Given a convex $ n $-gon ($ n \geq 5 $). Prove that the number of triangles of area $1$ with vertices at the vertices of the $ n $-gon does not exceed $ \frac{1}{3} n (2n-5) $.

Find all positive integers $n$ such that $n^3 - 18n^2 + 115n - 391$ is the cube of a positive intege

Evaluate $\int_0^{\frac{\pi}{12}} \frac{\tan ^ 2 x-3}{3\tan ^ 2 x-1}dx$. created by kunny

Let $ABC$ be a non-isosceles,aqute triangle with $AB<AC$ inscribed in circle $c(O,R)$.The circle $c_{1}(B,AB)$ crosses $AC$ at $K$ and $c$ at $E$. $KE$ crosses $c$ at $F$ and $BO$ crosses $KE$ at $L$ and $AC$ at $M$ while $AE$ crosses $BF$ at $D$.Prove that: i)$D,L,M,F$ are concyclic. ii)$B,D,K,M,E$ are concyclic.

A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score? $ \textbf{(A) } 10 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 17 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 24$

A set of points in the plane is called rigid if each point is equidistant from the three (or more) points nearest to it. (a) Does there exist a rigid set of $9$ points? (b) Does there exist a rigid set of $11$ points?

Let $$a_k = 0.\overbrace{0 \ldots 0}^{k - 1 \: 0's} 1 \overbrace{0 \ldots 0}^{k - 1 \: 0's} 1$$ The value of $\sum_{k = 1}^\infty a_k$ can be expressed as a rational number $\frac{p}{q}$ in simplest form. Find $p + q$.

Let $n$ be a positive integer and $p$ a fixed prime. We have a deck of $n$ cards, numbered $1,\ 2,\ldots,\ n$ and $p$ boxes for put the cards on them. Determine all posible integers $n$ for which is possible to distribute the cards in the boxes in such a way the sum of the numbers of the cards in each box is the same.

If $x\in\mathbb{R}$ satisfy $sin$ $x+tan$ $x\in\mathbb{Q}$, $cos$ $x+cot$ $x\in\mathbb{Q}$ Prove that $sin$ $2x$ is a root of an integral coefficient quadratic function

The circles $k_1$ and $k_2$ intersect at points $D$ and $P$. The common tangent of the two circles on the side of $D$ touches $k_1$ at $A$ and $k_2$ at $B$. The straight line $AD$ intersects $k_2$ for a second time at $C$. Let $M$ be the center of the segment $BC$. Show that $ \angle DPM = \angle BDC$ .

How many ordered pairs of real numbers $ (a, b) $ satisfy equality $\lim_{x \to 0} \frac{\sin^2x}{e^{ax}-2bx-1}= \frac{1}{2}$?

Let $(x_i)$ be a decreasing sequence of positive reals, then show that: (a) for every positive integer $n$ we have $\sqrt{\sum^n_{i=1}{x_i^2}} \leq \sum^n_{i=1}\frac{x_i}{\sqrt{i}}$. (b) there is a constant C for which we have $\sum^{\infty}_{k=1}\frac{1}{\sqrt{k}}\sqrt{\sum^{\infty}_{i=k}x_i^2} \le C\sum^{\infty}_{i=1}x_i$.

[b]p1. [/b] A clock has a long hand for minutes and a short hand for hours. A placement of those hands is [i]natural [/i] if you will see it in a correctly functioning clock. So, having both hands pointing straight up toward $12$ is natural and so is having the long hand pointing toward $6$ and the short hand half-way between $2$ and $3$. A natural placement of the hands is symmetric if you get another natural placement by interchanging the long and short hands. One kind of symmetric natural placement is when the hands are pointed in exactly the same direction. Are there symmetric natural placements of the hands in which the two hands are not pointed in exactly the same direction? If so, describe one such placement. If not, explain why none are possible. [b]p2.[/b] Let $\frac{m}{n}$ be a fraction such that when you write out the decimal expansion of $\frac{m}{n}$ , it eventually ends up with the four digits $2001$ repeated over and over and over. Prove that $101$ divides $n$. [b]p3.[/b] Consider the following two questions: Question $1$: I am thinking of a number between $0$ and $15$. You get to ask me seven yes-or-no questions, and I am allowed to lie at most once in answering your questions. What seven questions can you ask that will always allow you to determine the number? Note: You need to come up with seven questions that are independent of the answers that are received. In other words, you are not allowed to say, "If the answer to question $1$ is yes, then question $2$ is XXX; but if the answer to question $1$ is no, then question $2$ is YYY." Question $2$: Consider the set $S$ of all seven-tuples of zeros and ones. What sixteen elements of $S$ can you choose so that every pair of your chosen seven-tuples differ in at least three coordinates? a. These two questions are closely related. Show that an answer to Question $1$ gives an answer to Question $2$. b. Answer either Question $1$ or Question $2$. [b]p4.[/b] You may wish to use the angle addition formulas for the sine and cosine functions: $\sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$ $\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ a) Prove the identity $(\sin x)(1 + 2 \cos 2x) = \sin (3x)$. b) For any positive integer $n$, prove the identity $$(sin x)(1 + 2 \cos 2x + 2\cos 4x + ... +2\cos 2nx) = \sin ((2n +1)x)$$ [b]p5.[/b] Define the set $\Omega$ in the $xy$-plane as the union of the regions bounded by the three geometric figures: triangle $A$ with vertices $(0.5, 1.5)$, $(1.5, 0.5)$ and $(0.5,-0.5)$, triangle $B$ with vertices $(-0.5,1.5)$, $(-1.5,-0.5)$ and $(-0.5, 0.5)$, and rectangle $C$ with corners $(0.5, 1.0)$, $(-0.5, 1.0)$, $(-0.5,-1.0)$, and $(0.5,-1.0)$. a. Explain how copies of $\Omega$ can be used to cover the $xy$-plane. The copies are obtained by translating $\Omega$ in the $xy$-plane, and copies can intersect only along their edges. b. We can define a transformation of the plane as follows: map any point $(x, y)$ to $(x + G, x + y + G)$, where $G = 1$ if $y < -2x$, $G = -1$ if $y > -2x$, and $G = 0$ if $y = -2x$. Prove that every point in $\Omega$ is transformed into another point in $\Omega$, and that there are at least two points in $\Omega$ that are transformed into the same point. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For postive integer $n$ consider the hyperplane \[ R_0^n = {x=(x_1x_2...x_n)\in\mathbb{R}^n : \sum\limits^n_{i=1}x_i=0} \] and the lattice \[ Z_0^n = \{y \in R^n_0 : \ (\forall i: y_i \in \mathbb{N})\} \] Define the quasi-norm in $\mathbb{R}^n$ by $\|x\|_p= \sqrt[p]{\sum\limits^{n}_{i=1}|x_i|^p}$ if $0<p<\infty$ and $\|x\|_{\infty} = \max\limits_i |x_i|$. (a) If $x\in R^n_0$ so that $\max x_i - \min x_i \le 1$ then prove that $\forall p \in [1,\infty], \forall y \in Z^n_0$ we have $\|x\|_p\le\|x+y\|_p$ (b) Prove that for every $p\in ]0,1[$, there exist $n \in \mathbb{N}, x\in R^n_0, y\in Z^n_0$ with $\max x_i - \min x_i \le 1$ and $\|x\|_p>\|x+y\|_p$

Decide if there is a permutation $a_1,a_2,\cdots,a_{6666}$ of the numbers $1,2,\cdots,6666$ with the property that the sum $k+a_k$ is a perfect square for all $k=1,2,\cdots,6666$

We are given the natural numbers $1=a_1,\, \, a_2,...,a_n$, for which $a_i\leq a_{i+1}\leq 2a_i$ for $i=1,2,...,n-1$ and the sum $\sum_{i=1}^n a_i$ is even. Prove that these numbers can be partitioned into two groups with equal sum.

How many perfect squares are divisors of the product $ 1!\cdot 2!\cdot 3!\cdots 9!$? $ \textbf{(A)}\ 504 \qquad \textbf{(B)}\ 672 \qquad \textbf{(C)}\ 864 \qquad \textbf{(D)}\ 936 \qquad \textbf{(E)}\ 1008$

In an acute-angled triangle $ABC$, $A_1$ and $B_1$ are the feet of the altitudes from $A$ and $B$ respectively, and $M$ is the midpoint of $AB$. a) Prove that $MA_1$ is tangent to the circumcircle of triangle $A_1B_1C$. b) Prove that the circumcircles of triangles $A_1B_1C,BMA_1$, and $AMB_1$ have a common point.

Nine points are placed on a circle. Show that it is possible to colour the $36$ chords connecting them using four colours so that for any set of four points, each of the four colours is used for at least one of the six chords connecting the given points

Isosceles trapezoid $ABCD$ with bases $AB$ and $CD$ has a point $P$ on $AB$ with $AP=11, BP=27,$ $CD=34,$ and $\angle{CPD}=90^\circ.$ Compute the height of isosceles trapezoid $ABCD.$

Let $a,b,c\ge 0$ and $a+b+c=\sqrt2$. Show that \[\frac1{\sqrt{1+a^2}}+\frac1{\sqrt{1+b^2}}+\frac1{\sqrt{1+c^2}} \ge 2+\frac1{\sqrt3}\] [hide] In general if $a_1, a_2, \cdots , a_n \ge 0$ and $\sum_{i=1}^n a_i=\sqrt2$ we have \[\sum_{i=1}^n \frac1{\sqrt{1+a_i^2}} \ge (n-1)+\frac1{\sqrt3}\] [/hide]

Kobar and Borah are playing on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board, either from $P$ and $Q$ to $2P-Q$ and $2Q-P$, or from $P$ and $Q$ to $5P-4Q$ and $5Q-4P$. The game ends if a player writes an integer that is not positive. That player is declared to lose, and the opponent is declared the winner. At the beginning of the game, the two numbers on the board are $2024$ and $A$. If it is known that Kobar does not lose on his first move, determine the largest possible value of $A$ so that Borah can win this game.

Prove that in a convex quadrilateral inscribed in a circle, straight lines passing through the midpoints of the sides and perpendicular to the opposite sides intersect at one point.

The pressure $ (P)$ of wind on a sail varies jointly as the area $ (A)$ of the sail and the square of the velocity $ (V)$ of the wind. The pressure on a square foot is $ 1$ pound when the velocity is $ 16$ miles per hour. The velocity of the wind when the pressure on a square yard is $ 36$ pounds is: $ \textbf{(A)}\ 10\frac {2}{3} \text{ mph} \qquad\textbf{(B)}\ 96 \text{ mph} \qquad\textbf{(C)}\ 32\text{ mph} \qquad\textbf{(D)}\ 1\frac {2}{3} \text{ mph} \qquad\textbf{(E)}\ 16 \text{ mph}$

Find $\displaystyle n \in \mathbb N$, $\displaystyle n \geq 2$, and the digits $\displaystyle a_1,a_2,\ldots,a_n$ such that \[ \displaystyle \sqrt{\overline{a_1 a_2 \ldots a_n}} - \sqrt{\overline{a_1 a_2 \ldots a_{n-1}}} = a_n . \]