Found problems: 85335
What is the positive difference between the sum of the first $5$ positive even integers and the first $5$ positive odd integers?
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$
If the line $L$ in the $xy$-plane has half the slope and twice the y-intercept of the line $y = \frac{2}{3} x + 4$, then an equation for $L$ is:
$ \textbf{(A)}\ y = \frac{1}{3} x + 8 \qquad
\textbf{(B)}\ y = \frac{4}{3} x + 2 \qquad
\textbf{(C)}\ y = \frac{1}{3} x + 4 \qquad\\
\textbf{(D)}\ y = \frac{4}{3} x + 4 \qquad
\textbf{(E)}\ y = \frac{1}{3} x + 2 $
Let be a real number $ a, $ and a nondecreasing function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ Prove that $ f $ is continuous in $ a $ if and only if there exists a sequence $ \left( a_n \right)_{n\ge 1} $ of real positive numbers such that
$$ \int_a^{a+a_n} f(x)dx+\int_a^{a-a_n} f(x)dx\le\frac{a_n}{n} , $$
for all natural numbers $ n. $
[i]Dan Marinescu[/i]
Consider a regular triangular pyramid with base $\vartriangle ABC$ and apex $D$. If we have $AB = BC =AC = 6$ and $AD = BD = CD = 4$, calculate the surface area of the circumsphere of the pyramid.
Let $l$ be a line and $A$ be a point such that $A$ is not on $l.$ Let $P$ be a point on $l$ such that segment $AP$ and line $l$ for a $60^{\circ}$ angle and $AP=1.$ Extend segment $AP$ past $P$ to a point $B$ on the other side of $l.$ Then, let the perpendicular from $B$ to $l$ have foot $M,$ and extend $BM$ past $M$ to $C.$ Finally, extend $CP$ past $P$ to $D.$ Given that $\frac{BP}{AP}=\frac{CM}{BM}=\frac{DP}{CP}=2,$ determine the are of triangle $BPD.$
Let $n$ be a natural number. The leader of the math team invites $n$ girls for winter training, and each leaves her two gloves in a common box upon entry. The mischievous little brother randomly pairs the gloves into pairs, where each pair consists of one left glove and one right glove. A pairing is called [i]weak[/i] if there is a set of $k < \frac{n}{2}$ pairs containing gloves of exactly $k$ girls. Find the probability that the pairing is not weak.
The lengths of the sides $a, b$ and $c$ of a right triangle satisfy the relations $a <b <c$, and $\alpha$ is the measure of the smallest angle of the triangle. For which real values $k$ the equation $ax^2 + bx + kc = 0$ has real solutions for any measure of the angle $\alpha$ not exceeding $18^o$
Show that for all real numbers $x$ and $y$ with $x > -1$ and $y > -1$ and $x + y = 1$ the inequality
$$\frac{x}{y + 1} +\frac{y}{x + 1} \ge \frac23$$
holds. When does equality apply?
[i](Walther Janous)[/i]
The Pell numbers $P_n$ satisfy $P_0 = 0$, $P_1 = 1$, and $P_n=2P_{n-1}+P_{n-2}$ for $n\geq 2$. Find $$\sum \limits_{n=1}^{\infty} \left (\tan^{-1}\frac{1}{P_{2n}}+\tan^{-1}\frac{1}{P_{2n+2}}\right )\tan^{-1}\frac{2}{P_{2n+1}}$$
If $x^{x^4}=4 $ what is the value of $x^{x^2}+x^{x^8} $ ?
Find $0/1 + 1/1 + 0/2 + 1/2 + 2/2 + 0/3 + 1/3 + 2/3 + 3/3 + 0/4 + 1/4 + 2/4 + 3/4 + 4/4 + 0/5 + 1/5 + 2/5 + 3/5 + 4/5 + 5/5 + 0/6 + 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6$
Does there exist an integer $A$ such that each of the ten digits $0, 1, . . . , 9$ appears exactly once as a digit in exactly one of the numbers $A, A^2, A^ 3$ ?
Let $a,b$ positive real numbers. Prove that $$\frac{(a+b)^3}{a^2b}\ge \frac{27}{4}.$$ When does equality occur?
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for positive integers $a$ and $b,$ \[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]
Consider a $25 \times 25$ grid of unit squares. Draw with a red pen contours of squares of any size on the grid. What is the minimal number of squares we must draw in order to colour all the lines of the grid?
Let be nonnegative real numbers $ a,b,c, $ holding the inequality: $ \sum_{\text{cyc}} \frac{a}{b+c+1} \le 1. $
Prove that $ \sum_{\text{cyc}} \frac{1}{b+c+1} \ge 1. $
There are $n$ ($n \ge 4$) straight lines on the plane. For two straight lines $a$ and $b$, if there are at least two straight lines among the remaining $n-2$ lines that intersect both straight lines $a$ and $b$, then $a$ and $b$ are called a [i]congruent [/i] pair of staight lines, otherwise it is called a [i]separated[/i] pair of straight lines. If the number of [i]congruent [/i] pairs of straight line among $n$ straight lines is $2012$ more than the number of [i]separated[/i] pairs of straight line , find the smallest possible value of $n$ (the order of the two straight lines in a pair is not counted).
Three bags of jelly beans contain 26, 28, and 30 beans. The ratios of yellow beans to all beans in each of these bags are $50\%$, $25\%$, and $20\%$, respectively. All three bags of candy are dumped into one bowl. Which of the following is closest to the ratio of yellow jelly beans to all beans in the bowl?
$\textbf{(A)}\ 31\% \qquad \textbf{(B)}\ 32\% \qquad \textbf{(C)}\ 33\% \qquad \textbf{(D)}\ 35\% \qquad \textbf{(E)}\ 95\%$
Find all the pairs $(x,y)$ of positive integers that satisfy the equation $x^2-xy+2x-3y=2013$.
Show that the set of positive integers which cannot be represented as a sum of distinct perfect squares is finite.
We have a set of 2n positive integers whose sum is a multiple of n. One operation consists of choosing n of them and adding the same positive integer to all of them.
Show that, starting from the initial 2n numbers, we can get all
are equal, performing a maximum of 2n - 1 operations.
Let $ABC$ be a triangle. Find all the triangles $XYZ$ with vertices inside triangle $ABC$ such that $XY,YZ,ZX$ and six non-intersecting segments from the following $AX, AY, AZ, BX, BY, BZ, CX, CY, CZ$ divide the triangle $ABC$ into seven regions with equal areas.
Given a positive integer $n\ge 3$, colour each cell of an $n\times n$ square array with one of $\lfloor (n+2)^2/3\rfloor$ colours, each colour being used at least once. Prove that there is some $1\times 3$ or $3\times 1$ rectangular subarray whose three cells are coloured with three different colours.
[i](Russia) Ilya Bogdanov, Grigory Chelnokov, Dmitry Khramtsov[/i]
Each of the 12 edges of a cube is coloured either red or green. Every face of the cube has at least one
red edge. What is the smallest number of red edges?
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$
A semicircle has diameter $AB$ and center $S$,with a point $M$ on the circumference.$U,V$ are the incircles of sectors $ASM$ and $BSM$.Prove that circles $U,V$ can be seperated by a line perpendicular to $AB$.