Found problems: 85335
Determine all positive integer numbers $n$ satisfying the following condition:
the sum of the squares of any $n$ prime numbers greater than $3$ is divisible by $n$.
The diagram shows a large circular dart board with four smaller shaded circles each internally tangent to the larger circle. Two of the internal circles have half the radius of the large circle, and are, therefore, tangent to each other. The other two smaller circles are tangent to these circles. If a dart is thrown so that it sticks to a point randomly chosen on the dart board, then the probability that the dart sticks to a point in the shaded area is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Let $n\geq 3$ be an odd integer. On a line, $n$ points are marked in such a way that the distance between any two of them is an integer. It turns out that each marked point has an even sum of distances to the remaining $n-1$ marked points. Prove that the distance between any two marked points is even.
Two players in turn put two or three coins into their own hats (before the game starts, the hats are empty). Each time, after the second player duplicated the move of the first player, they exchange hats. The player wins, if after his move his hat contains one hundred or more coins. Which player has a winning strategy?
As shown in the figure, $\odot O$ is the circumcircle of $\vartriangle ABC$, $\odot J$ is inscribed in $\odot O$ and is tangent to $AB$, $AC$ at points $D$ and E respectively, line segment $FG$ and $\odot O$ are tangent to point $A$, and $AF =AG=AD$, the circumscribed circle of $\vartriangle AFB$ intersects $\odot J$ at point $S$. Prove that the circumscribed circle of $\vartriangle ASG$ is tangent to $\odot J$.
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Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays $40$ coins in toll to Rabbit after each crossing. The payment is made after the doubling, Fox is excited about his good fortune until he discovers that all his money is gone after crossing the bridge three times. How many coins did Fox have at the beginning?
$\textbf{(A)}\ 20 \qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 35\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 45$
Suppose $a$, $b$, and $c$ are positive integers such that \[\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.\] Which of the following statements are necessarily true?
I. If $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both, then $\gcd(c,210)=1$.
II. If $\gcd(c,210)=1$, then $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both.
III. $\gcd(c,210)=1$ if and only if $\gcd(a,14)=\gcd(b,15)=1$.
$\textbf{(A)}~\text{I, II, and III}\qquad\textbf{(B)}~\text{I only}\qquad\textbf{(C)}~\text{I and II only}\qquad\textbf{(D)}~\text{III only}\qquad\textbf{(E)}~\text{II and III only}$
Find the number of permutations $x_1, x_2, x_3, x_4, x_5$ of numbers $1, 2, 3, 4, 5$ such that the sum of five products
$$x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2$$
is divisible by $3$.
Find all pairs of non-negative integers $(m,n)$ satisfying $\frac{n(n+2)}{4}=m^4+m^2-m+1$
A cyclic quadrilateral $ABCD$ has side lengths $AB = 3, BC = AD = 5$, and $CD = 8$. The radius of its circumcircle can be written in the form $a\sqrt{b}/c$, where $a, b, c$ are positive integers, $a, c$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a + b + c$.
The arithmetic mean (average) of the first $n$ positive integers is:
$ \textbf{(A)}\ \frac{n}{2} \qquad\textbf{(B)}\ \frac{n^2}{2}\qquad\textbf{(C)}\ n\qquad\textbf{(D)}\ \frac{n-1}{2}\qquad\textbf{(E)}\ \frac{n+1}{2} $
[b]T[/b]wo ordered pairs $(a,b)$ and $(c,d)$, where $a,b,c,d$ are real numbers, form a basis of the coordinate plane if $ad \neq bc$. Determine the number of ordered quadruples $(a,b,c)$ of integers between $1$ and $3$ inclusive for which $(a,b)$ and $(c,d)$ form a basis for the coordinate plane.
Prove that for any positive integers $k$ and $N$, \[\left(\frac{1}{N}\sum\limits_{n=1}^{N}(\omega (n))^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q},\] where $\sum\limits_{q\leq N}\frac{1}{q}$ is the summation over of prime powers $q\leq N$ (including $q=1$).
Note: For integer $n>1$, $\omega (n)$ denotes number of distinct prime factors of $n$, and $\omega (1)=0$.
Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.
We will say that a set $ A $ of points is [i]disastrous [/i] if it meets the following conditions:
$\bullet$ There are no $ 3 $ collinear points
$\bullet$ There is not a trio of mutually equal distances between points.
If $ P $ and $ Q $ are points in $ A $, then there are $ M $, $ N $, $ R $ and $ T $ in $ A $ such that: $$ d (P, Q) = \frac {d (M, N) + d (R, T)} {2} $$
Show that all disastrous sets are infinite.
[hide=original wording of second condition]No existe ni un trío de distancias entre puntos mutuamente iguales.
[/hide]
A father and two sons went to visit their grandmother, who Raya lives $33$ km from the city. My father has a motor roller, the speed of which $25$ km/h, and with a passenger - $20$ km/h (with two passengers on a scooter It’s impossible to move). Each of the brothers walks along the road at a speed of $5$ km/h. Prove that all three can get to grandma's in $3$ hours
Evaluate $ \int_1^{\sqrt{2}} \frac{x^2\plus{}1}{x\sqrt{x^4\plus{}1}}\ dx$.
The numbers $x, y, z, t, a$ and $b$ are positive integers, so that $xt-yz = 1$ and $$\frac{x}{y} \ge \frac{a}{b} \ge \frac{z}{t} .$$Prove that $$ab \le (x + z) (y +t)$$
Let $ k$ and $ k'$ two concentric circles centered at $ O$, with $ k'$ being larger than $ k$. A line through $ O$ intersects $ k$ at $ A$ and $ k'$ at $ B$ such that $ O$ seperates $ A$ and $ B$. Another line through $ O$ intersects $ k$ at $ E$ and $ k'$ at $ F$ such that $ E$ separates $ O$ and $ F$.
Show that the circumcircle of $ \triangle{OAE}$ and the circles with diametres $ AB$ and $ EF$ have a common point.
Let $ABCD$ be a quadrilateral with a incircle $\omega$. Let $I$ be the center of $\omega$, suppose that the lines $AD$ and $BC$ intersect at $Q$ and the lines $AB$ and $CD$ intersect at $P$ with $B$ is in the segment $AP$ and $D$ is in the segment $AQ$. Let $X$ and $Y$ the incenters of $\triangle PBD$ and $\triangle QBD$ respectively. Let $R$ be the intersection of $PY$ and $QX$. Prove that the line $IR$ is perpendicular to $BD$.
Let $(m,n)$ be the pairs of integers satisfying $2(8n^3+m^3)+6(m^2-6n^2)+3(2m+9n)=437$. Find the sum of all possible values of $mn$.
Find all the triples of prime numbers $(p, q, r)$ such that $$p | 2qr + r \,\,\,, \,\,\,q |2pr + p \,\,\, and \,\,\, r | 2pq + q.$$
Let $ f$ be a differentiable real function and let $ M$ be a positive real number. Prove that if \[ |f(x\plus{}t)\minus{}2f(x)\plus{}f(x\minus{}t)| \leq Mt^2 \; \textrm{for all}\ \;x\ \; \textrm{and}\ \;t\ , \] then \[ |f'(x\plus{}t)\minus{}f'(x)| \leq M|t|.\]
[i]J. Szabados[/i]
Points $X$ and $Y$ are the midpoints of arcs $AB$ and $BC$ of the circumscribed circle of triangle $ABC$. Point $T$ lies on side $AC$. It turned out that the bisectors of the angles $ATB$ and $BTC$ pass through points $X$ and $Y$ respectively. What angle $B$ can be in triangle $ABC$?
Let $ABCD$ be a parallelogram. Points $E, F$ lie on the sides $AB, CD$ respectively,
such that $\angle EDC = \angle FBC$ and $\angle ECD = \angle FAD$. Prove that $AB \geq 2BC$.
[i]Proposed by Pouria Mahmoudkhan Shirazi - Iran[/i]