Found problems: 85335
Given an integer $n$, an integer $1 \le a \le n$ is called $n$-[i]well[/i] if \[ \left\lfloor\frac{n}{\left\lfloor n/a \right\rfloor}\right\rfloor = a. \] Let $f(n)$ be the number of $n$-well numbers, for each integer $n \ge 1$. Compute $f(1) + f(2) + \ldots + f(9999)$.
[i]Proposed by Ashwin Sah[/i]
Let $m$ be a positive integer and $a$ a positive real number. Find the greatest value of $a^{m_1}+a^{m_2}+\ldots+a^{m_p}$ where $m_1+m_2+\ldots+m_p=m, m_i\in\mathbb{N},i=1,2,\ldots,p;$ $1\leq p\leq m, p\in\mathbb{N}$.
Suppose $M$ is an infinite set of natural numbers such that, whenever the sum of two natural numbers is in $M$, one of these two numbers is in $M$ as well. Prove that the elements of any finite set of natural numbers not belonging to $M$ have a common divisor greater than $1$.
There are some balls, on each of which a positive integer not exceeding $14$ (and not necessarily distinct) is written, and the sum of the numbers on all balls is $S$. Find the greatest possible value of $S$ such that, regardless of what the integers are, one can ensure that the balls can be divided into two piles so that the sum of the numbers on the balls in each pile does not exceed $129$.
The set $\{1, 2, . . . , 10\}$ is partitioned to three subsets $A, B$ and $C.$ For each subset the sum of its elements, the product of its elements and the sum of the digits of all its elements are calculated.
Is it possible that $A$ alone has the largest sum of elements, $B$ alone has the largest product of elements, and $C$ alone has the largest sum of digits?
Does there exists a function $f : \mathbb{N} \longrightarrow \mathbb{N}$
\begin{align*} f \left( m+ f(n) \right) = f(m) +f(n) + f(n+1) \end{align*}
for all $m,n \in \mathbb{N}$ ?
Show that \[ \prod_{1\leq x < y \leq \frac{p\minus{}1}{2}} (x^2\plus{}y^2) \equiv (\minus{}1)^{\lfloor\frac{p\plus{}1}{8}\rfloor} \;(\textbf{mod}\;p\ ) \] for every prime $ p\equiv 3 \;(\textbf{mod}\;4\ )$. [J. Suranyi]
The dimensions of a rectangular box in inches are all positive integers and the volume of the box is $2002\text{ in}^3$. Find the minimum possible sum in inches of the three dimensions.
$\textbf{(A) }36\qquad\textbf{(B) }38\qquad\textbf{(C) }42\qquad\textbf{(D) }44\qquad\textbf{(E) }92$
Let $A$ be an arbitrary angle,let $B$ and $C$ be acute angles. Is there an angle $x$ such that $$\sin x =\frac{\sin B \cdot \sin C}{1 - \cos B \cdot \cos C \cdot \cos A} ?$$
In a square $ABCD$, $E$ is the midpoint of $BC$ and $F$ is the midpoint of $CD$. Prove that $AF$ and $AE$ divide the diagonal $BD$ in three equal segments.
Let $P (x) = x^{3}-3x+1.$ Find the polynomial $Q$ whose roots are the fifth powers of the roots of $P$.
Find all triangles $ABC$ for which $\angle ACB$ is acute and the interior angle bisector of $BC$ intersects the trisectors $(AX, (AY$ of the angle $\angle BAC$ in the points $N,P$ respectively, such that $AB=NP=2DM$, where $D$ is the foot of the altitude from $A$ on $BC$ and $M$ is the midpoint of the side $BC$.
Let $ABC$ be an acute-angled triangle with $AB < AC$, and let points $D$ and $E$ be chosen on the side $AC$ and $BC$ respectively in such a way that $AD = AE = AB$. The circumcircle of $ABE$ intersects the line $AC$ at $A$ and $F$ and the line $DE$ at $E$ and $P$. Prove that $P$ is the circumcentre of $BDF$.
Let $ABCD$ be a cyclic quadrilateral with circumcircle $\Gamma$. Let $M$ be the midpoint of the arc $ABC$. The circle with center $M$ and radius $MA$ meets $AD, AB$ at $X, Y$. The point $Z \in XY$ with $Z \neq Y$ satisfies $BY=BZ$. Show that $\angle BZD=\angle BCD$.
Define a function $f$ on positive real numbers to satisfy
\[f(1)=1 , f(x+1)=xf(x) \textrm{ and } f(x)=10^{g(x)},\]
where $g(x) $ is a function defined on real numbers and for all real numbers $y,z$ and $0\leq t \leq 1$, it satisfies
\[g(ty+(1-t)z) \leq tg(y)+(1-t)g(z).\]
(1) Prove: for any integer $n$ and $0 \leq t \leq 1$, we have
\[t[g(n)-g(n-1)] \leq g(n+t)-g(n) \leq t[g(n+1)-g(n)].\]
(2) Prove that \[\frac{4}{3} \leq f(\frac{1}{2}) \leq \frac{4}{3} \sqrt{2}.\]
Andy creates a 3 sided dice with a side labeled $7$, a side labeled $17$, and a side labeled $27$. He then asks Anthony to roll the dice $3$ times. The probability that the product of Anthony's rolls is greater than $2023$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by Andy Xu[/i]
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.
Let $W,X,Y$ and $Z$ be four different digits selected from the set
$\{ 1,2,3,4,5,6,7,8,9\}.$
If the sum $\dfrac{W}{X} + \dfrac{Y}{Z}$ is to be as small as possible, then $\dfrac{W}{X} + \dfrac{Y}{Z}$ must equal
$\text{(A)}\ \dfrac{2}{17} \qquad \text{(B)}\ \dfrac{3}{17} \qquad \text{(C)}\ \dfrac{17}{72} \qquad \text{(D)}\ \dfrac{25}{72} \qquad \text{(E)}\ \dfrac{13}{36}$
Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that
\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]
Let $P(x) \in \mathbb{R}[x]$ be a monic, non-constant polynomial. Determine all continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$f(f(P(x))+y+2023f(y))=P(x)+2024f(y),$$
for all reals $x,y$.
Let $S$ be the set of all composite positive odd integers less than $79$. [list=a] [*] Show that $S$ may be written as the union of three (not necessarily disjoint) arithmetic progressions.[*] Show that $S$ cannot be written as the union of two arithmetic progressions.[/list]
$O$ is the vertex of a parabola, $F$ is its focus. $PQ$ is a chord of the parabola. If $|OF|=a,|PQ|=b$, find the area of $\triangle OPQ$.
Prove that on the coordinate plane it is impossible to draw a closed broken line such that
[i](i)[/i] the coordinates of each vertex are rational;
[i](ii)[/i] the length each of its edges is 1;
[i](iii)[/i] the line has an odd number of vertices.
A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day?
$ \textbf{(A)}\ \frac{1}{10}\qquad
\textbf{(B)}\ \frac{1}{9}\qquad
\textbf{(C)}\ \frac{1}{3}\qquad
\textbf{(D)}\ \frac{4}{9}\qquad
\textbf{(E)}\ \frac{5}{9}$
Given a circle with radius 1 and 2 points C, D given on it. Given a constant l with $0<l\le 2$. Moving chord of the circle AB=l and ABCD is a non-degenerated convex quadrilateral. AC and BD intersects at P. Find the loci of the circumcenters of triangles ABP and BCP.