Let all positive integers $n$ satisfy the following condition: for each non-negative integers $k, m$ with $k + m \le n$, the numbers $\binom{n-k}{m}$ and $\binom{n-m}{k}$ leave the same remainder when divided by $2$. (Poland) PS. The translation was done using Google translate and in case it is not right, there is the original text in Slovak

Kunal and Arnab play a game as follows. Initially there are $2$ piles of coins with $x$ and $y$ coins respectively. The game starts with Kunal. In each turn a player chooses one pile and removes as many coins as he wants from that pile. The game goes on and the last one to remove a coin loses. Determine all possible values of $(x,y)$ which ensure Kunal's victory against Arnab given both os them play optimally. \\ [i]You are required to find an exhaustive set of solutions[/i]

The sequence of positive integers $a_1,a_2,a_3,\dots$ is [i]brazilian[/i] if $a_1=1$ and $a_n$ is the least integer greater than $a_{n-1}$ and $a_n$ is [b]coprime[/b] with at least half elements of the set $\{a_1,a_2,\dots, a_{n-1}\}$. Is there any odd number which does [b]not[/b] belong to the brazilian sequence?

Given three parallel straight lines. Construct a square three of whose vertices belong to these lines.

It is known that the bisector of the angle $\angle ADC$ of the inscribed quadrilateral $ABCD$ passes through the center of the circle inscribed in the triangle $ABC$. Let $M$ be an arbitrary point of the arc $ABC$ of the circle circumscribed around $ABCD$. Denote by $P$ and $Q$ the centers of the circles inscribed in the triangles $ABM$ and $BCM$. Prove that all triangles $DPQ$ obtained by moving point $M$ are similar to each other. Find the angle $\angle PDQ$ and ratio $BP : PQ$ if $\angle BAC = \alpha$, $\angle BCA = \beta$

Let $n,k$ be positive integers such that $n$ is not divisible by $3$ and $k\ge n$. Prove that there is an integer $m$ divisible by $n$ whose sum of digits in base $10$ equals $k$.

Circle centred at point $O$ intersects sides $AC, AB$ of triangle $\triangle ABC$ at points $B_1$ and $C_1$ respectively and passes through points $B,C$. It is known that lines $AO, CC_1, BB_1 $ are concurrent. Prove that $\triangle ABC$ is isosceles.

With a rational denominator, the expression $ \frac {\sqrt {2}}{\sqrt {2} \plus{} \sqrt {3} \minus{} \sqrt {5}}$ is equivalent to: $ \textbf{(A)}\ \frac {3 \plus{} \sqrt {6} \plus{} \sqrt {15}}{6} \qquad\textbf{(B)}\ \frac {\sqrt {6} \minus{} 2 \plus{} \sqrt {10}}{6} \qquad\textbf{(C)}\ \frac {2 \plus{} \sqrt {6} \plus{} \sqrt {10}}{10}$ $ \textbf{(D)}\ \frac {2 \plus{} \sqrt {6} \minus{} \sqrt {10}}{6} \qquad\textbf{(E)}\ \text{none of these}$

Let $\vartriangle ABC$ be an acute triangle with $AB < AC$. Let $H$ be its orthocentre and $M$ be the midpoint of arc $BAC$ on the circumcircle. It is given that $B$, $H$, $M$ are collinear, the length of the altitude from $M$ to $AB$ is $1$, and the length of the altitude from $M$ to $BC$ is $6$. Determine all possible areas for $\vartriangle ABC$ .

Let $ABC$ be an isosceles right triangle with $\angle A=90^o$. Point $D$ is the midpoint of the side $[AC]$, and point $E \in [AC]$ is so that $EC = 2AE$. Calculate $\angle AEB + \angle ADB$ .

Let $ a_{1},a_{2},\cdots,a_{n}$ be positive real numbers satisfying $ a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{n} \equal{} 1$. Prove that \[\left(a_{1}a_{2} \plus{} a_{2}a_{3} \plus{} \cdots \plus{} a_{n}a_{1}\right)\left(\frac {a_{1}}{a_{2}^2 \plus{} a_{2}} \plus{} \frac {a_{2}}{a_{3}^2 \plus{} a_{3}} \plus{} \cdots \plus{} \frac {a_{n}}{a_{1}^2 \plus{} a_{1}}\right)\ge\frac {n}{n \plus{} 1}\]

Let $c$ be a real number, and let $z_1, z_2$ be the two complex numbers satisfying the quadratic $z^2 - cz + 10 = 0$. Points $z_1, z_2, \frac{1}{z_1}$, and $\frac{1}{z_2}$ are the vertices of a (convex) quadrilateral $Q$ in the complex plane. When the area of $Q$ obtains its maximum value, $c$ is the closest to which of the following? $\textbf{(A)}~4.5\qquad\textbf{(B)}~5\qquad\textbf{(C)}~5.5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~6.5$

Consider the set $A=\{1,2,3,4,5,6,7,8,9\}$.A partition $\Pi $ of $A$ is collection of disjoint sets whose union is $A$. For example, $\Pi_1=\{\{1,2\},\{3,4,5\},\{6,7,8,9\}\}$ and $\Pi _2 =\{\{1\},\{2,5\},\{3,7\},\{4,5,6,7,8,9\}\}$ can be considered as partitions of $A$. For, each $\Pi$ partition ,we consider the function $\pi$ defined on the elements of$A$. $\pi (x)$ denotes the cardinality of the subset in $\Pi$ which contains $x$. For, example in case of $\Pi_1$ , $\pi_1(1)=\pi_1(2)=2$, $\pi_1(3)=\pi_1(4)=\pi_1 (5)=3$, and $\pi_1(6)=\pi_1(7)=\pi_1(8)=\pi_1(9)=4$. For $\Pi_2$ we have $\pi_2(1)=1$, $\pi_2(2)=\pi_2(5)=2$, $\pi_2(3)=\pi_2(7)=2$ and $\pi_2(4)=\pi_2(6)=\pi_2(8)=\pi_2(9)=4$ Given any two partitions $\Pi$ and $\Pi '$, show that there are two numbers $x$ and $y$ in $A$, such that $\pi (x)= \pi '(x)$ and $ \pi (y)= \pi'(y)$.[[b]Hint:[/b] Consider the case where there is a block of size greater than or equal to 4 in a partition and the alternative case]

For positive real numbers $a_1, a_2, ..., a_n$ with product 1 prove: $$\left(\frac{a_1}{a_2}\right)^{n-1}+\left(\frac{a_2}{a_3}\right)^{n-1}+...+\left(\frac{a_{n-1}}{a_n}\right)^{n-1}+\left(\frac{a_n}{a_1}\right)^{n-1} \geq a_1^{2}+a_2^{2}+...+a_n^{2}$$ Proposed by Andrei Eckstein

Every day, Heesu talks to Sally with some probability $p$. One day, after not talking to Sally the previous day, Heesu resolves to ask Sally out on a date. From now on, each day, if Heesu has talked to Sally each of the past four days, then Heesu will ask Sally out on a date. Heesu’s friend remarked that at this rate, it would take Heesu an expected $2800$ days to finally ask Sally out. Suppose $p=\tfrac{m}{n}$, where $\gcd(m, n) = 1$ and $m, n > 0$. What is $m + n$?

How many seven-digit numbers are multiples of $388$ and end in $388$?

A circle with center $O$ passes through the ends of the hypotenuse of a right-angled triangle and intersects its legs at points $M$ and $K$. Prove that the distance from point $O$ to line $MK$ is half the hypotenuse.

Let $X$ be the set of consisting of twenty positive integers $n,n+2,...,n+38$. The smallest value of $n$ for which any three numbers $a,b,c \in X$, not necessarily distinct, form the sides of an acute-angled triangle is:

Find the sum of squares of the digits of the least positive integer having $72$ positive divisors. $\textbf{(A)}\ 41 \qquad\textbf{(B)}\ 65 \qquad\textbf{(C)}\ 110 \qquad\textbf{(D)}\ 123 \qquad\textbf{(E)}\ \text{None}$

For which rational $ x$ is the number $ 1 \plus{} 105 \cdot 2^x$ the square of a rational number?

Let $f(x)=x^4+2x+1$. Find the slope of the tangent line to the curve at $(0,1)$.

We say that a positive integer $n$ is [i]fantastic[/i] if there exist positive rational numbers $a$ and $b$ such that $$ n = a + \frac 1a + b + \frac 1b.$$ [b](a)[/b] Prove that there exist infinitely many prime numbers $p$ such that no multiple of $p$ is fantastic. [b](b)[/b] Prove that there exist infinitely many prime numbers $p$ such that some multiple of $p$ is fantastic. [i]Proposed by Walther Janous, Austria[/i]

a) Prove that, whatever the real number x would be, the following inequality takes place ${{x}^{4}}-{{x}^{3}}-x+1\ge 0.$ b) Solve the following system in the set of real numbers: ${{x}_{1}}+{{x}_{2}}+{{x}_{3}}=3,x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=x_{1}^{4}+x_{2}^{4}+x_{3}^{4}$. The Mathematical Gazette

All of the following are condensation polymers except: $ \textbf{(A) }\text{Nylon} \qquad\textbf{(B) }\text{Polyethylene}\qquad\textbf{(C) }\text{Protein} \qquad\textbf{(D) }\text{Starch}\qquad $

A particle is launched from the surface of a uniform, stationary spherical planet at an angle to the vertical. The particle travels in the absence of air resistance and eventually falls back onto the planet. Spaceman Fred describes the path of the particle as a parabola using the laws of projectile motion. Spacewoman Kate recalls from Kepler’s laws that every bound orbit around a point mass is an ellipse (or circle), and that the gravitation due to a uniform sphere is identical to that of a point mass. Which of the following best explains the discrepancy? (A) Because the experiment takes place very close to the surface of the sphere, it is no longer valid to replace the sphere with a point mass. (B) Because the particle strikes the ground, it is not in orbit of the planet and therefore can follow a nonelliptical path. (C) Kate disregarded the fact that motions around a point mass may also be parabolas or hyperbolas. (D) Kepler’s laws only hold in the limit of large orbits. (E) The path is an ellipse, but is very close to a parabola due to the short length of the flight relative to the distance from the center of the planet.