The altitudes of an acute triangle $ABC$ with $AB<AC$ intersect at a point $H$, and $O$ is the center of the circumcircle $\Omega$. The segment $OH$ intersects the circumcircle of $BHC$ at a point $X$, different from $O$ and $H$. The circumcircle of $AOX$ intersects the smaller arc $AB$ of $\Omega$ at point $Y$. Prove that the line $XY$ bisects the segment $BC$. [i]Proposed by A. Tereshin[/i]

$AD$ is the altitude on side $BC$ of triangle $ABC$. If $BC+AD-AB-AC = 0$, find the range of $\angle BAC$. [i]Alternative formulation.[/i] Let $AD$ be the altitude of triangle $ABC$ to the side $BC$. If $BC+AD=AB+AC$, then find the range of $\angle{A}$.

Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. The lines tangent to $\Gamma$ through $B$ and $C$ meet at $P$. Let $M$ be a point on the arc $AC$ that does not contain $B$ such that $M \neq A$ and $M \neq C$, and $K$ be the point where the lines $BC$ and $AM$ meet. Let $R$ be the point symmetrical to $P$ with respect to the line $AM$ and $Q$ the point of intersection of lines $RA$ and $PM$. Let $J$ be the midpoint of $BC$ and $L$ be the intersection point of the line $PJ$ and the line through $A$ parallel to $PR$. Prove that $L, J, A, Q,$ and $K$ all lie on a circle.

Let $P(z)$ be a polynomial with real coefficients whose roots are covered by a disk of radius R. Prove that for any real number $k$, the roots of the polynomial $nP(z)-kP'(z)$ can be covered by a disk of radius $R+|k|$, where $n$ is the degree of $P(z)$, and $P'(z)$ is the derivative of $P(z)$. can anyone help me? It would also be extremely helpful if anyone could tell me where they've seen this type of problems.............Has it appeared in any mathematics competitions? Or are there any similar questions for me to attempt? Thanks in advance!

Two concentric circles $\omega, \Omega$ with radii $8,13$ are given. $AB$ is a diameter of $\Omega$ and the tangent from $B$ to $\omega$ touches $\omega$ at $D$. What is the length of $AD$.

Let $ d_n$ be the determinant of the $ n\times n$ matrix whose entries, from left to right and then from top to bottom, are $ \cos 1,\cos 2,\dots,\cos n^2.$ (For example, $ d_3 \equal{} \begin{vmatrix}\cos 1 & \cos2 & \cos3 \\ \cos4 & \cos5 & \cos 6 \\ \cos7 & \cos8 & \cos 9\end{vmatrix}.$ The argument of $ \cos$ is always in radians, not degrees.) Evaluate $ \lim_{n\to\infty}d_n.$

Given positive integer $n (n \geq 2)$, find the largest positive integer $\lambda$ satisfying : For $n$ bags, if every bag contains some balls whose weights are all integer powers of $2$ (the weights of balls in a bag may not be distinct), and the total weights of balls in every bag are equal, then there exists a weight among these balls such that the total number of balls with this weight is at least $\lambda$.

$A$ and $B$ are $2\times 2$ real valued matrices satisfying $$\det A = \det B = 1,\quad \text{tr}(A)>2,\quad \text{tr}(B)>2,\quad \text{tr}(ABA^{-1}B^{-1}) = 2$$ Prove that $A$ and $B$ have a common eigenvector.

Let $x_1,\ldots,x_n$ be a sequence with each term equal to $0$ or $1$. Form a triangle as follows: its first row is $x_1,\ldots,x_n$ and if a row if $a_1, a_2, \ldots, a_m$, then the next row is $a_1 + a_2, a_2 + a_3, \ldots, a_{m-1} + a_m$ where the addition is performed modulo $2$ (so $1+1=0$). For example, starting from $1$, $0$, $1$, $0$, the second row is $1$, $1$, $1$, the third one is $0$, $0$ and the fourth one is $0$. A sequence is called good it is the same as the sequence formed by taking the last element of each row, starting from the last row (so in the above example, the sequence is $1010$ and the corresponding sequence from last terms is $0010$ and they are not equal in this case). How many possibilities are there for the sequence formed by taking the first element of each row, starting from the last row, which arise from a good sequence?

We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.

Initially $A$ selects a graph with \( 2221 \) vertices such that each vertex is incident to at least one edge. Then $B$ deletes some of the edges (possibly none) from the chosen graph. Finally, $A$ pays $B$ one lev for each vertex that is incident to an odd number of edges. What is the maximum amount that $B$ can guarantee to earn?

The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\tfrac{3}{2}$ and center $(0,\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$. What is the area of the shark's fin falcata? [asy] import cse5;pathpen=black;pointpen=black; size(1.5inch); D(MP("x",(3.5,0),S)--(0,0)--MP("\frac{3}{2}",(0,3/2),W)--MP("y",(0,3.5),W)); path P=(0,0)--MP("3",(3,0),S)..(3*dir(45))..MP("3",(0,3),W)--(0,3)..(3/2,3/2)..cycle; draw(P,linewidth(2)); fill(P,gray); [/asy] $\textbf{(A) } \dfrac{4\pi}{5} \qquad\textbf{(B) } \dfrac{9\pi}{8} \qquad\textbf{(C) } \dfrac{4\pi}{3} \qquad\textbf{(D) } \dfrac{7\pi}{5} \qquad\textbf{(E) } \dfrac{3\pi}{2} $

Let $\displaystyle s_n=\int_0^1 \text{sin}^n(nx) \,dx$. (a) Prove that $s_n \leq \dfrac 2n$ for all odd $n$. (b) Find all the limit points of the sequence $s_1, s_2, s_3, \dots$. [i]Proposed by Cristi Calin[/i]

Find all positive integers n such that : $-2^{0}+2^{1}-2^{2}+2^{3}-2^{4}+...-(-2)^{n}=4^{0}+4^{1}+4^{2}+...+4^{2010}$

$(x_1, y_1), (x_2, y_2), (x_3, y_3)$ lie on a straight line and on the curve $y^2 = x^3$. Show that $\frac{x_1}{y_1} + \frac{x_2}{y_2}+\frac{x_3}{y_3} = 0$.

In triangle $ABC$, in which $AB = BC$, on side $AB$ is selected point $D$, and the ciscumcircles of triangles $ADC$ and $BDC$ , $S1$ and $S2$ respectively. The tangent drawn to $S_1$ at point $D$ intersects $S_2$ for second time at point $M$. Prove that $BM \parallel AC$.

Let $ABCD$ be a convex quadrilateral inscribed in a circle and satisfying $DA < AB = BC < CD$. Points $E$ and $F$ are chosen on sides $CD$ and $AB$ such that $BE \perp AC$ and $EF \parallel BC$. Prove that $FB = FD$. [i]Milan Haiman[/i]

$ T$ is a subset of $ {1,2,...,n}$ which has this property : for all distinct $ i,j \in T$ , $ 2j$ is not divisible by $ i$ . Prove that : $ |T| \leq \frac {4}{9}n + \log_2 n + 2$

In $ \triangle ABC$ we have $ AB \equal{} 7$, $ AC \equal{} 8$, and $ BC \equal{} 9$. Point $ D$ is on the circumscribed circle of the triangle so that $ \overline{AD}$ bisects $ \angle BAC$. What is the value of $ AD/CD$? $ \textbf{(A)}\ \frac{9}{8}\qquad \textbf{(B)}\ \frac{5}{3}\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ \frac{17}{7}\qquad \textbf{(E)}\ \frac{5}{2}$

How many ways are there to arrange the $6$ permutations of the tuple $(1, 2, 3)$ in a sequence, such that each pair of adjacent permutations contains at least one entry in common? For example, a valid such sequence is given by $(3, 2, 1) - (2, 3, 1) - (2, 1, 3) - (1, 2, 3) - (1, 3, 2) - (3, 1, 2)$.

Let $k\leq n$ be two positive integers. IMO-nation has $n$ villages, some of which are connected by a road. For any two villages, the distance between them is the minimum number of toads that one needs to travel from one of the villages to the other, if the traveling is impossible, then the distance is set as infinite. Alice, who just arrived IMO-nation, is doing her quarantine in some place, so she does not know the configuration of roads, but she knows $n$ and $k$. She wants to know whether the furthest two villages have finite distance. To do so, for every phone call she dials to the IMO office, she can choose two villages, and ask the office whether the distance between them is larger than, equal to, or smaller than $k$. The office answers faithfully (infinite distance is larger than $k$). Prove that Alice can know whether the furthest two villages have finite distance between them in at most $2n^2/k$ calls. [i]Proposed by usjl and Cheng-Ying Chang[/i]

Let $ z_1,z_2,z_3\in\mathbb{C} $ such that $\text{(i)}\quad \left|z_1\right| = \left|z_2\right| = \left|z_3\right| = 1$ $\text{(ii)}\quad z_1+z_2+z_3\neq 0 $ $\text{(iii)}\quad z_1^2 +z_2^2+z_3^2 =0. $ Show that $ \left| z_1^3+z_2^3+z_3^3\right| = 1. $

Let $a,b,c$ be sidelengths of a triangle and $r,R,s$ be the inradius, the circumradius and the semiperimeter respectively of the same triangle. Prove that: $$\frac{1}{a + b} + \frac{1}{a + c} + \frac{1}{b + c} \leq \frac{r}{16Rs}+\frac{s}{16Rr} + \frac{11}{8s}$$ (Albania)

[b]p22.[/b] Find the unique ordered pair $(m, n)$ of positive integers such that $x = \sqrt[3]{m} -\sqrt[3]{n}$ satisfies $x^6 + 4x^3 - 36x^2 + 4 = 0$. [b]p23.[/b] Jenny plays with a die by placing it flat on the ground and rolling it along any edge for each step. Initially the face with $1$ pip is face up. How many ways are there to roll the dice for $6$ steps and end with the $1$ face up again? [b]p24.[/b] There exists a unique positive five-digit integer with all odd digits that is divisible by $5^5$. Find this integer. PS. You should use hide for answers.

A mouse is on the below grid: \begin{center} \begin{asy} unitsize(2cm); filldraw(circle((0,0),0.07), black); filldraw(circle((0,1),0.07), black); filldraw(circle((1,0),0.07), black); filldraw(circle((0.5,0.5),0.07), black); filldraw(circle((1,1),0.07), black); draw((0,0)--(1,0)); draw((0,0)--(0,1)); draw((1,0)--(1,1)); draw((0,1)--(1,1)); draw((0,1)--(0.5,0.5)); draw((1,0)--(0.5,0.5)); draw((1,1)--(0.5,0.5)); draw((0,0)--(0.5,0.5)); \end{asy} \end{center} The paths connecting each node are the possible paths the mouse can take to walk from a node to another node. Call a ``turn" the action of a walk from one node to another. Given the mouse starts off on an arbitrary node, what is the expected number of turns it takes for the mouse to return to its original node? [i]Lightning 4.2[/i]