, $r_5$, and $r_6$. What is the value of the sum
\[
\frac{1}{\tan^2 r_1} +
\frac{1}{\tan^2 r_2} +
\frac{1}{\tan^2 r_3} +
\frac{1}{\tan^2 r_4} +
\frac{1}{\tan^2 r_5} +
\frac{1}{\tan^2 r_6} \, ?
\]
An [i]Egyptian number[/i] is a positive integer that can be expressed as a sum of positive integers, not necessarily distinct, such that the sum of their reciprocals is $1$. For example, $32 = 2 + 3 + 9 + 18$ is Egyptian because $\frac 12 +\frac 13 +\frac 19 +\frac{1}{18}=1$ . Prove that all integers greater than $23$ are [i]Egyptian[/i].
$ABCD$ is a square with side length 20. A light beam is radiated from $A$ and intersects sides $BC,CD,DA$ respectively and reaches the midpoint of side $AB$. What is the length of the path that the beam has taken?
[img]https://s8.uupload.ir/files/photo14908575660_2r3g.jpg[/img]
[i]Proposed by Mahdi Etesamifard - Iran[/i]
Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$
[i]Michael Ren and Ankan Bhattacharya, USA[/i]
An equilateral triangle $ABE$ is built inside the square $ABCD$ on the side $AB$, and an equilateral triangle $AFC$ is built on the diagonal $AC$ ($D$ is inside this triangle). The segment $EF$ intersects $CD$ at point $P$. Prove that the lines $AP$, $BE$ and $CF$ intersect at the same point.
Let $z_1,z_2,...,z_n$ be complex numbers satisfying $|z_i - 1| \leq r$ for some $r$ in $(0,1)$. Show that
\[ \left | \sum_{i=1}^n z_i \right | \cdot \left | \sum_{i=1}^n \frac{1}{z_i} \right | \geq n^2(1-r^2).\]
Let $a, b, c$ be positive real numbers such that
$$a^2 + b^2 + c^2 = \frac{1}{4}.$$
Prove that
$$\frac{1}{\sqrt{b^2 + c^2}} + \frac{1}{\sqrt{c^2 + a^2}} + \frac{1}{\sqrt{a^2 + b^2}} \le \frac{\sqrt{2}}{(a + b)(b + c)(c + a)}.$$
[i]Proposed by Petar Filipovski, Macedonia[/i]
Four right triangles, each with the sides $1$ and $2$, are assembled to a figure as shown.
How large a fraction does the area of the small circle make up of that of the big one?
[img]https://1.bp.blogspot.com/-XODK1XKCS0Q/XzXDtcA-xAI/AAAAAAAAMWA/zSLPpf3IcX0rgaRtOxm_F2begnVdUargACLcBGAsYHQ/s0/2010%2BMohr%2Bp1.png[/img]
Medians $AA_1, BB_1, CC_1$ and altitudes $AA_2, BB_2, CC_2$ are drawn in triangle $ABC$ . Prove that the length of the broken line $A_1B_2C_1A_2B_1C_2A_1$ is equal to the perimeter of triangle $ABC$.
Some unit cubes are stacked atop a flat 4 by 4 square. The figures show views of the stacks from two different sides. Find the maximum and minimum number of cubes that could be in the stacks. Also give top views of a maximum arrangement and a minimum arrangement with each stack marked with its height.
[asy]
string s = "1010101010111111";
defaultpen(linewidth(0.7));
for(int x=0;x<4;++x) {
for(int y=0;y<4;++y) {
if(hex(substr(s,4*(3-y)+x,1))==1) {
draw((x,y)--(x,y+1)--(x+1,y+1)--(x+1,y)--cycle);
}
}}
label("South View",(2,4),N);
s = "0101110111111111";
for(int x=0;x<4;++x) {
for(int y=0;y<4;++y) {
if(hex(substr(s,4*(3-y)+x,1))==1) {
x=x+5;
draw((x,y)--(x,y+1)--(x+1,y+1)--(x+1,y)--cycle);
x=x-5;
}
}}
label("East View",(7,4),N);[/asy]
$10$ cities are connected by one-way air routes in a way so that each city can be reached from any other by several connected flights. Let $n$ be the smallest number of flights needed for a tourist to visit every city and return to the starting city. Clearly $n$ depends on the flight schedule. Find the largest $n$ and the corresponding flight schedule.
Prove that for all positive integers $n$, $169$ divides $21n^2 + 89n + 44$ if $13$ divides $n^2 + 3n + 51$.
What are the last two digits of ${7^{7^{7^7}}}$?