Found problems: 85335
Two polygons are cut from the cardboard. Is it possible that for any disposition of these polygons on the plane they have either common inner points or only a finite number of common points on the boundary?
Some of the values produced by two functions, $f(x)$ and $g(x)$, are shown below. Find $f(g(3))$
\begin{tabular}{c||c|c|c|c|c}
$x$ & 1 & 3 & 5 & 7 & 9 \\ \hline\hline
$f(x)$ & 3 & 7 & 9 & 13 & 17 \\ \hline
$g(x)$ & 54 & 9 & 25 & 19 & 44
\end{tabular}
$\textbf{(A) }3\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }13\qquad\textbf{(E) }17$
A triangle is given whose altitudes' lengths are $\frac{1}{5},\frac{1}{5},\frac{1}{8}$. Evaluate the triangle's area.
On two intersecting lines $\ell_1$ and $\ell_2$, segments $AB$ and $CD$ of a given length are selected, respectively. Prove that the volume of the tetrahedron $ABCD$ does not depend on the position of the segments $AB$ and $CD$ on the lines $\ell_1$ and $\ell_2$.
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x,y$, we have $$f(xf(x)+2y)=f(x)^2+x+2f(y)$$
$X$ has $n$ elements. $F$ is a family of subsets of $X$ each with three elements, such that any two of the subsets have at most one element in common. Show that there is a subset of $X$ with at least $\sqrt{2n}$ members which does not contain any members of $F$.
Let $ n\ge2$ be a positive integer and denote by $ S_n$ the set of all permutations of the set $ \{1,2,\ldots,n\}$. For $ \sigma\in S_n$ define $ l(\sigma)$ to be $ \displaystyle\min_{1\le i\le n\minus{}1}\left|\sigma(i\plus{}1)\minus{}\sigma(i)\right|$. Determine $ \displaystyle\max_{\sigma\in S_n}l(\sigma)$.
Two different prime numbers $p$ and $q$ differ in less than $2$ times. Prove that exists two consecutive natural numbers, such that largest prime divisor of first number is $p$, and largest prime divisor of second number is $q$.
Recall that an icosahedron is a $3$-dimensional solid characterized by its $20$ congruent faces, each of which is an equilateral triangle. Determine the number of rigid rotations that preserve the symmetry of the icosahedron. (Each vertex moves to the location of another vertex.)
The numbers from 1 to $n$ are arranged in some way on a circle. What’s the smallest value of $n$, for which no matter how the numbers are arranged there exist ten consecutively increasing numbers $A_1<A_2<A_3…<A_{10}$ such that $A_1 A_2…A_{10}$ is a convex decagon?
Sasha wants to bake $6$ cookies in his $8$ inch $\times$ $8$ inch square baking sheet. With a cookie cutter, he cuts out from the dough six circular shapes, each exactly $3$ inches in diameter. Can he place these six dough shapes on the baking sheet without the shapes touching each other? If yes, show us how. If no, explain why not. (Assume that the dough does not expand during baking.)
Let $ABC$ be a triangle, and let $O$ be its circumcenter. Let $\overline{CO}\cap AB\equiv D$. Let $\angle BAC=\alpha$, and $\angle CBA=\beta$. Prove that $$\dfrac{OD}{OC}=\Bigg|\dfrac{\cos(\alpha+\beta)}{\cos(\alpha-\beta)}\Bigg|$$\\
For clarification, $\overline{CO}$ represents the line $CO$, and $AC$ represents the segment $AC$. Cases in which $D$ doesn't exist should be ignored.
Given an equilateral triangle $ABC$ and a point $P$ so that the distances $P$ to $A$ and to $C$ are not farther than the distances $P$ to $B$. Prove that $PB = PA + PC$ if and only if $P$ lies on the circumcircle of $\vartriangle ABC$.
Let $ABCD$ be a convex quadrilateral such that $m \left (\widehat{DAB} \right )=m \left (\widehat{CBD} \right )=120^{\circ}$, $|AB|=2$, $|AD|=4$ and $|BC|=|BD|$. If the line through $C$ which is parallel to $AB$ meets $AD$ at $E$, what is $|CE|$?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 5
\qquad\textbf{(E)}\ \text{None of the preceding}
$
Let $x, y, z$ be non-negative numbers, so that $x + y + z = 1$. Prove that
$$\sqrt{x+\frac{(y-z)^2}{12}}+\sqrt{y+\frac{(x-z)^2}{12}}+\sqrt{z+\frac{(x-y)^2}{12}}\le \sqrt{3}$$
Compute $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}.$
Consider $N$ points in the plane such that the area of a triangle formed by any three of the points does not exceed $1$. Prove that there is a triangle of area not more than $4$ that contains all $N$ points.
Compute $\displaystyle\int_0^1\dfrac{dx}{\sqrt{x}+\sqrt[3]{x}}$.
Lisa writes a positive whole number in the decimal system on the blackboard and now makes in each turn the following:
The last digit is deleted from the number on the board and then the remaining shorter number (or 0 if the number was one digit) becomes four times the number deleted number added. The number on the board is now replaced by the result of this calculation.
Lisa repeats this until she gets a number for the first time was on the board.
(a) Show that the sequence of moves always ends.
(b) If Lisa begins with the number $53^{2022} - 1$, what is the last number on the board?
Example: If Lisa starts with the number $2022$, she gets $202 + 4\cdot 2 = 210$ in the first move and overall the result $$2022 \to 210 \to 21 \to 6 \to 24 \to 18 \to 33 \to 15 \to 21$$.
Since Lisa gets $21$ for the second time, the turn order ends.
[i](Stephan Pfannerer)[/i]
Evaluate
\[ \int_0^{\pi} (\sqrt[2009]{\cos x}\plus{}\sqrt[2009]{\sin x}\plus{}\sqrt[2009]{\tan x})\ dx.\]
For a sequence of $10$ coin flips, each pair of consecutive flips and count the number of “Heads-Heads”, “Heads-Tails”, “Tails-Heads”, and “Tails-Tails” sequences is recorded. These four numbers are then multiplied to get the [i]Tiger number[/i] of the sequence of flips. How many such sequences have a [i]Tiger number [/i] of $24$?
In an acute-angled triangle $ABC$, let $M$ be the midpoint of $AB$ and $P$ the foot of the altitude to $BC$. Prove that if $AC+BC = \sqrt{2}AB$, then the circumcircle of triangle $BMP$ is tangent to $AC$.
Let $A$, $B$, $C$, $D$ be four points in the plane, with $C$ and $D$ on the same side of the line $AB$, such that $AC \cdot BD = AD \cdot BC$ and $\angle ADB = 90^{\circ}+\angle ACB$. Find the ratio
\[\frac{AB \cdot CD}{AC \cdot BD}, \]
and prove that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicuar. Thus, proving that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal is equivalent to proving that the tangents to the circumcircles of the triangles $ACD$ and $BCD$ at the point $C$ are perpendicular.)
For arbitrary natural number $a$, show that $\gcd(a^3+1, a^7+1)=a+1$.
If a certain number is doubled and the result is increased by $11$, the final number is $23$. What is the
original number?