In triangle $ABC$, the incircle touches sides $BC,CA,AB$ at $D,E,F$ respectively. Assume there exists a point $X$ on the line $EF$ such that \[\angle{XBC} = \angle{XCB} = 45^{\circ}.\] Let $M$ be the midpoint of the arc $BC$ on the circumcircle of $ABC$ not containing $A$. Prove that the line $MD$ passes through $E$ or $F$. United Kingdom

Given $7$ points inside or on a regular hexagon, show that three of them form a triangle with area $\le 1/6$ the area of the hexagon.

$a,b$ are positive integers, $a>b$. $\sin\theta=\frac{2ab}{a^2+b^2}(0<\theta<\frac{\pi}{2})$. If $A_n=(a^2+b^2)\sin n\theta$, prove that $A_n$ is an integer for all $n\in\mathbb{Z}_+$

BdMO National 2018 Higher Secondary P3 Nazia rolls four fair six-sided dice. She doesn’t see the results. Her friend Faria tells her that the product of the numbers is $144$. Faria also says the sum of the dice, $S$ satisfies $14\leq S\leq 18$ . Nazia tells Faria that $S$ cannot be one of the numbers in the set {$14,15,16,17,18$} if the product is $144$. Which number in the range {$14,15,16,17,18$} is an impossible value for $S$ ?

Suppose the sequence of nonnegative integers $a_1, a_2, \ldots, a_{1997}$ satisfies \[ a_i + a_j \leq a_{i+j} \leq a_i + a_j + 1 \] for all $i,j \geq 1$ with $i + j \leq 1997$. Show that there exists a real number $x$ such that $a_n = \lfloor nx \rfloor$ (the greatest integer $\leq nx$) for all $1 \leq n \leq 1997$.

Find all triplets $(x, y, z)$ of real numbers for which $$\begin{cases}x^2- yz = |y-z| +1 \\ y^2 - zx = |z-x| +1 \\ z^2 -xy = |x-y| + 1 \end{cases}$$

In the convex $(2n+2) $-gon are drawn $n^2$ diagonals. Prove that one of these of diagonals cuts the $(2n+2)$ -gon into two polygons, each of which has an odd number vertices.

Let $ABCDE$ be a regular pentagon with center $M$. A point $P\neq M$ is chosen on the line segment $MD$. The circumcircle of $ABP$ intersects the line segment $AE$ in $A$ and $Q$ and the line through $P$ perpendicular to $CD$ in $P$ and $R$. Prove that $AR$ and $QR$ are of the same length.

Let $M$ be the midpoint of the arc $BAC$ of the circumcircle of the triangle $ABC,$ $I$ the incenter of the triangle $ABC$ and $L$ a point on the side $BC$ such that $AL$ is bisector. The line $MI$ cuts the circumcircle again at $K.$ The circumcircle of the triangle $AKL$ cuts the line $BC$ again at $P.$ Prove that $\angle AIP = 90^{\circ}.$

Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$, and $CA$ (including $A$, $B$, and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.

Find all real numbers $x$ for which the number $$a =\frac{2x + 1}{x^2 + 2x + 3}$$ is an integer.

Find $ax^5 + by^5$ if the real numbers $a$, $b$, $x$, and $y$ satisfy the equations \begin{eqnarray*} ax + by &=& 3, \\ ax^2 + by^2 &=& 7, \\ ax^3 + by^3 &=& 16, \\ ax^4 + by^4 &=& 42. \end{eqnarray*}

Jacob flips five coins, exactly three of which land heads. What is the probability that the first two are both heads?

Holding a rectangular sheet of paper $ABCD$, Prair folds triangle $ABD$ over diagonal $BD$, so that the new location of point $A$ is $A'$. She notices that $A'C =\frac13 BD$. If the area of $ABCD$ is $27\sqrt2$, find $BD$.

Let $ABCD$ be a convex quadrilateral. Prove that the incircles of the triangles $ABC$, $BCD$, $CDA$ and $DAB$ have a point in common if, and only if, $ABCD$ is a rhombus.

Suppose that $n$ squares of an infinite square grid are colored grey, and the rest are colored white. At each step, a new grid of squares is obtained based on the previous one, as follows. For each location in the grid, examine that square, the square immediately above, and the square immediately to the right. If there are two or three grey squares among these three, then in the next grid, color that location grey, otherwise, color it white. Prove that after at most n steps all the squares in the grid will be white. Below is an example with $n = 4$. The first grid shows the initial configuration, and the second grid shows the configuration after one step. [img]https://cdn.artofproblemsolving.com/attachments/1/a/87f7e3892cdb45fb3529127234aae2cea08749.png[/img]

Let $F_1=F_2=1$. We define inductively $F_{n+1}=F_n+F_{n-1}$ for all $n\geq 2$. Then the sum $$F_1+F_2+\cdots+F_{2017}$$ is [list=1] [*] Even but not divisible by 3 [*] Odd but divisible by 3 [*] Odd and leaves remainder 1 when divisible by 3 [*] None of these [/list]

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.

Convex quadrilaterals \(ABCD\), \(A_1B_1C_1D_1\), and \(A_2B_2C_2D_2\) are similar with vertices in order. Points \(A\), \(A_1\), \(B_2\), \(B\) are collinear in order, points \(B\), \(B_1\), \(C_2\), \(C\) are collinear in order, points \(C\), \(C_1\), \(D_2\), \(D\) are collinear in order, and points \(D\), \(D_1\), \(A_2\), \(A\) are collinear in order. Diagonals \(AC\) and \(BD\) intersect at \(P\), diagonals \(A_1C_1\) and \(B_1D_1\) intersect at \(P_1\), and diagonals \(A_2C_2\) and \(B_2D_2\) intersect at \(P_2\). Prove that points \(P\), \(P_1\), and \(P_2\) are collinear. [i]Proposed by Holden Mui[/i]

Let $n$ be a positive integer and let $x_1$, $\ldots$, $x_n$ be positive real numbers. Show that: \[ \min\left ( x_1,\frac{1}{x_1}+x_2, \cdots,\frac{1}{x_{n-1}}+x_n,\frac{1}{x_n} \right )\leq 2\cos \frac{\pi}{n+2} \leq\max\left ( x_1,\frac{1}{x_1}+x_2, \cdots,\frac{1}{x_{n-1}}+x_n,\frac{1}{x_n} \right ). \]

a) Two legs of an angle $\alpha$ on a plane are mirrors. Prove that after several reflections in the mirrors any ray leaves in the direction opposite the one from which it came if and only if $\alpha = \frac{90^o}{n}$ for an integer $n$. Find the number of reflections. b) Given three planar mirrors in space forming an octant (trihedral angle with right planar angles), prove that any ray of light coming into this mirrored octant leaves it, after several reflections in the mirrors, in the direction opposite to the one from which it came. Find the number of reflections.

If ${si}(x) =- \int \limits_{x}^{\infty}\left(\frac{\sin t}{t}\right)dt; x>0$ then $$\int \limits_{e}^{e^2} \left(\frac{1}{x}\left(si\left(e^4x\right)-si\left(e^3x\right)\right)\right)\,dx=\int \limits_{3}^{e^4} \left(\frac{1}{x}\left(\operatorname{si}\left(e^2x\right)-si\left(ex\right)\right)\right)dx$$

Malmer Pebane's apartment uses a six-digit access code, with leading zeros allowed. He noticed that his fingers leave that reveal which digits were pressed. He decided to change his access code to provide the largest number of possible combinations for a burglar to try when the digits are known. For each number of distinct digits that could be used in the access code, calculate the number of possible combinations when the digits are known but their order and frequency are not known. For example, if there are smudges on $3$ and $9,$ two possible codes are $393939$ and $993999.$ Which number of distinct digits in the access code offers the most combinations?

At the Lexington High School, each student is given a unique five-character ID consisting of uppercase letters. Compute the number of possible IDs that contain the string "LMT". [i]Proposed by Alex Li[/i]

Given positive integer $ n > 1$ and define \[ S \equal{} \{1,2,\dots,n\}. \] Suppose \[ T \equal{} \{t \in S: \gcd(t,n) \equal{} 1\}. \] Let $ A$ be arbitrary non-empty subset of $ A$ such thar for all $ x,y \in A$, we have $ (xy\mod n) \in A$. Prove that the number of elements of $ A$ divides $ \phi(n)$. ($ \phi(n)$ is Euler-Phi function)