Found problems: 85335
Let $a, b$ and $c$ be positive real numbers such that min $\{ab, bc, ca\} \ge 1$. Prove that
$$\sqrt[3]{(a^2 + 1)(b^2 + 1)(c^2 + 1)} \le (\frac{a+b+c}{3} )^2 + 1 $$
$AB$ is the chord of the circle $S$. Circles $S_1$ and $S_2$ touch the circle $S$ at points $P$ and $Q$, respectively, and the segment $AB$ at point $K$. It turned out that $\angle{PBA}=\angle{QBA}$. Prove that $AB$ is the diameter of the circle $S$.
A jobber buys an article at $\$24$ less $12\frac{1}{2}\%$. He then wishes to sell the article at a gain of $33\frac{1}{3}\%$ of his cost after allowing a $20\%$ discount on his marked price. At what price, in dollars, should the article be marked?
${{ \textbf{(A)}\ 25.20 \qquad\textbf{(B)}\ 30.00 \qquad\textbf{(C)}\ 33.60 \qquad\textbf{(D)}\ 40.00 }\qquad\textbf{(E)}\ \text{none of these} } $
Let $b$ and $c$ be any two positive integers. Define an integer sequence $a_n$, for $n\geq 1$, by $a_1=1$, $a_2=1$, $a_3=b$ and $a_{n+3}=ba_{n+2}a_{n+1}+ca_n$.
Find all positive integers $r$ for which there exists a positive integer $n$ such that the number $a_n$ is divisible by $r$.
The triangle ABC has BC=1 and $ \angle BAC \equal{} a$. Find the shortest distance between its incenter and its centroid. Denote this shortest distance by $ f(a)$. When a varies in the interval $ (\frac {\pi}{3},\pi)$, find the maximum value of $ f(a)$.
The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD \equal{} \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP \equal{} \angle DAQ$.
[i]Author: Vyacheslav Yasinskiy, Ukraine[/i]
Find all polynomials with real coefficients $P$ such that
\[ P(x)P(2x^2-1)=P(x^2)P(2x-1)\]
for every $x\in\mathbb{R}$.
Let be a circumcircle of radius $ 1 $ of a triangle whose centered representation in the complex plane is given by the affixes of $ a,b,c, $ and for which the equation $ a+b\cos x +c\sin x=0 $ has a real root in $ \left( 0,\frac{\pi }{2} \right) . $ prove that the area of the triangle is a real number from the interval $ \left( 1,\frac{1+\sqrt 2}{2} \right] . $
[i]Gheorghe Iurea[/i]
Let $ ABCD$ be a quadrilateral inscribed in a circle. The diagonal $ BD$ bisects $ AC$. If $ AB = 10$, $ AD = 12$ and $ DC = 11$, find $ BC$.
How many permutations $(a_1,a_2,\dots,a_{10})$ of $1,2,3,4,5,6,7,8,9,10$ satisfy $|a_1-1|+|a_2-2|+\dots+|a_{10}-10|=4$ ?
$ \textbf{(A)}\ 60 \qquad \textbf{(B)}\ 52 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 44 \qquad \textbf{(E)}\ 36$
Determine the remainder when
\[\prod_{i=1}^{2016} (i^4+5)\]
is divided by $2017$.
The sequence \( (a_n)_{n\geq 1} \) of positive integers is such that \( a_1 = 1 \) and \( a_{m+n} \) divides \( a_m + a_n \) for any positive integers \( m \) and \( n \).
a) Prove that if the sequence is unbounded, then \( a_n = n \) for all \( n \).
b) Does there exist a non-constant bounded sequence with the above properties?
(A sequence \( (a_n)_{n\geq 1} \) of positive integers is bounded if there exists a positive integer \( A \) such that \( a_n \leq A \) for all \( n \), and unbounded otherwise.)
Let $n\geq 2$ and consider the functions $f,g:\{1,2,\dots ,n\}\rightarrow\{1,2,\dots ,n\}$ such that $$g(k)=|\{i\mid f(i)\leq f(k)\}|$$ for all $1\leq k\leq n$.
[list=a]
[*] Show that $f$ is bijective if and only if $g$ is bijective.
[*] If $g$ is a given function, find how many functions $f$ (in terms of $g$) satisfy the hypothesis.
[/list]
[i]Silviu Cristea[/i]
If, instead, the graph is a graph of ACCELERATION vs. TIME and the squirrel starts from rest, then the squirrel has the greatest speed at what time(s) or during what time interval?
(A) at B
(B) at C
(C) at D
(D) at both B and D
(E) From C to D
Let $n$ be an odd natural number. We consider an $n\times n$ grid which is made up of $n^2$ unit squares and $2n(n+1)$ edges. We colour each of these edges either $\color{red} \textit{red}$ or $\color{blue}\textit{blue}$. If there are at most $n^2$ $\color{red} \textit{red}$ edges, then show that there exists a unit square at least three of whose edges are $\color{blue}\textit{blue}$.
[list=a]
[*]Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ be a function such that
$g\colon\mathbb{R}\rightarrow\mathbb{R}$, $g(x)=f(x)+f(2x)$, and
$h\colon\mathbb{R}\rightarrow\mathbb{R}$, $h(x)=f(x)+f(4x)$, are continuous
functions. Prove that $f$ is also continuous.
[*]Give an example of a discontinuous function $f\colon\mathbb{R}
\rightarrow\mathbb{R}$, with the following property: there exists an interval
$I\subset\mathbb{R}$, such that, for any $a$ in $I$, the function $g_{a}
\colon\mathbb{R}\rightarrow\mathbb{R}$, $g_{a}(x)=f(x)+f(ax)$, is continuous.[/list]
Eighteen people are standing in a (socially-distanced) line to enter a grocery store. Five people are wearing a black mask, $6$ are wearing a gray mask, and $7$ are wearing a white mask. Suppose that these $18$ people got on line in a random order. The expected number of pairs of adjacent people wearing different-colored masks can be given by $\tfrac{a}{b}$, where $\gcd(a, b) = 1$. Compute $a+b$.
There is a grid of equilateral triangles with a distance 1 between any two neighboring grid points. An equilateral triangle with side length $n$ lies on the grid so that all of its vertices are grid points, and all of its sides match the grid. Now, let us decompose this equilateral triangle into $n^2$ smaller triangles (not necessarily equilateral triangles) so that the vertices of all these smaller triangles are all grid points, and all these small triangles have equal areas.
Prove that there are at least $n$ equilateral triangles among these smaller triangles.
Let $O$ be the circumcenter of triangle $ABC$. A line through $O$ intersects the sides $CA$ and $CB$ at points $D$ and $E$ respectively, and meets the circumcircle of $ABO$ again at point $P \neq O$ inside the triangle. A point $Q$ on side $AB$ is such that $\frac{AQ}{QB}=\frac{DP}{PE}$. Prove that $\angle APQ = 2\angle CAP$.
[i]Proposed by Dusan Djukic[/i]
Construct outer squares $ABMN$, $BCKL$, $ACPQ$ on sides $[AB]$, $[BC]$, $[CA]$ of triangle $ABC$, respectively. Construct squares $NQZT$ and $KPYX$ on segments $[NQ]$ and $[KP]$. If $Area(ABMN) - Area(BCKL)=1$, what is $Area(NQZT)-Area(KPYX)$?
$
\textbf{(A)}\ \dfrac 34
\qquad\textbf{(B)}\ \dfrac 53
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ 4
$
Is it possible to tile space using a combination of regular tetrahedra and regular octahedra?
(A Belov)
Let $A=\{1,2,\ldots,2n\}$ and let the function $g:A\to A$ be defined by $g(k)=2n-k+1$. Does there exist a function $f:A\to A$ such that $f(k)\ne g(k)$ and $f(f(f(k)))=g(k)$ for all $k\in A$, if (a) $n=999$; (b) $n=1000$?
Let $b \geq 2$ be a positive integer.
(a) Show that for an integer $N$, written in base $b$, to be equal to the sum of the squares of its digits, it is necessary either that $N = 1$ or that $N$ have only two digits.
(b) Give a complete list of all integers not exceeding $50$ that, relative to some base $b$, are equal to the sum of the squares of their digits.
(c) Show that for any base b the number of two-digit integers that are equal to the sum of the squares of their digits is even.
(d) Show that for any odd base $b$ there is an integer other than $1$ that is equal to the sum of the squares of its digits.
Let $p$ be a constant number such that $0<p<1$.
Evaluate
\[\sum_{k=0}^{2004} \frac{p^k (1-p)^{2004-k}}{\displaystyle \int_0^1 x^k (1-x)^{2004-k} dx}\]
Given a pentagon $ABCDE$ with all its interior angles less than $180^\circ$. Prove that if $\angle ABC = \angle ADE$ and $\angle ADB = \angle AEC$, then $\angle BAC = \angle DAE$.