Found problems: 85335
Let $ f(x) \equal{} e^{(p \plus{} 1)x} \minus{} e^x$ for real number $ p > 0$. Answer the following questions.
(1) Find the value of $ x \equal{} s_p$ for which $ f(x)$ is minimal and draw the graph of $ y \equal{} f(x)$.
(2) Let $ g(t) \equal{} \int_t^{t \plus{} 1} f(x)e^{t \minus{} x}\ dx$. Find the value of $ t \equal{} t_p$ for which $ g(t)$ is minimal.
(3) Use the fact $ 1 \plus{} \frac {p}{2}\leq \frac {e^p \minus{} 1}{p}\leq 1 \plus{} \frac {p}{2} \plus{} p^2\ (0 < p\leq 1)$ to find the limit $ \lim_{p\rightarrow \plus{}0} (t_p \minus{} s_p)$.
A sequence of integers $(x_n)_{n=1}^{\infty}$ satisfies $x_1 = 1$ and $x_n < x_{n+1} \le 2n$ for all $n$.
Show that for every positive integer $k$ there exist indices $r, s$ such that $x_r-x_s = k$.
Given five points in the plane, no three of which are collinear, prove that there can be found at least two obtuse-angled triangles with vertices at the given points. Construct an example in which there are exactly two such triangles.
Compute the smallest positive integer $n$ for which $$\sqrt{100+\sqrt{n}}+\sqrt{100-\sqrt{n}}$$ is an integer.
In Cartesian space, let $\Omega = \{(a, b, c) : a, b, c$ are integers between $1$ and $30\}$.
A point of $\Omega$ is said to be [i]visible [/i] from the origin if the segment that joins said point with the origin does not contain any other elements of $\Omega$. Find the number of points of $\Omega$ that are [i]visible [/i] from the origin.
Let $ABC$ be a right angled triangle with $\angle A = 90^o$and $BC = a$, $AC = b$, $AB = c$. Let $d$ be a line passing trough the incenter of triangle and intersecting the sides $AB$ and $AC$ in $P$ and $Q$, respectively.
(a) Prove that $$b \cdot \left( \frac{PB}{PA}\right)+ c \cdot \left( \frac{QC}{QA}\right) =a$$
(b) Find the minimum of $$\left( \frac{PB}{PA}\right)^ 2+\left( \frac{QC}{QA}\right)^ 2$$
For each natural number $x$, let $S(x)$ be the sum of its digits. Find the smallest natural number $n$ such that $9S(n) = 16S(2n)$.
Prove that no matter how we number the vertices of a cube with integers from 1 to 8, there exists two opposite vertices in the cube (e.g. they are the endpoints of a large diagonal of the cube), united through a broken line formed with 3 edges of the cube, such that the sum of the 4 numbers written in the vertices of this broken lines is at least 21.
Let real numbers $x_1, x_2, \cdots , x_n$ satisfy $0 < x_1 < x_2 < \cdots< x_n < 1$ and set $x_0 = 0, x_{n+1} = 1$. Suppose that these numbers satisfy the following system of equations:
\[\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \quad \text{where } i = 1, 2, . . ., n.\]
Prove that $x_{n+1-i} = 1- x_i$ for $i = 1, 2, . . . , n.$
Two players $A$ and $B$ play the following game taking alternate moves. In each move, a player writes one digit on the blackboard. Each new digit is written either to the right or left of the sequence of digits already written on the blackboard. Suppose that $A$ begins the game and initially the blackboard was empty. $B$ wins the game if ,after some move of $B$, the sequence of digits written in the blackboard represents a perfect square. Prove that $A$ can prevent $B$ from winning.
Kymbrea's comic book collection currently has $30$ comic books in it, and she is adding to her collection at the rate of $2$ comic books per month. LaShawn's comic book collection currently has $10$ comic books in it, and he is adding to his collection at the rate of $6$ comic books per month. After how many months will LaShawn's collection have twice as many comic books as Kymbrea's?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 25$
Three numbers have sum $k$ (where $k\in \mathbb{R}$) such that the numbers are arethmetic progression.If First of two numbers remain the same and to the third number we add $\frac{k}{6}$ than we have geometry progression.
Find those numbers?
In triangle $ABC$, $AB=125,AC=117$, and $BC=120$. The angle bisector of angle $A$ intersects $\overline{BC}$ at point $L$, and the angle bisector of angle $B$ intersects $\overline{AC}$ at point $K$. Let $M$ and $N$ be the feet of the perpendiculars from $C$ to $\overline{BK}$ and $\overline{AL}$, respectively. Find $MN$.
Let $a$, $b$, $c$, $d$, $e$, $f$ and $g$ be seven distinct positive integers not bigger than $7$. Find all primes which can be expressed as $abcd+efg$
Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.
Let $Z$ shape be a shape such that it covers $(i,j)$, $(i,j+1)$, $(i+1,j+1)$, $(i+2,j+1)$ and $(i+2,j+2)$ where $(i,j)$ stands for cell in $i$-th row and $j$-th column on an arbitrary table. At least how many $Z$ shapes is necessary to cover one $8 \times 8$ table if every cell of a $Z$ shape is either cell of a table or it is outside the table (two $Z$ shapes can overlap and $Z$ shapes can rotate)?
A circle of radius 1 has four circles $\omega_1, \omega_2, \omega_3$, and $\omega_4$ of equal radius internally tangent to it, so that $\omega_1$ is tangent to $\omega_2$, which is tangent to $\omega_3$, which is tangent to $\omega_4$, which is tangent to $\omega_1$, as shown. The radius of the circle externally tangent to $\omega_1, \omega_2, \omega_3$, and $\omega_4$ has radius r. If $r = a -\sqrt{b}$ for positive integers $a$ and $b$, compute $a + b$.
[img]https://cdn.artofproblemsolving.com/attachments/e/3/c23f66333c0b4c0bf31b704cec665e50816149.png[/img]
Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.
Given parallelogram $ABCD$ with $E$ the midpoint of diagonal $BD$. Point $E$ is connected to a point $F$ in $DA$ so that $DF=\frac{1}{3}DA$. What is the ratio of the area of triangle $DFE$ to the area of quadrilateral $ABEF$?
$ \textbf{(A)}\ 1:2 \qquad\textbf{(B)}\ 1:3 \qquad\textbf{(C)}\ 1:5 \qquad\textbf{(D)}\ 1:6 \qquad\textbf{(E)}\ 1:7 $
Suppose that $S$ is a finite set of points in the plane such that the area of triangle $\triangle ABC$ is at most $1$ whenever $A,B,$ and $C$ are in $S.$ Show that there exists a triangle of area $4$ that (together with its interior) covers the set $S.$
Given that $\text{1 mile} = \text{8 furlongs}$ and $\text{1 furlong} = \text{40 rods}$, the number of rods in one mile is
$\text{(A)}\ 5 \qquad \text{(B)}\ 320 \qquad \text{(C)}\ 660 \qquad \text{(D)}\ 1760 \qquad \text{(E)}\ 5280$
If an integer of two digits is $ k$ times the sum of its digits, the number formed by interchanging the digits is the sum of the digits multiplied by:
$ \textbf{(A)}\ (9 \minus{} k) \qquad\textbf{(B)}\ (10 \minus{} k) \qquad\textbf{(C)}\ (11 \minus{} k) \qquad\textbf{(D)}\ (k \minus{} 1) \qquad\textbf{(E)}\ (k \plus{} 1)$
For each positive integer $n \ge 3$, define $A_n$ and $B_n$ as
\[A_n = \sqrt{n^2 + 1} + \sqrt{n^2 + 3} + \cdots + \sqrt{n^2+2n-1}\]
\[B_n = \sqrt{n^2 + 2} + \sqrt{n^2 + 4} + \cdots + \sqrt{n^2 + 2n}.\]
Determine all positive integers $n\ge 3$ for which $\lfloor A_n \rfloor = \lfloor B_n \rfloor$.
Note. For any real number $x$, $\lfloor x\rfloor$ denotes the largest integer $N\le x$.
[i]Anant Mudgal and Navilarekallu Tejaswi[/i]
$d$ and $n$ are positive integers such that $d \mid n$. The n-number sets $(x_1, x_2, \cdots x_n)$ satisfy the following condition:
(1) $0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq n$
(2) $d \mid (x_1+x_2+ \cdots x_n)$
Prove that in all the n-number sets that meet the conditions, there are exactly half satisfy $x_n=n$.
Let $n \geq 2$ be an integer. A lead soldier is moving across the unit squares of a $n \times n$ grid, starting from the corner square. Before each move to the neighboring square, the lead soldier can (but doesn't need to) turn left or right. Determine the smallest number of turns, which it needs to do, to visit every square of the grid at least once.
At the beginning the soldier's back is faced at the edge of the grid.