Found problems: 85335
Let $ f[0,\infty )\longrightarrow\mathbb{R} $ be a convex and differentiable function with $ f(0)=0. $
[b]a)[/b] Prove that $ \int_0^x f(t)dt\le \frac{x^2}{2}f'(x) , $ for any nonnegative $ x. $
[b]b)[/b] Determine $ f $ if the above inequality is actually an equality.
[i]Dorin Andrica[/i] and [i]Mihai Piticari[/i]
One of two circumferences of radius $R$ comes through $A$ and $B$ vertices of the $ABCD$ parallelogram. Another comes through $B$ and $D$. Let $M$ be another point of circumferences intersection. Prove that the circle circumscribed around $AMD$ triangle has radius $R$.
A figure with area $1$ is cut out of paper. We divide this figure into $10$ parts and color them in $10$ different colors. Now, we turn around the piece of paper, divide the same figure on the other side of the paper in $10$ parts again (in some different way). Show that we can color these new parts in the same $10$ colors again (hereby, different parts should have different colors) such that the sum of the areas of all parts of the figure colored with the same color on both sides is $\geq \frac{1}{10}.$
Natural numbers are written in the cells of of a $2014\times2014$ regular square grid such that every number is the average of the numbers in the adjacent cells. Describe and prove how the number distribution in the grid can be.
Calculate $f(\sqrt[3]{2}-1) $, where
$$f(x) = x^{1999} + 3x^{1998} + 4x^{1997} + 2x^{1996} + 4x^{1995} + 2x^{1994} + ...$$
$$... + 4x^3 + 2x^2 + 3x+ 1.$$
Let $X =\{x_0, x_1,\ldots , x_n\}$ be the basis set of a finite metric space, where the points are inductively listed such that $x_k$ maximizes the product of the distances from the points $\{x_0, x_1,\ldots , x_{k-1}\}$ for each $1\leqslant k\leqslant n.$ Prove that if for each $x\in X$ we let $\Pi_x$ be the product of the distances from $x{}$ to every other point, then $\Pi_{x_n}\leqslant 2^{n-1}\Pi_x$ for any $x\in X.$
A function $ f:\mathbb{Q}_{>0}\longrightarrow\mathbb{Q} $ has the following property:
$$ f(xy)=f(x)+f(y),\quad x,y\in\mathbb{Q}_{>0} $$
[b]a)[/b] Demonstrate that there are no injective functions with this property.
[b]b)[/b] Do exist surjective functions having this property?
Find all positive integers $a, b, c, d,$ and $n$ satisfying $n^a + n^b + n^c = n^d$ and prove that these are the only such solutions.
A number is said to be [i]regular[/i] if when a digit $k$ appears in that number, the digit appears exactly $k$ times. For example, the number $3133$ is a regular number because the digit $1$ appears exactly once and the digit $3$ appears exactly three times. How many regular six-digit numbers are there?
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
Let $ABC$ be a triangle. An arbitrary circle which passes through the points $B,C$ intersects the sides $AC,AB$ for the second time in $D,E$ respectively. The line $BD$ intersects the circumcircle of the triangle $AEC$ at $P{}$ and $Q{}$ and the line $CE$ intersects the circumcircle of the triangle $ABD$ at $R{}$ and $S{}$ such that $P{}$ is situated on the segment $BD{}$ and $R{}$ lies on the segment $CE.$ Prove that:
[list=a]
[*]The points $P,Q,R$ and $S{}$ are concyclic.
[*]The triangle $APQ$ is isosceles.
[/list]
[i]Petru Braica[/i]
Let $ABC$ be a triangle with ${AB\ne BC}$; and let ${BD}$ be the internal bisector of $\angle ABC,\ $, $\left( D\in AC \right)$. Denote by ${M}$ the midpoint of the arc ${AC}$ which contains point ${B}$. The circumscribed circle of the triangle ${\vartriangle BDM}$ intersects the segment ${AB}$ at point ${K\neq B}$. Let ${J}$ be the reflection of ${A}$ with respect to ${K}$. If ${DJ\cap AM=\left\{O\right\}}$, prove that the points ${J, B, M, O}$ belong to the same circle.
Let $BE$ and $CF$ be altitudes in triangle $ABC$. If $AE = 24$, $EC = 60$, and $BF = 31$, determine $AF$.
There is a grid of height $2$ stretching infinitely in one direction. Between any two edge-adjacent cells of the grid, there is a door that is locked with probability $\frac12$ independent of all other doors. Philip starts in a corner of the grid (in the starred cell). Compute the expected number of cells that Philip can reach, assuming he can only travel between cells if the door between them is unlocked.
[img]https://cdn.artofproblemsolving.com/attachments/f/d/fbf9998270e16055f02539bb532b1577a6f92a.png[/img]
For any two points $A (x_1 , y_1)$ and $B (x_2, y_2)$, the distance $r (A, B)$ between them is determined by the equality $r(A, B) = max\{| x_1- x_2 | , | y_1 - y_2 |\}$.
Prove that the triangle inequality $r(A, C) + r(C, B) \ge r(A, B)$. holds for the distance introduced in this way .
Let $A$ and $B$ be two points of the plane . Find the locus of points $C$ for which
a) $r(A, C) + r(C, B) = r(A, B)$
b) $r(A, C) = r(C, B).$
Let $\lambda$ be a positive real number. Inequality $|\lambda xy+yz|\le \dfrac{\sqrt5}{2}$ holds for arbitrary real numbers $x, y, z$ satisfying $x^2+y^2+z^2=1$. Find the maximal value of $\lambda$.
Let $a$ and $b$ be real numbers such that $\max_{0\le x\le 1} |x^3 - ax - b|$ is as small as possible. Find $a + b$ in simplest radical form.
(Hint: If $f(x) = x^3 - cx - d$, then the maximum (or minimum) of $f(x)$ either occurs when $x = 0$ and/or $x = 1$ and/or when x satisfies $3x^2 - c = 0$).
What is the least integer $n>2003$ such that $5^n + n^5$ is a multiple of $11$?
$
\textbf{(A)}\ 2010
\qquad\textbf{(B)}\ 2011
\qquad\textbf{(C)}\ 2012
\qquad\textbf{(D)}\ 2014
\qquad\textbf{(E)}\ \text{None of the preceding}
$
Consider the triangle $ABC$, in which $AB > AC$. Let $P$ and $Q$ be the feet of the perpendiculars dropped from the vertices $B$ and $C$ on the bisector of the angle $BAC$, respectively. On the line $BC$ note point $B$ such that $AD \perp AP.$ Prove that the lines $BQ, PC$ and $AD$ intersect at one point.
Inside the equilateral triangle $ABC$, points $P$ and $Q$ are chosen so that the quadrilateral $APQC$ is convex, $AP = PQ = QC$ and $\angle PBQ = 30^o$. Prove that $AQ = BP$.
Suppose there are $2019$ distinct points in a plane and the distances between pairs of them attain $k$ different values. Prove that $k$ is at least $44$.
Suppose $a_{1} < a_{2}< \cdots < a_{2024}$ is an arithmetic sequence of positive integers, and $b_{1} <b_{2} < \cdots <b_{2024}$ is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.
[i]Ray Li[/i]
Prove that for an arbitrary $\Delta ABC$ the following inequality holds:
$\frac{l_a}{m_a}+\frac{l_b}{m_b}+\frac{l_c}{m_c} >1$,
Where $l_a,l_b,l_c$ and $m_a,m_b,m_c$ are the lengths of the bisectors and medians through $A$, $B$, and $C$.
In $\Delta ABC$, $AD \perp BC$ at $D$. $E,F$ lie on line $AB$, such that $BD=BE=BF$. Let $I,J$ be the incenter and $A$-excenter. Prove that there exist two points $P,Q$ on the circumcircle of $\Delta ABC$ , such that $PB=QC$, and $\Delta PEI \sim \Delta QFJ$ .
John has a string of paper where $n$ real numbers $a_i \in [0, 1]$, for all $i \in \{1, \ldots, n\}$, are written in a row.
Show that for any given $k < n$, he can cut the string of paper into non-empty $k$ pieces, between adjacent numbers, in such a way that the sum of the numbers on each piece does not differ from any other sum by more than $1$.