Found problems: 85335
Any positive integer $N$ which can be expressed as the sum of three squares can obviously be written as \[N=\frac{a^2+b^2+c^2+d^2}{1+abcd}\]where $a,b,c,d$ are nonnegative integers. Is the mutual assertion true?
$k$ and $n$ are positive integers that are greater than $1$. $N$ is the set of positive integers. $A_1, A_2, \cdots A_k$ are pairwise not-intersecting subsets of $N$ and $A_1 \cup A_2 \cup \cdots \cup A_k = N$.
Prove that for some $i \in \{ 1,2,\cdots,k \}$, there exsits infinity many non-factorable n-th degree polynomials so that coefficients of one polynomial are pairwise distinct and all the coeficients are in $A_i$.
Let $\alpha$, $\beta$, $\gamma$ be the angles of a triangle. If $a$, $b$, $c$ are the side length of the triangle and $R$ is the circumradius, show that
\[
\cot \alpha + \cot \beta +\cot \gamma =\frac{R\left(a^2+b^2+c^2\right)}{abc}
\]
Let $a,b,c$ be elements of finite order in some group. Prove that if $a^{-1}ba=b^2$, $b^{-2}cb^2=c^2$, and $c^{-3}ac^3=a^2$ then $a=b=c=e$, where $e$ is the unit element.
Find all triplets $ (x,y,z) $ of real numbers such that
\[ 2x^3 + 1 = 3zx \]\[ 2y^3 + 1 = 3xy \]\[ 2z^3 + 1 = 3yz \]
There is a safe that can be opened by entering a secret code consisting of $n$ digits, each of them is $0$ or $1$. Initially, $n$ zeros were entered, and the safe is closed (so, all zeros is not the secret code).
In one attempt, you can enter an arbitrary sequence of $n$ digits, each of them is $0$ or $1$. If the entered sequence matches the secret code, the safe will open. If the entered sequence matches the secret code in more positions than the previously entered sequence, you will hear a click. In any other cases the safe will remain locked and there will be no click.
Find the smallest number of attempts that is sufficient to open the safe in all cases.
Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.
In an acute $\Delta ABC$, $AH_a$ and $BH_b$ are altitudes and $M$ is the middle point of $AB$. The circumscribed circles of $\Delta AMH_a$ and $\Delta BMH_b$ intersect for a second time in $P$. Prove that point $P$ lies on the circumscribed circle of $\Delta ABC$.
Let the function $f(x,y,t)=\frac{x^2-y^2}{2}-\frac{(x-yt)^2}{1-t^2}$ for all real values $x,y$ and $t\not=\pm1$
a) Evaluate $f(2,0,3)$ and $f(0,2,3)$.
b) Show that $f(x,y,0)=f(y,x,0)$ for any values of $(x,y)$.
c) Show that $f(x,y,t)=f(y,x,t)$ for any values of $(x,y)$ and $t\not=\pm1$.
d) Given
$$g(x,y,s)=\frac{(x^2-y^2)(1+\sin(s))}{2} -\frac{(x-y\sin(s))^2}{1-\sin(s)}$$
for all real values $x,y$ and $s\not=\frac{\pi}{2}+2\pi k$, where $k$ is an integer number, show that $g(x,y,s)=g(y,x,s)$ for any values of $(x,y)$ and $s$ in the domain of $g(x,y,s)$.
Find all natural numbers $ n > 1$ such that $ n^{2}$ does $ \text{not}$ divide $ (n \minus{} 2)!$.
Find the smallest $n$ such that $2^{2000}$ divides $n!$.
$N$ lines lie on a plane, no two of which are parallel and no three of which are concurrent. Prove that there exists a non-self-intersecting broken line $A_0A_1A_2A_3...A_N$ with $N$ parts, such that on each of the $N$ lines lies exactly one of the $N$ segments of the line.
Let $X$ be a point lying in the interior of the acute triangle $ABC$ such that
\begin{align*}
\sphericalangle BAX = 2\sphericalangle XBA \ \ \ \ \hbox{and} \ \ \ \ \sphericalangle XAC = 2\sphericalangle ACX.
\end{align*}
Denote by $M$ the midpoint of the arc $BC$ of the circumcircle $(ABC)$ containing $A$. Prove that $XM=XA$.
Yesterday Han drove $ 1$ hour longer than Ian at an average speed $ 5$ miles per hour faster than Ian. Jan drove $ 2$ hours longer than Ian at an average speed $ 10$ miles per hour faster than Ian. Han drove $ 70$ miles more than Ian. How many more miles did Jan drive than Ian?
$ \textbf{(A)}\ 120 \qquad \textbf{(B)}\ 130 \qquad \textbf{(C)}\ 140 \qquad \textbf{(D)}\ 150 \qquad \textbf{(E)}\ 160$
Let $ABC$ be a triangle and $H$ and $D$ be the feet of the height and bisector relative to $A$ in $BC$, respectively. Let $E$ be the intersection of the tangent to the circumcircle of $ABC$ by $A$ with $BC$ and $M$ be the midpoint of $AD$. Finally, let $r$ be the line perpendicular to $BC$ that passes through $M$. Show that $r$ is tangent to the circumcircle of $AHE$.
Call a sequence of positive integers $\{a_n\}$ good if for any distinct positive integers $m,n$, one has
$$\gcd(m,n) \mid a_m^2 + a_n^2 \text{ and } \gcd(a_m,a_n) \mid m^2 + n^2.$$
Call a positive integer $a$ to be $k$-good if there exists a good sequence such that $a_k = a$. Does there exists a $k$ such that there are exactly $2019$ $k$-good positive integers?
Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed-a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?
$\text{(A)}\ \frac{1}{2} \qquad \text{(B)}\ \frac{2}{3} \qquad \text{(C)}\ \frac{3}{4} \qquad \text{(D)}\ \frac{4}{5} \qquad \text{(E)}\ \frac{5}{6}$
[asy]
draw((0,0)--(0,8)--(6,8)--(6,0)--cycle);
draw((0,8)--(5,9)--(5,8));
draw((3,-1.5)--(3,10.3),dashed);
draw((0,5.5)..(-.75,4.75)..(0,4));
draw((0,4)--(1.5,4),EndArrow);
[/asy]
Three circumferences are tangent to each other, as shown in the figure. The region of the outer circle that is not covered by the two inner circles has an area equal to $2 \pi$. Determine the length of the $PQ$ segment .
[img]https://cdn.artofproblemsolving.com/attachments/a/e/65c08c47d4d20a05222a9b6cf65e84a25283b7.png[/img]
The diagonals of convex quadrilateral $BSCT$ meet at the midpoint $M$ of $\overline{ST}$. Lines $BT$ and $SC$ meet at $A$, and $AB = 91$, $BC = 98$, $CA = 105$. Given that $\overline{AM} \perp \overline{BC}$, find the positive difference between the areas of $\triangle SMC$ and $\triangle BMT$.
[i]Proposed by Evan Chen[/i]
A semicircular paper is folded along a chord such that the folded circular arc is tangent to the diameter of the semicircle. The radius of the semicircle is $4$ units and the point of tangency divides the diameter in the ratio $7 :1$. If the length of the crease (the dotted line segment in the figure) is $\ell$ then determine $ \ell^2$.
[img]https://cdn.artofproblemsolving.com/attachments/5/6/63fed83742c8baa92d9e63962a77a57d43556f.png[/img]
For any positive integer $n$ let $f(n)$ be the number of divisors of $n$ ending with $1$ or $9$ in base $10$ and let $g(n)$ be the number of divisors of $n$ ending with digit $3$ or $7$ in base $10$. Prove that $f(n)\geqslant g(n)$ for all nonnegative integers $n$.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 9)[/i]
Let $\mathcal{L}$ be a finite collection of lines in the plane in general position (no two lines in $\mathcal{L}$ are parallel and no three are concurrent). Consider the open circular discs inscribed in the triangles enclosed by each triple of lines in $\mathcal{L}$. Determine the number of such discs intersected by no line in $\mathcal{L}$, in terms of $|\mathcal{L}|$.
[i]B. Aronov et al.[/i]
A man has $\$$2.73 in pennies, nickels, dimes, quarters and half dollars. If he has an equal number of coins of each kind, then the total number of coins he has is
\[ \text{(A)}\ 3 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 15 \]
On a circular board there are $19$ squares numbered in order from $1$ to $19$ (to the right of $1$ is $2$, to the right of it is $3$, and so on, until $1$ is to the right of $19$). In each box there is a token. Every minute each checker moves to its right the number of the box it is in at that moment plus one; for example, the piece that is in the $7$th place leaves the first minute $7 + 1$ places to its right until the $15$th square; the second minute that same checker moves to your right $15 + 1$ places, to square $12$, etc. Determine if at some point all the tokens reach the place where they started and, if so, say how many minutes must elapse.
[hide=original wording]En un tablero circular hay 19 casillas numeradas en orden del 1 al 19 (a la derecha del 1 está el 2, a la derecha de éste está el 3 y así sucesivamente, hasta el 1 que está a la derecha del 19). En cada casilla hay una ficha. Cada minuto cada ficha se mueve a su derecha el número de la casilla en que se encuentra en ese momento más una; por ejemplo, la ficha que está en el lugar 7 se va el primer minuto 7 + 1 lugares a su derecha hasta la casilla 15; el segundo minuto esa misma ficha se mueve a su derecha 15 + 1 lugares, hasta la casilla 12, etc. Determinar si en algún momento todas las fichas llegan al lugar donde empezaron y, si es así, decir cuántos minutos deben transcurrir.[/hide]
$x^2 + 2y^2 = 1$ solve in integers.