Found problems: 85335
Let $\Omega$ be the circumcircle of triangle $ABC$ such that the tangents to $\Omega$ in points $B$ and $C$ intersect in $P$. The squares $ABB_1B_2$ and $ACC_1C_2$ are constructed on the sides $AB$ and $AC$ in the exterior of triangle $ABC$, such that the lines $B_1B_2$ and $C_1C_2$ intersect in point $Q$. Prove that $P$, $A$, and $Q$ are collinear.
A semicircle $(O)$ is drawn with the center $O$, where $O$ lies on a line $\ell$. $C$ and $D$ lie on the circle $(O)$, and the tangent lines of $(O)$ at points $C$ and $D$ intersects the line $\ell$ at points $B$ and $A$, respectively, such that $O$ lies between points $B$ and $A$. Let $E$ be the intersection point between $AC$ and $BD$, and $F$ the point on $\ell$ so that $EF $ is perpendicular to line $\ell$. Prove that $EF$ bisects the angle $\angle CFD$.
Let $H$ be the orthocenter of an acute-angled triangle $ABC$. The circumcircle of triangle $AHC$ meets segments $AB$ and $BC$ at points $P$ and $Q$. Lines $PQ$ and $AC$ meet at point $R$. A point $K$ lies on the line $PH$
in such a way that $\angle KAC = 90^{\circ}$. Prove that $KR$ is perpendicular to one of
the medians of triangle $ABC$.
Find the smallest number $a$ such that a square of side $a$ can contain five disks of radius $1$, so that no two of the disks have a common interior point.
Two concentric circles have differing radii such that a chord of the outer circle which is tangent to the inner circle has length $18$. Compute the area inside the bigger circle which lies outside of the smaller circle.
Solve the system $\begin{cases}
10x_1 + 3x_2 + 4x_3 + x_4 + x_5 = 0 \\
11x_2 + 2x_3 + 2x_4 + 3x_5 + x_6 = 0 \\
15x_3 + 4x_4 + 5x_5 + 4x_6 + x_7 = 0 \\
2x_1 + x_2 - 3x_3 + 12x_4 - 3x_5 + x_6 + x_7 = 0 \\
6x_1 - 5x_2 + 3x_3 - x_4 + 17x_5 + x_6 = 0 \\
3x_1 + 2x_2 - 3x_3 + 4x_4 + x_5 - 16x_6 + 2x_7 = 0\\
4x_1 - 8x_2 + x_3 + x_4 + 3x_5 + 19x_7 = 0 \end{cases}$
Determine the volume of the body obtained by cutting the ball of radius $R$ by the trihedron with vertex in the center of that ball, it its dihedral angles are $\alpha, \beta, \gamma.$
Let $ABC$ be a triangle and let $a,b,c$ be the lengths of the sides $BC, CA, AB$ respectively. Consider a triangle $DEF$ with the side lengths $EF=\sqrt{au}$, $FD=\sqrt{bu}$, $DE=\sqrt{cu}$. Prove that $\angle A >\angle B >\angle C$ implies $\angle A >\angle D >\angle E >\angle F >\angle C$.
Given $ABC$ triangle with incircle $L_1$ and circumcircle $L_2$. If points $X, Y, Z$ lie on $L_2$, such that $XY, XZ$ are tangent to $L_1$, then prove that $YZ$ is also tangent to $L_1$.
Let $h_a, h_b, h_c$ be the lengths of the altitudes of a triangle $ABC$ from $A, B, C$ respectively. Let $P$ be any point inside the triangle. Show that
\[\frac{PA}{h_b+h_c} + \frac{PB}{h_a+h_c} + \frac{PC}{h_a+h_b} \ge 1.\]
In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent
to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more
in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$.
Prove that $|AR|\cdot |BQ|=|P I|^2$
For any positive integers $x,y,z$ and $w,$ prove that $x^2,y^2,z^2$ and $w^2$ cannot be four consecutive terms of arithmetic sequence.
2018 little ducklings numbered 1 through 2018 are standing in a line, with each holding a slip of paper with a nonnegative number on it; it is given that ducklings 1 and 2018 have the number zero. At some point, ducklings 2 through 2017 change their number to equal the average of the numbers of the ducklings to their left and right. Suppose the new numbers on the ducklings sum to 1000. What is the maximum possible sum of the original numbers on all 2018 slips?
Let $a,b,c,d$ be positive real numbers such that
\[(a+b+c+d)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)=20. \]
Prove that
\[\left(a^{2}+b^{2}+c^{2}+d^{2}\right)\left(\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+\frac{1}{d^{2}}\right)\ge 36. \]
There are two solutions, one by Phan Thanh Nam, one by me, which are very nice.
Let $P$ be a point in a square whose side are mirror. A ray of light comes from $P$ and with slope $\alpha$. We know that this ray of light never arrives to a vertex. We make an infinite sequence of $0,1$. After each contact of light ray with a horizontal side, we put $0$, and after each contact with a vertical side, we put $1$. For each $n\geq 1$, let $B_{n}$ be set of all blocks of length $n$, in this sequence.
a) Prove that $B_{n}$ does not depend on location of $P$.
b) Prove that if $\frac{\alpha}{\pi}$ is irrational, then $|B_{n}|=n+1$.
[list]
[*] A power grid with the shape of a $3\times 3$ lattice with $16$ nodes (vertices of the lattice) joined by wires (along the sides of squares. It may have happened that some of the wires have burned out. In one test technician can choose any two nodes and check if electrical current circulates between them (i.e there is a chain of intact wires joining the chosen nodes) . Technicial knows that current will circulate from any node to another node. What is the least number of tests required to demonstrate this?
[*] Previous problem for the grid of $7\times 7$ lattice.[/list]
The vertices of a regular $4$-gon, $6$-gon and $12$-goncan be brought together in one point to form a complete angle of $360^o$ (see figure). [center][img]https://cdn.artofproblemsolving.com/attachments/b/1/e9245179b7e0f5acb98b226bdc6db87fd72ad5.png[/img] [/center]
Determine all triples $a, b, c \in N$ with $a < b < c$ for which the angles of a regular $a$-gon, $b$-gon and $c$-gon together also form $360^o$ .
Consider the set $A = \{1, 2, ..., n\}$, where $n \in N, n \ge 6$. Show that $A$ is the union of three pairwise disjoint sets, with the same cardinality and the same sum of their elements, if and only if $n$ is a multiple of $3$.
Prove that for any real numbers $ a,b,c, $ the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as
$$ f(x)=\sqrt{(x-c)^2+b^2} +\sqrt{(x+c)^2+b^2} $$ is decreasing on $ (-\infty ,0] $ and increasing on $ [0,\infty ) . $
Let $M$ denote the set of $m\times n$ matrices with entries in the set $\{0,1,2,3,4\}$ such that in each row and each column the sum of elements is divisible by $5$. Find the cardinality of set $M$.
Consider a right-angled triangle with $AB=1,\ AC=\sqrt{3},\ \angle{BAC}=\frac{\pi}{2}.$ Let $P_1,\ P_2,\ \cdots\cdots,\ P_{n-1}\ (n\geq 2)$ be the points which are closest from $A$, in this order and obtained by dividing $n$ equally parts of the line segment $AB$. Denote by $A=P_0,\ B=P_n$, answer the questions as below.
(1) Find the inradius of $\triangle{P_kCP_{k+1}}\ (0\leq k\leq n-1)$.
(2) Denote by $S_n$ the total sum of the area of the incircle for $\triangle{P_kCP_{k+1}}\ (0\leq k\leq n-1)$.
Let $I_n=\frac{1}{n}\sum_{k=0}^{n-1} \frac{1}{3+\left(\frac{k}{n}\right)^2}$, show that $nS_n\leq \frac {3\pi}4I_n$, then find the limit $\lim_{n\to\infty} I_n$.
(3) Find the limit $\lim_{n\to\infty} nS_n$.
Let $ABC$ be a triangle with incenter $I,$ circumcenter $O,$ and $A$-excenter $J_A.$ The incircle of $\triangle{ABC}$ touches side $BC$ at a point $D.$ Lines $OI$ and $J_AD$ meet at a point $K.$ Line $AK$ meets the circumcircle of $\triangle{ABC}$ again at a point $L \neq A.$ If $BD=11, CD=5,$ and $AO=10,$ the length of $DL$ can be expressed as $\tfrac{m\sqrt{p}}{n},$ where $m,n,p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p.$
Let $m$ and $n$ be natural numbers. Every one of the $m*n$ squares of the $m*n$ board is colored either black or white, so that no 2 neighbouring squares are the same color(the board is colored like in chess").In one step we can pick 2 neighbouring squares and change their colors like this:
[b]- [/b]a white square becomes black;
[b]-[/b]a black square becomes blue;
[b]-[/b]a blue square becomes white.
For which $m$ and $n$ can we ,in a finite sequence of these steps, switch the starting colors from white to black and vice versa.
Matthew Casertano and Fox Chyatte make a series of bets. In each bet, Matthew sets the stake (the amount he wins or loses) at half his current amount of money. He has an equal chance of winning and losing each bet. If he starts with \$256, find the probability that after 8 bets, he will have at least \$50.
[i]Proposed by Jeffrey Tong[/i]
Let $\left \lfloor x \right \rfloor$ denote the greatest integer less than or equal to $x$. Find the sum of all positive integer solutions to $$\left \lfloor \frac{n^3}{27} \right \rfloor - \left \lfloor \frac{n}{3} \right \rfloor ^3=10.$$
[i]Proposed by Jason Hsu[/i]