In arithmetic sequence $(a_n)$, $3a_8=5a_{13},a_1>0$. Define $S_n=\sum_{i=1}^n a_i$, then the largest number in $(S_n)$ is $\text{(A)}S_{10}\qquad\text{(B)}S_{11}\qquad\text{(C)}S_{20}\qquad\text{(D)}S_{21}$

In the quadrilateral $ABCD$, shown in fig. , the equations are true: $\angle ABC = \angle BCD$ and $2AB = CD$. On the side $BC$, a point $X$ is selected such that $\angle BAX = \angle CDA$. Prove that $AX = AD$. [img]https://cdn.artofproblemsolving.com/attachments/2/9/0884eb311d1e40300c1e5980fd53eaadfa7a25.png[/img]

In a class among 80 students number of boys is 40 and number of girls is 40. 50 of the students use spectacles. Which of the following is correct? [list=1] [*] Only 10 boys use spectacles [*] Only 20 girls use spectacles [*] At most 25 boys do not use spectacles [*] At most 30 girls do not use spectacles [/list]

A straight one-mile stretch of highway, $40$ feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at $5$ miles per hour, how many hours will it take to cover the one-mile stretch? Note: $1$ mile= $5280$ feet [asy]size(10cm); pathpen=black; pointpen=black; D(arc((-2,0),1,300,360)); D(arc((0,0),1,0,180)); D(arc((2,0),1,180,360)); D(arc((4,0),1,0,180)); D(arc((6,0),1,180,240)); D((-1.5,1)--(5.5,1)); D((-1.5,0)--(5.5,0),dashed); D((-1.5,-1)--(5.5,-1)); [/asy] $\textbf{(A) }\frac{\pi}{11}\qquad\textbf{(B) }\frac{\pi}{10}\qquad\textbf{(C) }\frac{\pi}{5}\qquad\textbf{(D) }\frac{2\pi}{5}\qquad \textbf{(E) }\frac{2\pi}{3}$

In the following, a rhombus is one with edge length $1$ and interior angles $60^o$ and $120^o$ . Now let $n$ be a natural number and $H$ a regular hexagon with edge length $n$, which is covered with rhombuses without overlapping has been. The rhombuses then appear in three different orientations. Prove that whatever the overlap looks exactly, each of these three orientations can be viewed at the same time.

There is a $2021 \times 2023$ board that has a white piece in the central square, on which Mich and Moka are going to play in turns. First Mich places a green token on any free space so that it is not in the same row or column as the white token, then Moka places a red token on any free space so that it is not in the same row or column as the white token. white or green. From now on, Mich will place green tokens and Moka will place red tokens alternately according to the following rules: $\bullet$ For the placed piece there must be another piece of the same color in its row or column, such that there is no other piece between both pieces. $\bullet$ If there is at least one box that meets the previous rule, then it is mandatory to place a token. When a token is placed, it changes all the tokens that are on squares adjacent to it to the same color. The game ends when one of the players can no longer place tiles. If when the game ends the board has more green tiles then Mich wins, and if it has more red tiles then Moka wins. Determine if either player has a winning strategy.

Show that we can find $\alpha>0$ such that, given any point $P$ inside a regular $2n+1$-gon which is inscribed in a circle radius $1$, we can find two vertices of the polygon whose distance from $P$ differ by less than $\frac1n-\frac\alpha{n^3}$.

Let $x$ and $y$ be positive integers such that $\frac{x^2-1}{y+1}+\frac{y^2-1}{x+1}$ is integer. Prove that numbers $\frac{x^2-1}{y+1}$ and $\frac{y^2-1}{x+1}$ are integers

a) Each point, on the perimeter of a square, is colored either red, or blue. Show that, there is a right-angled triangle where all the corners are on the square of the square and so that all the corners are on points of the same color. b) Show that, it is possible to color each point on the perimeter of one square, red, white, or blue so that, there is not a right-angled triangle where all the three corners are at points of same color.

A real-valued function $f$ has the property that for all real numbers $a$ and $b,$ $$f(a + b) + f(a - b) = 2f(a) f(b).$$ Which one of the following cannot be the value of $f(1)?$ $ \textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } -1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } -2$

There are $3n$ participants in the Mathematical Olympiad competition. They are assigned seats in three rows, with $n$ seats in each, and are admitted into the hall one at a time, after which they immediately take their seats. Calculate the probability that until the last competitor takes his seat, at any moment for each two rows the difference in the number of players sitting in them is no greater than 1.

Let $ABC$ be an acute triangle with $AB = 3$ and $AC = 4$. Suppose that $AH,AO$ and $AM$ are the altitude, the bisector and the median derived from $A$, respectively. If $HO = 3 MO$, then the length of $BC$ is [img]https://cdn.artofproblemsolving.com/attachments/e/c/26cc00629f4c0ab27096b8bdc562c56ff01ce5.png[/img] A. $3$ B. $\frac72$ C. $4$ D. $\frac92$ E. $5$

A cake has the form of an $ n$ x $ n$ square composed of $ n^{2}$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $\mathcal{A}$. Let $\mathcal{B}$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $\mathcal{B}$ than of arrangement $\mathcal{A}$. Prove that arrangement $\mathcal{B}$ can be obtained from $ \mathcal{A}$ by performing a number of switches, defined as follows: A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.

Let $P(n)$ be the number of ways to split a natural number $n$ to the sum of powers of two, when the order does not matter. For example $P(5) = 4$, as $5=4+1=2+2+1=2+1+1+1=1+1+1+1+1$. Prove that for any natural the identity $P(n) + (-1)^{a_1} P(n-1) + (-1)^{a_2} P(n-2) + \ldots + (-1)^{a_{n-1}} P(1) + (-1)^{a_n} = 0,$ is true, where $a_k$ is the number of units in the binary number record $k$ . [url=http://matol.kz/comments/2720/show]source[/url]

A semicircle of diameter $ 1$ sits at the top of a semicircle of diameter $ 2$, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune. [asy]unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); filldraw(Circle((0,.866),.5),grey,black); label("1",(0,.866),S); filldraw(Circle((0,0),1),white,black); draw((-.5,.866)--(.5,.866),linetype("4 4")); clip((-1,0)--(1,0)--(1,2)--(-1,2)--cycle); draw((-1,0)--(1,0)); label("2",(0,0),S);[/asy]$ \textbf{(A)}\ \frac {1}{6}\pi \minus{} \frac {\sqrt {3}}{4} \qquad \textbf{(B)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{12}\pi \qquad \textbf{(C)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{24}\pi\qquad\textbf{(D)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{24}\pi$ $ \textbf{(E)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{12}\pi$

The lengths of the sides of a triangle with positive area are $\log_{10} 12$, $\log_{10} 75$, and $\log_{10} n$, where $n$ is a positive integer. Find the number of possible values for $n$.