Show that the expressions $2x + 3y$, $9x + 5y$ are both divisible by $17$, for the same values of $x$ and $y$.

Let $n$ be a positive integer. Prove that $$\sum_{k=1}^n (-1)^{\lfloor k (\sqrt{2} - 1) \rfloor} \geq 0.$$ (As usual, $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)

A number of signal lights are equally spaced along a one-way railroad track, labeled in oder $ 1,2, \ldots, N, N \geq 2.$ As a safety rule, a train is not allowed to pass a signal if any other train is in motion on the length of track between it and the following signal. However, there is no limit to the number of trains that can be parked motionless at a signal, one behind the other. (Assume the trains have zero length.) A series of $ K$ freight trains must be driven from Signal 1 to Signal $ N.$ Each train travels at a distinct but constant spped at all times when it is not blocked by the safety rule. Show that, regardless of the order in which the trains are arranged, the same time will elapse between the first train's departure from Signal 1 and the last train's arrival at Signal $ N.$

$(YUG 2)$ A park has the shape of a convex pentagon of area $50000\sqrt{3} m^2$. A man standing at an interior point $O$ of the park notices that he stands at a distance of at most $200 m$ from each vertex of the pentagon. Prove that he stands at a distance of at least $100 m$ from each side of the pentagon.

As shown in the following figure, point $ P$ lies on the circumcicle of triangle $ ABC.$ Lines $ AB$ and $ CP$ meet at $ E,$ and lines $ AC$ and $ BP$ meet at $ F.$ The perpendicular bisector of line segment $ AB$ meets line segment $ AC$ at $ K,$ and the perpendicular bisector of line segment $ AC$ meets line segment $ AB$ at $ J.$ Prove that \[ \left(\frac{CE}{BF} \right)^2 \equal{} \frac{AJ \cdot JE}{AK \cdot KF}.\]

let m and n be natural numbers such that: $3m|(m+3)^n+1$ Prove that $\frac{(m+3)^n+1}{3m}$ is odd

Find the remainder when $2^{2019}$ is divided by $7$.

What is the sum of the prime factors of 20!?

Consider a $2^k$-tuple of numbers $(a_1,a_2,\dots,a_{2^k})$ all equal to $1$ or $-1$. In one step, we transform it to $(a_1a_2,a_2a_3,\dots,a_{2^k}a_1)$. Prove that eventually, we will obtain a $2^k$-tuple consisting only of $1$'s.

Does there exist a nowhere dense, nonempty compact set $C \subset [0,1]$ such that \[ \liminf_{h \to 0^+} \frac{\lambda(C \cap (x, x+h))}{h} > 0 \quad \text{or} \quad \liminf_{h \to 0^+} \frac{\lambda(C \cap (x-h, x))}{h} > 0 \] holds for every point $x \in C$, where $\lambda(A)$ denotes the Lebesgue measure of $A$?

Let $a$ and $b$ be positive real numbers with $a > b$. Find the smallest possible values of $$S = 2a +3 +\frac{32}{(a - b)(2b +3)^2}$$

Let $\mathbb{N}$ denote the set of positive integers. For how many positive integers $k\le 2018$ do there exist a function $f: \mathbb{N}\to \mathbb{N}$ such that $f(f(n))=2n$ for all $n\in \mathbb{N}$ and $f(k)=2018$? [i]Proposed by James Lin

Let $p$ be a prime number such that $p\equiv 1\pmod 9$. Show that there exist an integer $n$ such that $n^3-3n+1$ is divisible by $p$.

Find the minimum value of \[\max(a + b + c, b + c + d, c + d + e, d + e + f, e + f + g)\] subject to the constraints (i) $a, b, c, d, e, f, g \geq 0,$ (ii)$ a + b + c + d + e + f + g = 1.$

Suppose that $\alpha$ is a real number and $a_1<a_2<.....$ is a strictly increasing sequence of natural numbers such that for each natural number $n$ we have $a_n\le n^{\alpha}$. We call the prime number $q$ golden if there exists a natural number $m$ such that $q|a_m$. Suppose that $q_1<q_2<q_3<.....$ are all the golden prime numbers of the sequence $\{a_n\}$. [b]a)[/b] Prove that if $\alpha=1.5$, then $q_n\le 1390^n$. Can you find a better bound for $q_n$? [b]b)[/b] Prove that if $\alpha=2.4$, then $q_n\le 1390^{2n}$. Can you find a better bound for $q_n$? [i]part [b]a[/b] proposed by mahyar sefidgaran by an idea of this question that the $n$th prime number is less than $2^{2n-2}$ part [b]b[/b] proposed by mostafa einollah zade[/i]

A container is shaped like a square-based pyramid where the base has side length $23$ centimeters and the height is $120$ centimeters. The container is open at the base of the pyramid and stands in an open field with its vertex pointing down. One afternoon $5$ centimeters of rain falls in the open field partially filling the previously empty container. Find the depth in centimeters of the rainwater in the bottom of the container after the rain.

Initially, one of the two boxes on the table is empty and the other contains $29$ different colored marbles. By starting with the full box and performing moves in order, in each move, one or more marbles are selected from that box and transferred to the other box. At most, how many moves can be made without selecting the same set of marbles more than once?

Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=1$. Prove that\[ 6(a^3+b^3+c^3+d^3)\ge(a^2+b^2+c^2+d^2)+\frac{1}{8} \]

A game about passing barriers rules that in the $n$th barrier, you need to throw a dice for $n$ times. If the sum of points you get is larger than $2^n$, then you can pass this barrier. [b](a)[/b] How many barriers can you pass at most? [b](b)[/b] Find the probablity of passing the first three barriers.

$200$ sticks are given whose lengths are $1, 2, 4, \ldots , 2^{199}$. What is the smallest number of sticks needed to be broken so that out of all the resulting sticks, several triangles could be created, if each stick could be broken only once, and each triangle can be created out of only three sticks?

Consider a coin with a head toss probability $p$ where $0 <p <1$ is fixed. Toss the coin several times, the tosses should be independent of each other. Denote by $A_i$ the event that of the $i$-th, $(i + 1)$-th, $\ldots$ , the $(i+m-1)$-th throws, exactly $T$ is the tail. For $T = 1$, calculate the conditional probability $\mathbb{P}(\bar{A_2} \bar{A_3} \cdots \bar{A_m} | A_1)$, and for $T = 2$, prove that $\mathbb{P}(\bar{A_2} \bar{A_3} \cdots \bar{A_m} | A_1)$ has approximation in the form $a+ \tfrac{b}{m} + \mathcal{O}(p^m)$ as $m \to \infty$.

Compute $$\frac{2019! \cdot 2^{2019}}{(2020^2-2018^2)(2020^2-2016^2)\dots(2020^2-2^2)}.$$ [i]Proposed by Ada Tsui[/i]

$z$ is a complex number and $|z|=1$ and $z^2\neq 1$. Then $\frac{z}{1-z^2}$ lies on [list=1] [*] A line not passing through origin [*] $|z|=2$ [*] $x$-axis [*] $y$-axis [/list]

A real sequence $(a_n)_{n=0}^\infty$ is defined recursively by $a_0 = 2$ and the recursion formula $$ a_{n} = \begin{dcases} a_{n-1}^2 & \text{if $a_{n-1}<\sqrt3$} \\ \frac{a_{n-1}^2}{3} & \text{if $a_{n-1}\geq\sqrt 3$.} \end{dcases} $$ Another real sequence $(b_n)_{n=1}^\infty$ is defined in terms of the first by the formula $$ b_{n} = \begin{dcases} 0 & \text{if $a_{n-1}<\sqrt3$} \\ \frac{1}{2^{n}} & \text{if $a_{n-1}\geq\sqrt 3$,} \end{dcases} $$ valid for each $n\geq 1$. Prove that $$ b_1 + b_2 + \cdots + b_{2020} < \frac23. $$

Reimu and Sanae play a game using $4$ fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the four coins are tossed. If there are more red sides showing up, then Reimu wins, and if there are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides and green sides, then [i]neither[/i] of them wins. Given that both of them play optimally to maximize the probability of winning, what is the probability that Reimu wins?