Consider $n$ events, each of which has probability $\frac12$. We also know that the probability of any two both happening is $\frac14$. Prove the following. (a) The probability that none of these events happen is at most $\frac1{n+1}$. (b) We can reach equality in (a) for infinitely many $n$.

For how many ordered quadruplets $(a,b,c,d)$ of positive integers such that $2\leq a\leq b \leq c$ and $1 \leq d \leq 418$ do we have that $bcd+abd+acd=abc+abcd?$

Let real numbers $a,b,c$ such that $\left| a-b \right|\ge \left| c \right|,\left| b-c \right|\ge \left| a \right|,\left| c-a \right|\ge \left| b \right|.$ Prove that $a=b+c$ or $b=c+a$ or $c=a+b.$

Let $Q$ be a polynomial \[Q(x)=a_0+a_1x+\cdots+a_nx^n,\] where $a_0,\ldots,a_n$ are nonnegative integers. Given that $Q(1)=4$ and $Q(5)=152$, find $Q(6)$.

Let $M=\{1,2,\ldots,3 \cdot n\}$. Partition $M$ into three sets $A,B,C$ which $card$ $A$ $=$ $card$ $B$ $=$ $card$ $C$ $=$ $n .$ Prove that there exists $a$ in $A,b$ in $B, c$ in $C$ such that or $a=b+c,$ or $b=c+a,$ or $c=a+b$ [i]Edited by orl.[/i]

Let $ABCD$ be a trapezoid such that $AB \parallel CD$ and $AB > CD$. Points $X$ and $Y$ are on the side $AB$ such that $XY = AB-CD$ and $X$ lies between $A$ and $Y$. Prove that one intersection of the circumcircles of triangles $AYD$ and $BXC$ is on line $CD$.

Let $x, y$, and $z$ be positive integers satisfying the following system of equations: $$x^2 +\frac{2023}{x}= 2y^2$$ $$y +\frac{2028}{y^2} = z^2$$ $$2z +\frac{2025}{z^2} = xy$$ Find $x + y + z$.

In the right-angled $\Delta ABC$, with area $S$, a circle with area $S_1$ is inscribed and a circle with area $S_2$ is circumscribed. Prove the following inequality: $\pi \frac{S-S_1}{S_2} <\frac{1}{\pi-1}$.

Three students write on the blackboard next to each other three two-digit squares. In the end, they observe that the 6-digit number thus obtained is also a square. Find this number!

In a parallelogram $ABCD$, the perpendiculars from $A$ to $BC$ and $CD$ meet the line segments $BC$ and $CD$ at the points $E$ and $F$ respectively. Suppose $AC = 37$ cm and $EF = 35$ cm. Let $H$ be the orthocentre of $\vartriangle AEF$. Find the length of $AH$ in cm. Show the steps in your calculations.

Let $a, b, c$ be real numbers such that $a+b+c=abc$. Prove that $\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\ge \frac{3}{4}$.

Two circles $ \Omega_{1}$ and $ \Omega_{2}$ are externally tangent to each other at a point $ I$, and both of these circles are tangent to a third circle $ \Omega$ which encloses the two circles $ \Omega_{1}$ and $ \Omega_{2}$. The common tangent to the two circles $ \Omega_{1}$ and $ \Omega_{2}$ at the point $ I$ meets the circle $ \Omega$ at a point $ A$. One common tangent to the circles $ \Omega_{1}$ and $ \Omega_{2}$ which doesn't pass through $ I$ meets the circle $ \Omega$ at the points $ B$ and $ C$ such that the points $ A$ and $ I$ lie on the same side of the line $ BC$. Prove that the point $ I$ is the incenter of triangle $ ABC$. [i]Alternative formulation.[/i] Two circles touch externally at a point $ I$. The two circles lie inside a large circle and both touch it. The chord $ BC$ of the large circle touches both smaller circles (not at $ I$). The common tangent to the two smaller circles at the point $ I$ meets the large circle at a point $ A$, where the points $ A$ and $ I$ are on the same side of the chord $ BC$. Show that the point $ I$ is the incenter of triangle $ ABC$.

For any positive integer $n$ denote $S(n)$ the digital sum of $n$ when represented in the decimal system. Find every positive integer $M$ for which $S(Mk)=S(M)$ holds for all integers $1\le k\le M$.

Let $n$ be a positive integer and let $\alpha_n $ be the number of $1$'s within binary representation of $n$. Show that for all positive integers $r$, \[2^{2n-\alpha_n}\phantom{-1} \bigg|^{\phantom{0}}_{\phantom{-1}} \sum_{k=-n}^{n} \binom{2n}{n+k} k^{2r}.\]

Given a positive integer $m$, let $d(m)$ be the number of positive divisors of $m$. Determine all positive integers $n$ such that $d(n) +d(n+ 1) = 5$.

Suppose that a sequence $\{a_n\}_{n\ge 1}$ satisfies $0 < a_n \le a_{2n} + a_{2n+1}$ for all $n\in \mathbb{N}$. Prove that the series$\sum_{n=1}^{\infty} a_n$ diverges.