Found problems: 85335
Let $F$ be a finite subset of the integer numbers. We define a new subset $s(F)$ in that $a\in Z$ lies in $s (F)$ if and only if exactly one of the numbers $a$ and $a -1$ in $F$. In the same way one gets from $s (F)$ the set $s^2(F) = s (s (F))$ and by $n$-fold application of $s$ then iteratively further subsets $s^n (F)$.
Prove there are infinitely many natural numbers $n$ for which $s^n (F) = F\cup \{a + n|a \in F\}$.
Let $ a,b$ be two complex numbers. Prove that roots of $ z^{4}\plus{}az^{2}\plus{}b$ form a rhombus with origin as center, if and only if $ \frac{a^{2}}{b}$ is a non-positive real number.
There are two prime numbers $p$ so that $5p$ can be expressed in the form $\left\lfloor \dfrac{n^2}{5}\right\rfloor$ for some positive integer $n.$ What is the sum of these two prime numbers?
For $x^2+2x+5$ to be a factor of $x^4+px^2+q$, the values of $p$ and $q$ must be, respectively:
$ \textbf{(A)}\ -2, 5\qquad\textbf{(B)}\ 5, 25\qquad\textbf{(C)}\ 10, 20\qquad\textbf{(D)}\ 6, 25\qquad\textbf{(E)}\ 14, 25 $
Solve in the real numbers the equation $ |\sin 3x+\cos (7\pi /2 -5x)|=2. $
The orthocenter of an acute triangle $ABC$ is $H$. Let $K$ and $P$ be the midpoints of lines $BC$ and $AH$, respectively. The angle bisector drawn from the vertex $A$ of the triangle $ABC$ intersects with line $KP$ at $D$. Prove that $HD\perp AD$.
Let $P(x) = 2x^3-3x^2+2$, and the sets:
$$A =\{ P(n) | n \in N, n \le 1999\}, B=\{p^2+1 |p \in N\}, C=\{ q^2+2 | q \in N\}$$ Prove that the sets $A \cap B$ and $A\cap C$ have the same number of elements
Let \( n \geq 3 \) be an integer. For every two adjacent vertices \( A \) and \( B \) of a convex \( n \)-gon, we find a vertex \( C \) such that the angle \( \angle ACB \) is the largest, and write down the measure in degrees. Find the smallest possible value of the sum of the written \( n \) numbers.
Suppose that there is a set of $2016$ positive numbers, such that both their sum, and the sum of their reciprocals, are equal to $2017$. Let $x$ be one of those numbers. Find the maximum possible value of $x +\frac{1}{x}$.
.
Older television screens have an aspect ratio of $ 4: 3$. That is, the ratio of the width to the height is $ 4: 3$. The aspect ratio of many movies is not $ 4: 3$, so they are sometimes shown on a television screen by 'letterboxing' - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of $ 2: 1$ and is shown on an older television screen with a $ 27$-inch diagonal. What is the height, in inches, of each darkened strip?
[asy]unitsize(1mm);
defaultpen(linewidth(.8pt));
filldraw((0,0)--(21.6,0)--(21.6,2.7)--(0,2.7)--cycle,grey,black);
filldraw((0,13.5)--(21.6,13.5)--(21.6,16.2)--(0,16.2)--cycle,grey,black);
draw((0,2.7)--(0,13.5));
draw((21.6,2.7)--(21.6,13.5));[/asy]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 2.25 \qquad \textbf{(C)}\ 2.5 \qquad \textbf{(D)}\ 2.7 \qquad \textbf{(E)}\ 3$
In $\triangle ABC$, consider points $D$ and $E$ on $AC$ and $AB$, respectively, and assume that they do not coincide with any of the vertices $A$, $B$, $C$. If the segments $BD$ and $CE$ intersect at $F$, consider areas $w$, $x$, $y$, $z$ of the quadrilateral $AEFD$ and the triangles $BEF$, $BFC$, $CDF$, respectively.
[list=a]
[*] Prove that $y^2 > xz$.
[*] Determine $w$ in terms of $x$, $y$, $z$.
[/list]
[asy]
import graph; size(10cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(12); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -2.8465032978885407, xmax = 9.445649196374966, ymin = -1.7618066305534972, ymax = 4.389732795464592; /* image dimensions */
draw((3.8295013012181283,2.816337276198864)--(-0.7368327629589799,-0.5920813291311117)--(5.672613975760373,-0.636902634996282)--cycle, linewidth(0.5));
/* draw figures */
draw((3.8295013012181283,2.816337276198864)--(-0.7368327629589799,-0.5920813291311117), linewidth(0.5));
draw((-0.7368327629589799,-0.5920813291311117)--(5.672613975760373,-0.636902634996282), linewidth(0.5));
draw((5.672613975760373,-0.636902634996282)--(3.8295013012181283,2.816337276198864), linewidth(0.5));
draw((-0.7368327629589799,-0.5920813291311117)--(4.569287648059735,1.430279997142299), linewidth(0.5));
draw((5.672613975760373,-0.636902634996282)--(1.8844000180622977,1.3644681598392678), linewidth(0.5));
label("$y$",(2.74779188172294,0.23771684184669772),SE*labelscalefactor);
label("$w$",(3.2941097703568736,1.8657441499758196),SE*labelscalefactor);
label("$x$",(1.6660824622277512,1.0025618859342047),SE*labelscalefactor);
label("$z$",(4.288408327670633,0.8168138037986672),SE*labelscalefactor);
/* dots and labels */
dot((3.8295013012181283,2.816337276198864),dotstyle);
label("$A$", (3.8732067323088435,2.925600853925651), NE * labelscalefactor);
dot((-0.7368327629589799,-0.5920813291311117),dotstyle);
label("$B$", (-1.1,-0.7565817154670613), NE * labelscalefactor);
dot((5.672613975760373,-0.636902634996282),dotstyle);
label("$C$", (5.763466626982254,-0.7784344310124186), NE * labelscalefactor);
dot((4.569287648059735,1.430279997142299),dotstyle);
label("$D$", (4.692683565259744,1.5051743434774234), NE * labelscalefactor);
dot((1.8844000180622977,1.3644681598392678),dotstyle);
label("$E$", (1.775346039954538,1.4942479857047448), NE * labelscalefactor);
dot((2.937230516274804,0.8082418657164665),linewidth(4.pt) + dotstyle);
label("$F$", (2.889834532767763,0.954), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]
A scalene triangle $ABC$ is inscribed in a circle $\Gamma$. The bisector of angle $A$ meets $BC$ at $E$. Let $M$ be the midpoint of the arc $BAC$. The line $ME$ intersects $\Gamma$ again at $D$. Show that the circumcentre of triangle $AED$ coincides with the intersection point of the tangent to $\Gamma$ at $D$ and the line $BC$.
Given $ n$ points arbitrarily in the plane $ P_{1},P_{2},\ldots,P_{n},$ among them no three points are collinear. Each of $ P_{i}$ ($1\le i\le n$) is colored red or blue arbitrarily. Let $ S$ be the set of triangles having $ \{P_{1},P_{2},\ldots,P_{n}\}$ as vertices, and having the following property: for any two segments $ P_{i}P_{j}$ and $ P_{u}P_{v},$ the number of triangles having $ P_{i}P_{j}$ as side and the number of triangles having $ P_{u}P_{v}$ as side are the same in $ S.$ Find the least $ n$ such that in $ S$ there exist two triangles, the vertices of each triangle having the same color.
How many non-congruent scalene triangles with perimeter $21$ have integer side lengths that form an arithmetic sequence? (In an arithmetic sequence, successive terms differ by the same amount.)
$\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }6$
Let $A,B,C$ be angles of an acute triangle with
\begin{align*}
\cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\
\cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9}.
\end{align*}
There are positive integers $p$, $q$, $r$, and $s$ for which \[ \cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B = \frac{p-q\sqrt{r}}{s}, \] where $p+q$ and $s$ are relatively prime and $r$ is not divisible by the square of any prime. Find $p+q+r+s$.
[i]Note: due to an oversight by the exam-setters, there is no acute triangle satisfying these conditions. You should instead assume $ABC$ is obtuse with $\angle B > 90^{\circ}$.[/i]
Given a triangle $ABC$ inscribed in $(O)$. Choose points $M,N,P$ on the sides $AB,BC,CA$ such that $AMNP$ is a parallelogram. The segment $CM$ intersects $NP$ at $E$; the segment $BP$ intersects $NM$ at $F$; and the segment $BE$ intersects $CF$ at $D.$
a) Prove that: $A,D,N$ are collinear.
b) Let $I,J$ be the circumcenters of $\triangle MBF, \triangle PCE,$ respectively.
Prove that: $OD$ passes through the midpoint of $IJ.$
The angles of a triangle are $22.5^o, 45^o$ and $112.5^o$. Prove that inside this triangle there exists a point that is located on the median through one vertex, the angle bisector through another vertex and the altitude through the third vertex.
In a triangle $ ABC,$ let $ D$ and $ E$ be the intersections of the bisectors of $ \angle ABC$ and $ \angle ACB$ with the sides $ AC,AB,$ respectively. Determine the angles $ \angle A,\angle B, \angle C$ if $ \angle BDE \equal{} 24 ^{\circ},$ $ \angle CED \equal{} 18 ^{\circ}.$
$ $ $ $ $ $ $ $There is a group of more than three airports. For any two airports $A, B$ belonging to this group, if there is an aircraft from $A$ to $ $ $B$, there is an aircraft from $B$ to $ $ $A$.
For a list of different airports $A_0,A_1,...A_n$, define this list as a '[color=#00f]route[/color]' if there is an aircraft from $A_i$ to $A_{i+1}$ for each $i=0,1,...,n-1$. Also, define the beginning of this [color=#00f]route[/color] as $A_0$, the end as $A_n$, and the length as $n$. ($n\in \mathbb N$)
$ $ $ $ $ $ $ $Now, let's say that for any three different pairs of airports $(A,B,C)$, there is always a [color=#00f]route[/color] $P$ that satisfies the following condition.
$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ [b]Condition[/b]: $P$ begins with $A$ and ends with $B$, and does not include $C$.
$ $ $ $ $ $When the length of the longest of the existing [color=#00f]route[/color]s is $M$ ($\ge 2$), prove that any two [color=#00f]route[/color]s of length $M$ contain at least two different airports simultaneously.
Find all positive integers $n$ for which the following statement holds:
For any two polynomials $P(x)$ and $Q(x)$ of degree $n$ there exist monomials $ax^k$ and $bx^{ell}, 0 \le k,\ ell \le n$, such that the graphs of $P(x) + ax^k$ and $Q(x) + bx^{ell}$ have no common points.
Consider a convex $n$-gon $A_1A_2 \dots A_n$. (Note: In a convex polygon, all interior angles are less than $180 \circ$.) Let $h$ be a positive number. Using the sides of the polygon as bases, we draw $n$ rectangles, each of height $h$, so that each rectangle is either entirely inside the $n$-gon or partially overlaps the inside of the $n$-gon.
As an example, the left figure below shows a pentagon with a correct configuration of rectangles, while the right figure shows an incorrect configuration of rectangles (since some of the rectangles do not overlap with the pentagon):
Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?
$ \textbf{(A)}\ 56 \qquad \textbf{(B)}\ 58 \qquad \textbf{(C)}\ 60 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 64$
Let $n$ be a natural number. Suppose that $x_0=0$ and that $x_i>0$ for all $i\in\{1,2,\ldots ,n\}$. If $\sum_{i=1}^nx_i=1$ , prove that
\[1\leq\sum_{i=1}^{n} \frac{x_i}{\sqrt{1+x_0+x_1+\ldots +x_{i-1}}\sqrt{x_i+\ldots+x_n}} < \frac{\pi}{2} \]
Consider function $f:\mathbb{R}\to \mathbb{R}$ with $f(x)=\frac{4^x}{4^x+2},$ for any $x\in \mathbb{R}$
a) Prove that $f(x)+f(1-x)=1,$
b) Claculate the sum $$f\left(\frac{1}{1986} \right)+f\left(\frac{2}{1986} \right)+\cdots f\left(\frac{1986}{1986} \right).$$
The prime numbers $a,b$ and $c$ are such that $a+b^2=4c^2$. Determine the sum of all possible values of $a+b+c$.