For a group of 5 girls, denoted as $G_1,G_2,G_3,G_4,G_5$ and $12$ boys. There are $17$ chairs arranged in a row. The students have been grouped to sit in the seats such that the following conditions are simultaneously met: (a) Each chair has a proper seat. (b) The order, from left to right, of the girls seating is $G_1; G_2; G_3; G_4; G_5.$ (c) Between $G_1$ and $G_2$ there are at least three boys. (d) Between $G_4$ and $G_5$ there are at least one boy and most four boys. How many such arrangements are possible?

Let $p = 2017$. Given a positive integer $n$, an $n\times n$ matrix $A$ is formed with each element $a_{ij}$ randomly selected, with equal probability, from $\{0,1,\ldots,p - 1\}$. Let $q_n$ be probability that $\det A\equiv 1\pmod{p}$. Let $q=\displaystyle\lim_{n\rightarrow\infty} q_n$. If $d_1, d_2, d_3, \ldots$ are the digits after the decimal point in the base $p$ expansion of $q$, then compute the remainder when $\displaystyle\sum_{k = 1}^{p^2} d_k$ is divided by $10^9$. [i]Proposed by Ashwin Sah[/i]

In $\vartriangle ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be a point on $BC$ such that $BD = 6$. Let $E$ be a point on $CA$ such that $CE = 6$. Finally, let $F$ be a point on $AB$ such that $AF = 6$. Find the area of $\vartriangle DEF$.

(A.Myakishev, 9--10) Given triangle $ ABC$. One of its excircles is tangent to the side $ BC$ at point $ A_1$ and to the extensions of two other sides. Another excircle is tangent to side $ AC$ at point $ B_1$. Segments $ AA_1$ and $ BB_1$ meet at point $ N$. Point $ P$ is chosen on the ray $ AA_1$ so that $ AP\equal{}NA_1$. Prove that $ P$ lies on the incircle.

Let \( n > 1 \) be a positive integer. List in increasing order all the irreducible fractions in the interval \([0, 1]\) that have a positive denominator less than or equal to \( n \): \[ \frac{0}{1} = \frac{p_0}{q_0} < \frac{p_1}{q_1} < \cdots < \frac{p_M}{q_M} = \frac{1}{1}. \] Determine, in function of \( n \), the smallest possible value of \( q_{i-1} + q_i + q_{i+1} \), for \( 0 < i < M \). For example, if \( n = 4 \), the enumeration is \[ \frac{0}{1} < \frac{1}{4} < \frac{1}{3} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{1}{1}, \] where \( p_0 = 0, p_1 = 1, p_2 = 1, p_3 = 1, p_4 = 2, p_5 = 3, p_6 = 1, q_0 = 1, q_1 = 4, q_2 = 3, q_3 = 2, q_4 = 3, q_5 = 4, q_6 = 1 \), and the minimum is \( 1 + 4 + 3 = 3 + 2 + 3 = 3 + 4 + 1 = 8 \).

Let $ ABCD$ be a convex quadrilateral. We have that $ \angle BAC\equal{}3\angle CAD$, $ AB\equal{}CD$, $ \angle ACD\equal{}\angle CBD$. Find angle $ \angle ACD$

Two two-digit numbers $a$ and b satisfy that the product $a \cdot b$ divides the four-digit number one gets by writing the two digits in $a$ followed by the two digits in $b$. Determine all possible values of $a$ and $b$.

Let $a \ge b \ge 0$ be real numbers. Find the area of the region defined as; $K=\{(x,y): x\ge y\ge0$ and $\forall n$ positive integers satisfy $a^n+b^n\ge x^n+y^n\}$ in the cordinate plane.

We have $ n \geq 2$ lamps $ L_{1}, . . . ,L_{n}$ in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp $ L_{i}$ and its neighbours (only one neighbour for $ i \equal{} 1$ or $ i \equal{} n$, two neighbours for other $ i$) are in the same state, then $ L_{i}$ is switched off; – otherwise, $ L_{i}$ is switched on. Initially all the lamps are off except the leftmost one which is on. $ (a)$ Prove that there are infinitely many integers $ n$ for which all the lamps will eventually be off. $ (b)$ Prove that there are infinitely many integers $ n$ for which the lamps will never be all off.

Show that it is impossible to find a triangle in the plane with all integer coordinates such that the lengths of the sides are all odd.

Let $ a, b, c$ be positive real numbers such that $ bc +ca +b = 1,$ . Prove that $$ \frac {1 +b^2c^2}{(b +c)^2} + \frac {1+ c^2a^2}{(c + a)^2} +\frac {1 +a^2b^2}{(a +b)^2} \geq \frac {5}{2}.$$

Five men play several sets of dominoes (two against two) so that each player has each other player once as a partner and two times as an opponent. Find the number of sets and all ways to arrange the players.

The formula $N=8 \times 10^{8} \times x^{-3/2}$ gives, for a certain group, the number of individuals whose income exceeds $x$ dollars. The lowest income, in dollars, of the wealthiest $800$ individuals is at least: $ \textbf{(A)}\ 10^4\qquad\textbf{(B)}\ 10^6\qquad\textbf{(C)}\ 10^8\qquad\textbf{(D)}\ 10^{12} \qquad\textbf{(E)}\ 10^{16} $

Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $ 4 : 3$. The horizontal length of a “$ 27$-inch” television screen is closest, in inches, to which of the following? [asy]import math; unitsize(7mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((0,0)--(4,0)--(4,3)--(0,3)--(0,0)--(4,3)); fill((0,0)--(4,0)--(4,3)--cycle,mediumgray); label(rotate(aTan(3.0/4.0))*"Diagonal",(2,1.5),NW); label(rotate(90)*"Height",(4,1.5),E); label("Length",(2,0),S);[/asy]$ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 20.5 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 21.5 \qquad \textbf{(E)}\ 22$

Given two triangles and such that the lengths of the sides of the first triangle are the lengths of the medians of the second triangle. Determine the ratio of the areas of these triangles.

The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12$

If, instead, the graph is a graph of ACCELERATION vs. TIME and the squirrel starts from rest, then the squirrel has the greatest speed at what time(s) or during what time interval? (A) at B (B) at C (C) at D (D) at both B and D (E) From C to D

The Sequence $\{a_{n}\}_{n \geqslant 0}$ is defined by $a_{0}=1, a_{1}=-4$ and $a_{n+2}=-4a_{n+1}-7a_{n}$ , for $n \geqslant 0$. Find the number of positive integer divisors of $a^2_{50}-a_{49}a_{51}$.

Let $f(x)=\lim_{n\to\infty} \frac{\sin^{n+2}x+\cos^{n+2}x}{\sin^n x+\cos^n x}$ for $0\leq x\leq \frac{\pi}2.$ Evaluate $\int_0^{\frac{\pi}2} f(x)\ dx.$

A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.

Let $A,B$ be two points in the plane with integer coordinates $A=(x_1,y_1)$ and $B=(x_2,y_2)$. (Thus $x_i,y_i\in\mathbb Z$, for $i=1,2$.) A path $\pi:A\to B$ is a sequence of [b]down[/b] and [b]right[/b] steps, where each step has an integer length, and the initial step starts from $A$, the last step ending at $B$. In the figure below, we indicated a path from $A_1=(4,9)$ to $B1=(10,3)$. The distance $d(A,B)$ between $A$ and $B$ is the number of such paths. For example, the distance between $A=(0,2)$ and $B=(2,0)$ equals $6$. Consider now two pairs of points in the plane $A_i=(x_i,y_i)$ and $B_i=(u_i,z_i)$ for $i=1,2$, with integer coordinates, and in the configuration shown in the picture (but with arbitrary coordinates): $x_2<x_1$ and $y_1>y_2$, which means that $A_1$ is North-East of $A_2$; $u_2<u_1$ and $z_1>z_2$, which means that $B_1$ is North-East of $B_2$. Each of the points $A_i$ is North-West of the points $B_j$, for $1\le i$, $j\le2$. In terms of inequalities, this means that $x_i<\min\{u_1,u_2\}$ and $y_i>\max\{z_1,z_2\}$ for $i=1,2$. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi9hL2I4ODlmNDAyYmU5OWUyMzVmZmEzMTY1MGY3YjI3YjFlMmMxMTI2LnBuZw==&rn=VlRSTUMgMjAxNC5wbmc=[/img] (a) Find the distance between two points $A$ and $B$ as before, as a function of the coordinates of $A$ and $B$. Assume that $A$ is North-West of $B$. (b) Consider the $2\times2$ matrix $M=\begin{pmatrix}d(A_1,B_1)&d(A_1,B_2)\\d(A_2,B_1)&d(A_2,B_2)\end{pmatrix}$. Prove that for any configuration of points $A_1,A_2,B_1,B_2$ as described before, $\det M>0$.

Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron? $ \textbf{(A) } \frac 18 \qquad \textbf{(B) } \frac 16 \qquad \textbf{(C) } \frac 14 \qquad \textbf{(D) } \frac 13 \qquad \textbf{(E) } \frac 12$

Consider the system of equations $$a + 2b + 3c + \ldots + 26z = 2020$$ $$b + 2c + 3d + \ldots + 26a = 2019$$ $$\vdots$$ $$y + 2z + 3a + \ldots + 26x = 1996$$ $$z + 2a + 3b + \ldots + 26y = 1995$$ where each equation is a rearrangement of the first equation with the variables cycling and the coefficients staying in place. Find the value of $$z + 2y + 3x + \dots + 26a.$$ [i]Proposed by Joshua Hsieh[/i]

Each of the $48$ faces of eight $1\times 1\times 1$ cubes is randomly painted either blue or green. The probability that these eight cubes can then be assembled into a $2\times 2\times 2$ cube in a way so that its surface is solid green can be written $\frac{p^m}{q^n}$ , where $p$ and $q$ are prime numbers and $m$ and $n$ are positive integers. Find $p + q + m + n$.

(a) Is it true that, for arbitrary integer $n{}$ greater than $1$ and distinct positive integers $i{}$ and $j$ not greater than $n{}$, the set of any $n{}$ consecutive integers contains distinct numbers $i^{'}$ and $j^{'}$ whose product $i^{'}j^{'}$ is divisible by the product $ij$? (b) Is it true that, for arbitrary integer $n{}$ greater than $2$ and distinct positive integers $i, j, k$ not greater than $n{}$, the set of any $n{}$ consecutive integers contains distinct numbers $i^{'},j^{'},k^{'}$ whose product $i^{'}j^{'}k^{'}$ is divisible by the product $ijk$?