The base of the pyramid is a convex quadrangle. Is there necessarily a section of this pyramid that does not intersect the base and is an inscribed quadrangle? (M. Volchkevich)

Points $A_1, B_1, C_1$ inside an acute-angled triangle $ABC$ are selected on the altitudes from $A, B, C$ respectively so that the sum of the areas of triangles $ABC_1, BCA_1$, and $CAB_1$ is equal to the area of triangle $ABC$. Prove that the circumcircle of triangle $A_1B_1C_1$ passes through the orthocenter $H$ of triangle $ABC$.

Two players play the following game. They alternately write divisors of $100!$ on the blackboard, not repeating any of the numbers written before. The player after whose move the greatest common divisor of the written numbers equals $1,$ loses the game. Which player has a winning strategy?

For a positive integer $a$, define the sequence ($x_n$) by $x_1 = x_2 = 1$ and $x_{n+2 }= (a^4 +4a^2 +2)x_{n+1} -x_n -2a^2$ , for n $\ge 1$. Show that $x_n$ is a perfect square and that for $n > 2$ its square root equals the first entry in the matrix $\begin{pmatrix} a^2+1 & a \\ a & 1 \end{pmatrix}^{n-2}$

Given $3n$ points in the plane, no three collinear, is it always possible to form $n$ triangles (with vertices at the points), so that no point in the plane lies in more than one triangle?

If $ m$ men can do a job in $ d$ days, then $ m\plus{}r$ men can do the job in: $\textbf{(A)}\ d+r\text{ days} \qquad \textbf{(B)}\ d-r\text{ days} \qquad \textbf{(C)}\ \dfrac{md}{m+r}\text{ days} \qquad \textbf{(D)}\ \dfrac{d}{m+r}\text{ days} \qquad \textbf{(E)}\ \text{None of these}$

[b]Black hole thermodynamics [/b] The goal of this problem is to explore some interesting properties of Black Holes. The following equation was obtained by L. Smarr in 1973: \[ M^2 = \frac{1}{16\pi} A + \frac{4\pi}{A}\left(J^2 + \frac{1}{4}Q^4\right)+\frac{1}{2}Q^2\] where $M$, $J$, $Q$ and $A$ are the mass, angular momentum, charge and area of the event horizon of a black hole. To make contact with thermodynamics we write for the entropy of the Black Hole, \[S = \frac{1}{4}k_B A\] where $k_B$ is the Boltzmann constant. [list=1] [*] Work in natural units $G = \hbar = c = 1$ and show that the equation for the entropy is dimensionally correct. [/*] [*] Take $k_B = 1/8\pi$ (by choosing units) and derive an expression for $S(M,J,Q)$. Is this expression unique? (Hint: What is the entropy of the Schwarzschild Black Hole which corresponds to $J=Q=0$?) \item We suppose the mass-energy $M$ (since $c=1$) plays the role of internal energy. Show that $T,\Omega,\Phi$ defined via, \[ dM = T dS + \Omega dJ + \Phi dQ\] are given by, \begin{eqnarray*} & T = \frac{1}{M} \left[1- \frac{1}{16S^2}\left(J^2 + \frac{1}{4}Q^4\right)\right] \\ & \Omega = \frac{J}{8MS}\\ & \Phi = \frac{Q}{2M}\left[1+\frac{Q^2}{8S}\right]. \end{eqnarray*} This is the analog of the first law of thermodynamics. [/*] [*]Look at the expression for $M(S,J,Q)$ closely and derive the analog of the Gibbs-Duhem Relation familiar from Thermodynamics. [/*] [*] Show that, \[ S \to \frac{1}{4}M^2 - \frac{1}{8}Q^2 \] as $T \to 0$. What does this say about the third law of thermodynamics? Give reasons to support your answer. \item An alternative statement to the third law is that "it is impossible to reach absolute-zero in a finite number of steps". What can we conclude from part (e)? [/*] [/list]

Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees a second tail before she sees a second head? $\textbf{(A) } \frac{1}{36} \qquad \textbf{(B) } \frac{1}{24} \qquad \textbf{(C) } \frac{1}{18} \qquad \textbf{(D) } \frac{1}{12} \qquad \textbf{(E) } \frac{1}{6}$

A country has four political parties - the Blue Party, the Red Party, the Yellow Party and the Orange Party - and a parliament of 650 seats. (a) How many ways are there to divide the seats among the four parties so that none of the parties have a majority? (To have a majority that party must hold more than half of the seats.) The parliament is particularly worried about cyber security. They have decided that all login passwords must be of length exactly 6 and be a combination of a legal set of elements made up of the digits 0-9, the 52 upper and lower case letters (a-z and A-Z), and five special characters: \ICMC 2018/19 Round 1, Problem 1 $, £, *, &, %. For the password to be allowed, it must contain at least one letter or special character and any letter or special character in the password must be followed by a digit (so it must end in a digit). (b) The Blue members of parliament have decided to choose their password by selecting 6 elements from the legal set without replacement. What is the probability it is allowed? Note: you may leave your answers as combinatorial or factorial terms.

In $n$ transparent boxes there are red balls and blue balls. One needs to choose $50$ boxes such that, together, they contain at least half of the red balls and at least half of the blue balls. Is such a choice possible irrespective on the number of balls and on the way they are distributed in the boxes, if: a) $n = 100$ b) $n = 99$?

Let $C_1,C_2$, and $C_3$ be points inside a bounded convex planar set $M$. Rays $l_1,l_2,l_3$ emanating from $C_1,C_2,C_3$ respectively partition the complement of the set $M \cup l_1 \cup l_2 \cup l_3$ into three regions $D_1,D_2,D_3$. Prove that if the convex sets $A$ and $B$ satisfy $A\cap l_j =\emptyset = B\cap l_j$ and $A\cap D_j \ne \emptyset \ne B\cap D_j$ for $j = 1,2,3$, then $A\cap B \ne \emptyset$

Pairwise distinct real numbers $a, b, c$ satisfies the equality \[a +\frac 1b =b + \frac 1c =c+\frac 1a.\] Find all possible values of $abc .$

Let $a, b, c, d$ be real numbers for which $a^2 + b^2 + c^2 + d^2 = 1$. Show the following inequality: $$a^2 + b^2 - c^2 - d^2 \le \sqrt{2 + 4(ac + bd)}.$$

Andy's lawn has twice as much area as Beth's lawn and three times as much area as Carlos' lawn. Carlos' lawn mower cuts half as fast as Beth's mower and one third as fast as Andy's mower. If they all start to mow their lawns at the same time, who will finish first? $ \textbf{(A)}\ \text{Andy} \qquad \textbf{(B)}\ \text{Beth} \qquad \textbf{(C)}\ \text{Carlos} \qquad \textbf{(D)}\ \text{Andy and Carlos tie for first.}$ $\textbf{(E)}\ \text{All three tie.}$

Let $A = A(x,y)$ and $B = B(x,y)$ be two-variable polynomials with real coefficients. Suppose that $A(x,y)/B(x,y)$ is a polynomial in $x$ for infinitely many values of $y$, and a polynomial in $y$ for infinitely many values of $x$. Prove that $B$ divides $A$, meaning there exists a third polynomial $C$ with real coefficients such that $A = B \cdot C$. [i]Proposed by Victor Wang[/i]

Let the inscribed circle of the triangle $\vartriangle ABC$ touch side $BC$ at $M$ , side $CA$ at $N$ and side $AB$ at $P$ . Let $D$ be a point from $\left[ NP \right]$ such that $\frac{DP}{DN}=\frac{BD}{CD}$ . Show that $DM \perp PN$ .

Given an isosceles triangle $\triangle ABC$, $AB=AC$. A line passes through $M$, the midpoint of $BC$, and intersects segment $AB$ and ray $CA$ at $D$ and $E$, respectively. Let $F$ be a point of $ME$ such that $EF=DM$, and $K$ be a point on $MD$. Let $\Gamma_1$ be the circle passes through $B,D,K$ and $\Gamma_2$ be the circle passes through $C,E,K$. $\Gamma_1$ and $\Gamma_2$ intersect again at $L \neq K$. Let $\omega_1$ and $\omega_2$ be the circumcircle of $\triangle LDE$ and $\triangle LKM$. Prove that, if $\omega_1$ and $\omega_2$ are symmetric wrt $L$, then $BF$ is perpendicular to $BC$.

Let $n\ge 2$ be an integer. A sequence $\alpha = (a_1, a_2,..., a_n)$ of $n$ integers is called [i]Lima [/i] if $\gcd \{a_i - a_j \text{ such that } a_i> a_j \text{ and } 1\le i, j\le n\} = 1$, that is, if the greatest common divisor of all the differences $a_i - a_j$ with $a_i> a_j$ is $1$. One operation consists of choosing two elements $a_k$ and $a_{\ell}$ from a sequence, with $k\ne \ell $ , and replacing $a_{\ell}$ by $a'_{\ell} = 2a_k - a_{\ell}$ . Show that, given a collection of $2^n - 1$ Lima sequences, each one formed by $n$ integers, there are two of them, say $\beta$ and $\gamma$, such that it is possible to transform $\beta$ into $\gamma$ through a finite number of operations. Notes. The sequences $(1,2,2,7)$ and $(2,7,2,1)$ have the same elements but are different. If all the elements of a sequence are equal, then that sequence is not Lima.

Prove that if the quadratic $x^2 +ax+b$ is always positive (for all real $x$) then it can be written as the quotient of two polynomials whose coefficients are all positive.

Find the sum of all values of $x$ such that the set $\{107, 122,127, 137, 152,x\}$ has a mean that is equal to its median.

Find all pairs \((m,n)\) of positive integers such that there exists a polyhedron, with all faces being regular polygons, such that each vertex of the polyhedron is the vertex of exactly three faces, two of them having \(m\) sides, and the other having \(n\) sides.

When a positive integer $N$ is divided by $60$, the remainder is $49$. When $N$ is divided by $15$, the remainder is $ \mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 3 \qquad \mathrm {(C) \ } 4 \qquad \mathrm{(D) \ } 5 \qquad \mathrm{(E) \ } 8$

There are $ 6n \plus{} 4$ mathematicians participating in a conference which includes $ 2n \plus{} 1$ meetings. Each meeting has one round table that suits for $ 4$ people and $ n$ round tables that each table suits for $ 6$ people. We have known that two arbitrary people sit next to or have opposite places doesn't exceed one time. 1. Determine whether or not there is the case $ n \equal{} 1$. 2. Determine whether or not there is the case $ n > 1$.

In triangle $\vartriangle ABC$, $\angle BAC = 135^o$. $M$ is the midpoint of $BC$, and $N \ne M$ is on $BC$ such that $AN = AM$. The line $AM$ meets the circumcircle of $\vartriangle ABC$ at $D$. Point $E$ is chosen on segment $AN$ such that $AE = MD$. Show that $ME = BC$.

Let $P$ be a set of $n\ge 3$ points in the plane, no three of which are on a line. How many possibilities are there to choose a set $T$ of $\binom{n-1}{2}$ triangles, whose vertices are all in $P$, such that each triangle in $T$ has a side that is not a side of any other triangle in $T$?