Found problems: 85335
Positive numbers $x, y, z$ are such that the absolute value of the difference of any two of them are less than $2$. Prove that $$ \sqrt{xy +1}+\sqrt{yz + 1}+\sqrt{zx+ 1} > x+ y + z.$$
For a positive integer $k$, let $s(k)$ denote the number of $1$s in the binary representation of $k$. Prove that for any positive integer $n$,
\[\sum_{i=1}^{n}(-1)^{s(3i)} > 0.\]
[i]Holden Mui[/i]
On the sides $(AB)$, $(BC)$, $(CD)$ and $(DA)$ of the regular tetrahedron $ABCD$, one considers the points $M$, $N$, $P$, $Q$, respectively Prove that
$$MN \cdot NP \cdot PQ \cdot QM \ge AM \cdot BN \cdot CP \cdot DQ.$$
Find the largest prime number $p$ such that when $2012!$ is written in base $p$, it has at least $p$ trailing zeroes.
[i]Author: Alex Zhu[/i]
Let $ S$ be a sequence that:
\[ \left\{
\begin{array}{cc}
S_0\equal{}0\hfill\\
S_1\equal{}1\hfill\\
S_n\equal{}S_{n\minus{}1}\plus{}S_{n\minus{}2}\plus{}F_n& (n>1)
\end{array}
\right.\]
such that $ F_n$ is Fibonacci sequence such that $ F_1\equal{}F_2\equal{}1$. Find $ S_n$ in terms of Fibonacci numbers.
A sector with acute central angle $ \theta$ is cut from a circle of radius 6. The radius of the circle circumscribed about the sector is
$ \textbf{(A)}\ 3\cos\theta \qquad
\textbf{(B)}\ 3\sec\theta \qquad
\textbf{(C)}\ 3 \cos \frac12 \theta \qquad
\textbf{(D)}\ 3 \sec \frac12 \theta \qquad
\textbf{(E)}\ 3$
$18.$ A subset $B$ of the set of first $100$ positive integers has the property that no two elements of $B$ sum to $125.$ What is the maximum possible number of elements in $B ?$
The circumcircle of $\triangle ABC$ has radius $1$ and centre $O$ and $P$ is a point inside the triangle such that $OP=x$. Prove that
\[AP\cdot BP\cdot CP\le(1+x)^2(1-x),\]
with equality if and only if $P=O$.
For positive integers $m,n,$ define $f(m,n)$ as the number of ordered triples $(x,y,z)$ of integers such that
$$
\begin{cases}
xyz=x+y+z+m, \\
\max\{|x|,|y|,|z|\} \leq n
\end{cases}
$$
Does there exist positive integers $m,n,$ such that $f(m,n)=2018?$ Please prove your conclusion.
Let $A$ be the set of the $16$ first positive integers. Find the least positive integer $k$ satisfying the condition: In every $k$-subset of $A$, there exist two distinct $a, b \in A$ such that $a^2 + b^2$ is prime.
While Theophrastus was talking to Aristotle about the classification of plants, had a dog tied to a perfectly smooth cylindrical column of radius $r$, with a very fine rope that wrapped around the column and with a loop. The dog had the extreme free from the rope around his neck. In trying to reach Theophrastus, he put the rope tight and it broke. Find out how far from the column the knot was in the time to break the rope.
[hide=original wording]Mientras Teofrasto hablaba con Arist´oteles sobre la clasificaci´on de las plantas, ten´ıa un perro atado a una columna cil´ındrica perfectamente lisa de radio r, con una cuerda muy fina que envolv´ıa la columna y con un lazo. El perro ten´ıa el extremo libre de la cuerda cogido a su cuello. Al intentar alcanzar a Teofrasto, puso la cuerda tirante y ´esta se rompi´o. Averiguar a qu´e distancia de la columna estaba el nudo en el momento de romperse la cuerda.[/hide]
Prove that if $ a $ is an integer different from $ 1 $ and $ - 1 $, then $ a^4 + 4 $ is not a prime number.
We have garland with $n$ lights. Some lights are on, some are off. In one move we can take some turned on light (only turned on) and turn off it and also change state of neigbour lights. We want to turn off all lights after some moves.. For what $n$ is it always possible?
A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square?
[i]Proposed by Evan Chen[/i]
Andrew generates a finite random sequence $\{a_n\}$ of distinct integers according to the following criteria:
[list]
[*] $a_0 = 1$, $0 < |a_n| < 7$ for all $n$, and $a_i \neq a_j$ for all $i < j$.
[*] $a_{n+1}$ is selected uniformly at random from the set $\{a_n - 1, a_n + 1, -a_n\}$, conditioned on the above rule. The sequence terminates if no element of the set satisfies the first condition.
[/list]
For example, if $(a_0, a_1) = (1, 2)$, then $a_2$ would be chosen from the set $\{-2,3\}$, each with probability $\tfrac12$. Determine the probability that there exists an integer $k$ such that $a_k = 6$.
Cut pyramid $ABCD$ into $8$ equal and similar pyramids, if:
a) $AB = BC = CD$, $\angle ABC =\angle BCD = 90^o$, dihedral angle at edge $BC$ is right
b) all plane angles at vertex $B$ are right and $AB = BC = BD\sqrt2$.
Note. Whether there are other types of triangular pyramids that can be cut into any number similar to the original pyramids (their number is not necessarily $8$ and the pyramids are not necessarily equal to each other) is currently unknown
Let $ ABC$ be an isosceles triangle with $ AB \equal{} AC$. $ M$ is the midpoint of $ BC$ and $ O$ is the point on the line $ AM$ such that $ OB$ is perpendicular to $ AB$. $ Q$ is an arbitrary point on $ BC$ different from $ B$ and $ C$. $ E$ lies on the line $ AB$ and $ F$ lies on the line $ AC$ such that $ E, Q, F$ are distinct and collinear. Prove that $ OQ$ is perpendicular to $ EF$ if and only if $ QE \equal{} QF$.
A particle $(x,y)$ moves so that its angular velocities about $(1,0)$ and $(-1,0)$ are equal in magnitude but opposite in sign. Prove that
$$y(x^2 +y^2 +1)\; dx= x(x^2 +y^2 -1) \;dy,$$
and verify that this is the differential equation of the family of rectangular hyperbolas passing through $(1,0)$ and $(-1,0)$ and having the origin as center.
In a parallelogram $ABCD$ , $BD$ is the largest diagonal. By matching $B$ with $D$ by a bend, a regular pentagon is formed. Calculate the measures of the angles formed by the diagonal $BD$ with each of the sides of the parallelogram.
The set $M$ consists of integers, the smallest of which is $1$ and the greatest $100$. Each member of $M$, except $1$, is the sum of two (possibly identical) numbers in $M$. Of all such sets, find one with the smallest possible number of elements.
For a natural number $ n$, let $ a_n\equal{}\int_0^{\frac{\pi}{4}} (\tan x)^{2n}dx$.
Answer the following questions.
(1) Find $ a_1$.
(2) Express $ a_{n\plus{}1}$ in terms of $ a_n$.
(3) Find $ \lim_{n\to\infty} a_n$.
(4) Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{(\minus{}1)^{k\plus{}1}}{2k\minus{}1}$.
Find all functions $f:\mathbb{R^+}\rightarrow\mathbb{R^+}$ such that
$$\frac{x+f(y)}{xf(y)}=f(\frac{1}{y}+f(\frac{1}{x}))$$
for all positive real numbers $x$ and $y$.
In each unit square of square $100*100$ write any natural number. Called rectangle with sides parallel sides of square $good$ if sum of number inside rectangle divided by $17$. We can painted all unit squares in $good$ rectangle. One unit square cannot painted twice or more.
Find maximum $d$ for which we can guaranteed paint at least $d$ points.
Find real roots $x_1$, $x_2$ of equation $x^5-55x+21=0$, if we know $x_1\cdot x_2=1$
Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:
[list]
[*]In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin.
[*]In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are
two (not necessarily distinct) numbers from the first line.
[*]In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line.
[/list]
Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.