The equation $x-\frac{7}{x-3}=3-\frac{7}{x-3}$ has: $ \textbf{(A)}\ \text{infinitely many integral roots} \qquad\textbf{(B)}\ \text{no root} \qquad\textbf{(C)}\ \text{one integral root}\qquad$ $\textbf{(D)}\ \text{two equal integral roots} \qquad\textbf{(E)}\ \text{two equal non-integral roots} $

Prove that there exists a nonconstant function $ f:\mathbb{R}^2\longrightarrow\mathbb{R} $ verifying the following system of relations: $$ \left\{ \begin{matrix} f(x,x+y)=f(x,y) ,& \quad \forall x,y\in\mathbb{R} \\f(x,y+z)=f(x,y) +f(x,z) ,& \quad \forall x,y\in\mathbb{R} \end{matrix} \right. $$

Let $\Gamma$ be a circumference, and $A,B,C,D$ points of $\Gamma$ (in this order). $r$ is the tangent to $\Gamma$ at point A. $s$ is the tangent to $\Gamma$ at point D. Let $E=r \cap BC,F=s \cap BC$. Let $X=r \cap s,Y=AF \cap DE,Z=AB \cap CD$ Show that the points $X,Y,Z$ are collinear. Note: assume the existence of all above points.

Let $n\in\mathbb{N}$ and $0\leq a_1\leq a_2\leq\ldots\leq a_n\leq\pi$ and $b_1,b_2,\ldots ,b_n$ are real numbers for which the following inequality is satisfied : \[\left|\sum_{i\equal{}1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k}\] for all $ k\in\mathbb{N}$. Prove that $ b_1\equal{}b_2\equal{}\ldots \equal{}b_n\equal{}0$.

Find the units digit of the sum \[(1!)^2+(2!)^2+(3!)^2+(4!)^2+\cdots+(2007!)^2.\] $\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }3$ $\textbf{(D) }5\hspace{14em}\textbf{(E) }7\hspace{14em}\textbf{(F) }9$

Let $x_n=\binom{2n}{n}$ for all $n\in\mathbb{Z}^+$. Prove there exist infinitely many finite sets $A,B$ of positive integers, satisfying $A \cap B = \emptyset $, and \[\frac{{\prod\limits_{i \in A} {{x_i}} }}{{\prod\limits_{j\in B}{{x_j}} }}=2012.\]

Let $ABCD$ be a parallelogram and $P$ be a point on diagonal $BD$ such that $\angle PCB=\angle ACD$. Circumcircle of triangle $ABD$ intersects line $AC$ at points $A$ and $E$. Prove that \[\angle AED=\angle PEB.\]

Do there exist $6$ circles in the plane such that every circle passes through centers of exactly $3$ other circles? by Morteza Saghafian

Let $a_1,a_2,\ldots,a_7, b_1,b_2,\ldots,b_7\geq 0$ be real numbers satisfying $a_i+b_i\le 2$ for all $i=\overline{1,7}$. Prove that there exist $k\ne m$ such that $|a_k-a_m|+|b_k-b_m|\le 1$. Thanks for show me the mistake typing

Let $O$ be the center of the circle circumscribed around $\vartriangle ABC$, let $A_1, B_1, C_1$ be symmetric to $O$ through respective sides of $\vartriangle ABC$. Prove that all altitudes of $\vartriangle A_1B_1C_1$ pass through $O$, and all altitudes of $\vartriangle ABC$ pass through the center of the circle circumscribed around $\vartriangle A_1B_1C_1$.

A group of 6 math students is staying at a mathematical hotel to participate in a math tournament that will take place in the city in the coming days. This group, composed of 3 women and 3 men, was assigned rooms in a specific way by the hotel administration: in separate rooms and alternating between genders, specifically: woman, man, woman, man, woman, man, occupying the last 6 rooms in a corridor numbered from 101 to 110. \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline M & H & M & H & M & H & & & & \\ \hline 110 & 109 & 108 & 107 & 106 & 105 & 104 & 103 & 102 & 101 \\ \hline \end{tabular} Against the hotel's rules, the group devised the following game: A valid room exchange occurs when two students in consecutive rooms move to two empty rooms, such that the difference between their new room numbers and their original ones is the same. For example, if the students in rooms 105 and 106 move to rooms 101 and 102, this would be a valid exchange since both numbers decreased by 4 units. Determine if, following these rules, the students can manage to have rooms 101, 102, and 103 occupied by men and rooms 104, 105, and 106 occupied by women in just 3 valid exchanges.

Let $D, E$, and $F$ be respectively the midpoints of the sides $BC, CA$, and $AB$ of $\vartriangle ABC$. Let $H_a, H_b, H_c$ be the feet of perpendiculars from $A, B, C$ to the opposite sides, respectively. Let $P, Q, R$ be the midpoints of the $H_bH_c, H_cH_a$, and $H_aH_b$ respectively. Prove that $PD, QE$, and $RF$ are concurrent.

Define the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for all positive real numbers $x$. How many integers from $1$ to $2023$ inclusive are in the range of $f(x)$? Note that $\lfloor x\rfloor$ is known as the $\textit{floor}$ function, which returns the greatest integer less than or equal to $x$. [i]Proposed by Harry Kim[/i]

An $8$ by $8$ board (with $64$ $1$ by $1$ squares) is painted white. We are allowed to choose any rectangle consisting of $3$ of the $64$ squares and paint each of the $3$ squares in the opposite colour (the white ones black, the black ones white). Is it possible to paint the entire board black by means of such operations? (IS Rubanov, Kirov)

Let $ n$ be a positive integer such that $ n\geq 3$. Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2n$ positive real numbers satisfying the equations \[ a_1 \plus{} a_2 \plus{} ... \plus{} a_n \equal{} 1, \quad \text{and} \quad b_1^2 \plus{} b_2^2 \plus{} ... \plus{} b_n^2 \equal{} 1.\] Prove the inequality \[a_1\left(b_1 \plus{} a_2\right) \plus{} a_2\left(b_2 \plus{} a_3\right) \plus{} ... \plus{} a_{n \minus{} 1}\left(b_{n \minus{} 1} \plus{} a_n\right) \plus{} a_n\left(b_n \plus{} a_1\right) < 1.\]

Two positive numbers $x$ and $y$ are given. Show that $\left(1 +\frac{x}{y} \right)^3 + \left(1 +\frac{y}{x}\right)^3 \ge 16$.

For nonnegative integer $n$, the following are true: [list] [*]$f(0)=0$ [*]$f(1)=1$ [*]$f(n)=f(n-\tfrac{m(m-1)}2)-f(\tfrac{m(m+1)}2-n)$ for integer $m$ satisfying $m\geq 2$ and $\tfrac{m(m-1)}2<n\leq\tfrac{m(m+1)}2$.[/list] Find the smallest integer $n$ such that $f(n)=4$.

A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.

Given is a pyramid $SABCD$ whose base is a convex quadrilateral $ ABCD $ with perpendicular diagonals $ AC $ and $ BD $, and the orthogonal projection of vertex $S$ onto the base is the point $0$ of the intersection of the diagonals of the base. Prove that the orthogonal projections of point $O$ onto the lateral faces of the pyramid lie on the circle.

Let $ a,b,c$ be positive real numbers. Prove that \[ \frac{a^3}{c(a^2 + bc)} + \frac{b^3}{a(b^2 + ca)} + \frac{c^3}{b(c^2 +ab )} \ge \frac{3}{2} . \]

Let $ABCD$ be a trapezium such that $AB\parallel CD$. Let $E$ be the intersection of diagonals $AC$ and $BD$. Suppose that $AB = BE$ and $AC = DE$. Prove that the internal angle bisector of $\angle BAC$ is perpendicular to $AD$.

A list of integers has mode $ 32$ and mean $ 22$. The smallest number in the list is $ 10$. The median $ m$ of the list is a member of the list. If the list member $ m$ were replaced by $ m \plus{} 10$, the mean and median of the new list would be $ 24$ and $ m \plus{} 10$, respectively. If $ m$ were instead replaced by $ m \minus{} 8$, the median of the new list would be $ m \minus{} 4$. What is $ m$? $ \textbf{(A)}\ 16\qquad \textbf{(B)}\ 17\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 19\qquad \textbf{(E)}\ 20$

Let $a,b,c$ be the sidelengths of a triangle. Prove that $$2<\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}<\sqrt{6}.$$

Inside a circular column with a bottom surface with radius of $6$, there are two balls with radius of $6$ as well. The distance betwen their centers is $13$. Draw a plane that is tangent to both spherical surface, intersect the circular column at a curve $C$. $C$ is a ellipse, then the sum of its short axis and long axis is________.

Can You fill the squares of the infinite cross-lined paper with integers so, that the sum of the numbers in every $4\times 6$ fields rectangle would be a) $10$? b) $1$?