Prove that for all positive integers $n$, \[\lfloor \sqrt{n}+\sqrt{n+1}+\sqrt{n+2}\rfloor =\lfloor \sqrt{9n+8}\rfloor.\]

[b]p1[/b]. Prove that there is no interger $n$ such that $n^2 +1$ is divisible by $7$. [b]p2.[/b] Find all solutions of the system $$x^2-yz=1$$ $$y^2-xz=2$$ $$z^2-xy=3$$ [b]p3.[/b] A triangle with long legs is an isoceles triangle in which the length of the two equal sides is greater than or equal to the length of the remaining side. What is the maximum number, $n$ , of points in the plane with the property that every three of them form the vertices of a triangle with long legs? Prove all assertions. [b]p4.[/b] Prove that the area of a quadrilateral of sides $a, b, c, d$ which can be inscribed in a circle and circumscribed about another circle is given by $A=\sqrt{abcd}$ [b]p5.[/b] Prove that all of the squares of side length $$\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},...,\frac{1}{n},...$$ can fit inside a square of side length $1$ without overlap. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In $\vartriangle ABC, \angle A= 2 \angle B$. Let $a,b,c$ be the lengths of its sides $BC,CA,AB$, respectively. Prove that $a^2 = b(b + c)$.

The center of square $ABCD$ is $K$. The point $P$ is chosen such that $P \ne K$ and the angle $\angle APB$ is right . Prove that the line $PK$ bisects the angle between the lines $AP$ and $BP$.

Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by the numbers of the form $\ z_{k_1,k_2} = k_1r_1 + k_2r_2$ integers, i.e. for every number $x$ and every positive real number $p$ two integers $k_1$ and $k_2$ can be found so that $|x - (k_1r_1 + k_2r_2)| < p$ holds.

Let $\omega$ be the incircle of scalene triangle $ABC$. Let $\omega$ be tangent to $AB$ and $AC$ at points $X$ and $Y$. Construct points $X^\prime$ and $Y^\prime$ on line segments $AB$ and $AC$ respectively such that $AX^\prime=XB$ and $AY^\prime=YC$. Let line $CX^\prime$ intersects $\omega$ at points $P,Q$ such that $P$ is closer to $C$ than $Q$. Also let $R^\prime$ be the intersection of lines $CX^\prime$ and $BY^\prime$. Prove that $CP=RX^\prime$.

In triangle $ABC$ medians from $B$ and $C$ are perpendicular. Prove that $\frac{\sin(B+C)}{\sin B \cdot \sin C} \geq \frac{2}{3}.$

Find all the functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the functional equation $$f(xy^2)+f(x^2y)=y^2f(x)+x^2f(y)$$ for every $x,y\in\mathbb{R}$ and $f(2008) =f(-2008)$ [i](nooonuii)[/i]

Compute the product of positive integers $n$ such that $n^2 + 59n + 881$ is a perfect square.

Determine all infinite sequences $(a_1,a_2,\dots)$ of positive integers satisfying \[a_{n+1}^2=1+(n+2021)a_n\] for all $n \ge 1$.

The diagonals of the convex quadrilateral $ABCD$ are mutually perpendicular. Through the midpoint of the sides $AB$ and $AD$ draw lines, which are perpendicular to the opposite sides. Prove that they intersect on line $AC$.

Triangle $\triangle{ABC}$,O=circumcenter of (ABC),OA=R,the A-excircle intersect (AB),(BC),(CA) at points F,D,E. If the A-excircle has radius R prove that $OD\perp EF$

Evaluate \[\int_{-\pi}^{\pi} (\cos2x)(\cos 2^2x)\cdots (\cos 2^{2006}x)dx\]

Lucky Larry's teacher asked him to substitute numbers for $ a$, $ b$, $ c$, $ d$, and $ e$ in the expression $ a\minus{}(b\minus{}(c\minus{}(d\plus{}e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincedence. The numbers Larry substituted for $ a$, $ b$, $ c$, and $ d$ were $ 1$, $ 2$, $ 3$, and $ 4$, respectively. What number did Larry substitute for $ e$? $ \textbf{(A)}\ \minus{}5\qquad\textbf{(B)}\ \minus{}3\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 5$

A circle touches sides $AB, BC, CD$ of a parallelogram $ABCD$ at points $K, L, M$ respectively. Prove that the line $KL$ bisects the height of the parallelogram drawn from the vertex $C$ to $AB$.

Is it possible to find a set $A$ of eleven positive integers such that no six elements of $A$ have a sum which is divisible by $6$?

Let $(A_i)_{i\ge 1}$ be sequence of sets of two integer numbers, such that no integer is contained in more than one $A_i$ and for every $A_i$ the sum of its elements is $i$. Prove that there are infinitely many values of $k$ for which one of the elements of $A_k$ is greater than $13k/7$.

Let set $S$ contain all positive integers that are one less than a perfect square. Find the sum of all powers of $2$ that can be expressed as the product of two (not necessarily distinct) members of $S.$ [i]Proposed by Alex Li[/i]

Let $ABC$ be a triangle such that $AB \neq AC$. The incircle with centre $I$ touches $BC$ at $D$. Line $AI$ intersects the circumcircle $\Gamma$ of $ABC$ at $M$, and $DM$ again meets $\Gamma$ at $P$. Find $\angle API$

Let $x$ be a positive real number. Find the maximum possible value of \[\frac{x^2+2-\sqrt{x^4+4}}{x}.\]

The sequence \( (a_n)_{n\geq 1} \) of positive integers is such that \( a_1 = 1 \) and \( a_{m+n} \) divides \( a_m + a_n \) for any positive integers \( m \) and \( n \). a) Prove that if the sequence is unbounded, then \( a_n = n \) for all \( n \). b) Does there exist a non-constant bounded sequence with the above properties? (A sequence \( (a_n)_{n\geq 1} \) of positive integers is bounded if there exists a positive integer \( A \) such that \( a_n \leq A \) for all \( n \), and unbounded otherwise.)

Consider an $n \times n$ array of numbers $a_{ij}$ (standard notation). Suppose each row consists of the $n$ numbers $1,2,\ldots n$ in some order and $a_{ij} = a_{ji}$ for $i , j = 1,2, \ldots n$. If $n$ is odd, prove that the numbers $a_{11}, a_{22} , \ldots a_{nn}$ are $1,2,3, \ldots n$ in some order.

Let $\triangle{ABC}$ be an acute triangle with $BC > AC$. Let $S$ be the centroid of triangle $ABC$ and let $F$ be the foot of the perpendicular from $C$ to side $AB$. The median $CS$ intersects the circumcircle $\gamma$ of triangle $\triangle{ABC}$ at a second point $P$. Let $M$ be the point where $CS$ intersects $AB$. The line $SF$ intersects the circle $\gamma$ at a point $Q$, such that $F$ lies between $S$ and $Q$. Prove that the points $M,P,Q$ and $F$ lie on a circle. [i](Karl Czakler)[/i]

Prove that there exists a function $ f $ defined and differentiable in the set of all real numbers, satisfying the conditions $|f'(x) - f'(y)| \leq 4|x-y|$.

For $x\geq 0$ find the value of $x$ by which $f(x)=\int_0^x 3^t(3^t-4)(x-t)dt$ is minimized.