Let $a$, $b$, $c$, and $d$ be positive integers such that the following two inequalities hold: $a < 10^{20} \cdot c$ and $b > 10^{23} \cdot d$. Determine the minimum possible value of the total number of positive integer pairs $(n, m)$ for which $n \cdot m = 2^{2023}$ and $$ \frac {ab}{n} + \frac{cd}{m} < \frac{(a + c)(b + d)}{n + m}$$ [i]Proposed by Stijn Cambie, Belgium[/i]

Let $N_9$ be the answer to problem 9. In a rainforest, there is a row of nine rocks labeled from $1$ to $N_9$. A gecko is standing on rock $1$. The gecko jumps according to following rules: [list] [*] if it is on rock $1$, then it will jump with equal probability to any of the other rocks. [*] if it is on rock $R$ and $R$ is prime, then it will jump to rock $N_9$. [*] if it is on rock $4$, then it will jump with equal probability to rock $1$ or rock $6$. [*] if it is on rock $R$ and $R$ is composite with $4<R<N_9$, then it will jump with equal probability to rock $Q$ or $S$, where $Q$ is the greatest composite number less than $R$, and $S$ is the smallest composite number greater than $R$. [*] if it is on rock $N_9$, then it stops jumping. [/list] The expected number of jumps the gecko will take to reach rock $N_9$ is $\tfrac{p}{q}$, where $p$ are $q$ relatively prime positive integers. Compute $p+q$.

The Olcommunity consists of the next seven people: Christopher Took, Humberto Brandybuck, German son of Isildur, Leogolas, Argimli, Samzamora and Shago Baggins. This community needs to travel from the Olcomashire to Olcomordor to save the world. Each person can take with them a total of $4$ day-provisions, that can be transferred to other people that are on the same day of traveling, as long as nobody holds more than $4$ day-provisions at the time. If a person returns to Olcomashire, they will be too tired to go out again. What is the farthest away Olcomordor can be from Olcomashire, so that Shago Baggins can get to Olcomordor, and the rest of the Olcommunity can return save to Olcomashire? Note: All the members of the Olcommunity should eat exactly one day-provision while they are away from Olcomashire. The members only travel an integer number of days on each direction. Members of the Olcommunity may leave Olcomashire on different days.

An integer $n > 1$ and a prime $p$ are such that $n$ divides $p-1$, and $p$ divides $n^3 - 1$. Prove that $4p - 3$ is a perfect square.

The graph relates the distance traveled [in miles] to the time elapsed [in hours] on a trip taken by an experimental airplane. During which hour was the average speed of this airplane the largest? [asy] unitsize(12); for(int a=1; a<13; ++a) { draw((2a,-1)--(2a,1)); } draw((-1,4)--(1,4)); draw((-1,8)--(1,8)); draw((-1,12)--(1,12)); draw((-1,16)--(1,16)); draw((0,0)--(0,17)); draw((-5,0)--(33,0)); label("$0$",(0,-1),S); label("$1$",(2,-1),S); label("$2$",(4,-1),S); label("$3$",(6,-1),S); label("$4$",(8,-1),S); label("$5$",(10,-1),S); label("$6$",(12,-1),S); label("$7$",(14,-1),S); label("$8$",(16,-1),S); label("$9$",(18,-1),S); label("$10$",(20,-1),S); label("$11$",(22,-1),S); label("$12$",(24,-1),S); label("Time in hours",(11,-2),S); label("$500$",(-1,4),W); label("$1000$",(-1,8),W); label("$1500$",(-1,12),W); label("$2000$",(-1,16),W); label(rotate(90)*"Distance traveled in miles",(-4,10),W); draw((0,0)--(2,3)--(4,7.2)--(6,8.5)); draw((6,8.5)--(16,12.5)--(18,14)--(24,15));[/asy] $ \text{(A)}\ \text{first (0-1)}\qquad\text{(B)}\ \text{second (1-2)}\qquad\text{(C)}\ \text{third (2-3)}\qquad\text{(D)}\ \text{ninth (8-9)}\qquad\text{(E)}\ \text{last (11-12)} $

The lengths of the sides of a convex hexagon $ ABCDEF$ satisfy $ AB \equal{} BC$, $ CD \equal{} DE$, $ EF \equal{} FA$. Prove that: \[ \frac {BC}{BE} \plus{} \frac {DE}{DA} \plus{} \frac {FA}{FC} \geq \frac {3}{2}. \]

Solve the following system of equations, in which $a$ is a given number satisfying $|a| > 1$: $\begin{matrix} x_{1}^2 = ax_2 + 1 \\ x_{2}^2 = ax_3 + 1 \\ \ldots \\ x_{999}^2 = ax_{1000} + 1 \\ x_{1000}^2 = ax_1 + 1 \\ \end{matrix}$

Given a regular hexagon with a side of $100$. Each side is divided into one hundred equal parts. Through the division points and vertices of the hexagon, all sorts of straight lines parallel to its sides are drawn. These lines divided the hexagon into single regular triangles. Consider covering a hexagon with equal rhombuses. Each rhombus is made up of two triangles. (These rhombuses cover the entire hexagon and do not overlap.) Among the lines that form our grid, we select those that intersect exactly to the rhombuses (intersect diagonally). How many such lines will there be if: a) $k = 101$; b) $k = 100$; c) $k = 87$?

Let $ a$, $ b$, $ c$, $ d$, and $ e$ be positive integers with $ a\plus{}b\plus{}c\plus{}d\plus{}e\equal{}2010$, and let $ M$ be the largest of the sums $ a\plus{}b$, $ b\plus{}c$, $ c\plus{}d$, and $ d\plus{}e$. What is the smallest possible value of $ M$? $ \textbf{(A)}\ 670 \qquad \textbf{(B)}\ 671 \qquad \textbf{(C)}\ 802 \qquad \textbf{(D)}\ 803 \qquad \textbf{(E)}\ 804$

Find all triples $(x, y, z)$ of real numbers that satisfy $x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|$.

Find the smallest possible distance of points $ P$ and $ Q$ on a $ xy$-plane, if $ P$ lies on the line $ y \equal{} x$ and $ Q$ lies on the curve $ y \equal{} 2^x$.

Given a triangle $ABC$ with circumcircle $\Gamma$. Points $E$ and $F$ are the foot of angle bisectors of $B$ and $C$, $I$ is incenter and $K$ is the intersection of $AI$ and $EF$. Suppose that $T$ be the midpoint of arc $BAC$. Circle $\Gamma$ intersects the $A$-median and circumcircle of $AEF$ for the second time at $X$ and $S$. Let $S'$ be the reflection of $S$ across $AI$ and $J$ be the second intersection of circumcircle of $AS'K$ and $AX$. Prove that quadrilateral $TJIX$ is cyclic. [i]Proposed by Alireza Dadgarnia and Amir Parsa Hosseini[/i]

Bucket $A$ contains 3 litres of syrup. Bucket $B$ contains $n$ litres of water. Bucket $C$ is empty. We can perform any combination of the following operations: - Pour away the entire amount in bucket $X$, - Pour the entire amount in bucket $X$ into bucket $Y$, - Pour from bucket $X$ into bucket $Y$ until buckets $Y$ and $Z$ contain the same amount. [b](a)[/b] How can we obtain 10 litres of 30% syrup if $n = 20$? [b](b)[/b] Determine all possible values of $n$ for which the task in (a) is possible.

Let $a>0$ be a real number. Find the minimal constant $C_a$ for which the inequality$$\displaystyle C_a\sum_{k=1}^n \frac1{x_k-x_{k-1}} >\sum_{k=1}^n \frac{k+a}{x_k}$$holds for any positive integer $n$ and any sequence $0=x_0<x_1<\cdots <x_n$ of real numbers.

Consider the sequence of sets defined by $S_0=\{0,1\},S_1=\{0,1,2\}$, and for $n\ge2$, \[S_n=S_{n-1}\cup\{2^n+x\mid x\in S_{n-2}\}.\] For example, $S_2=\{0,1,2\}\cup\{2^2+0,2^2+1\}=\{0,1,2,4,5\}$. Find the $200$th smallest element of $S_{2016}$.

Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.

Which number is greater: $$A=\frac{2.00\ldots04}{1.00\ldots04^2+2.00\ldots04},\text{ or }B=\frac{2.00\ldots02}{1.00\ldots02^2+2.00\ldots02},$$where each of the numbers above contains $1998$ zeros?

On the board, the numbers from $1$ to $n$ are written. Achka (A) and Bavachka (B) play the following game. First, A erases one number, then B erases two consecutive numbers, then A erases three consecutive numbers, and finally B erases four consecutive numbers. What is the smallest $n$ such that B can definitely make her moves, no matter how A plays?

Eddie has a study block that lasts $1$ hour. It takes Eddie $25$ minutes to do his homework and $5$ minutes to play a game of Clash Royale. He can’t do both at the same time. How many games can he play in this study block while still completing his homework? [i]Proposed by Edwin Zhao[/i] [hide=Solution] [i]Solution.[/i] $\boxed{7}$ Study block lasts 60 minutes, thus he has 35 minutes to play Clash Royale, during which he can play $\frac{35}{5}=\boxed{7}$ games. [/hide]

Let $m$ and $n$ be positive integers integers such that $2m + 1 < n$, and let $S$ be the set of the $2^n$ subsets of $\{1,2,\ldots,n\}$. Prove that we can place the elements of $S$ on a circle, so that for any two adjacent elements $A$ and $B$, the set $A \Delta B$ has exactly $2m + 1$ elements. [b]Note[/b]: $A \Delta B = (A \cup B) - (A \cap B)$ is the set of elements that are exclusively in $A$ or exclusively in $B$.

$S$ is a set of positive integers with the following properties: (a) There are exactly $3$ positive integers missing from $S$. (b) If $a$ and $b$ are elements of $S$, then $a + b$ is an element of $S$. (We allow a and b to be the same.) Find all possibilities for the set $S$ (with proof).

Let $z_1 = 18 + 83i$, $z_2 = 18 + 39i, $ and $z_3 = 78 + 99i,$ where $i = \sqrt{-1}$. Let $z$ be the unique complex number with the properties that $\frac{z_3 - z_1}{z_2 - z_1} \cdot \frac{z - z_2}{z - z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.

A cake is prepared for a dinner party to which only $p$ or $q$ persons will come ($p$ and $q$ are given co-prime integers). Find the minimum number of pieces (not necessarily equal) into which the cake must be cut in advance so that the cake may be equally shared between the persons in either case. (D. Fomin, Leningrad)

The sides $AB, BC, CD$ and $DA$ of the convex quadrilateral $ABCD$ have midpoints $E, F, G$ and $H$. Prove that the triangles $EFB, FGC, GHD$ and $HEA$ can be put together into a parallelogram equal to $EFGH$.

Let $S$ be the set of all squarefree numbers and $n$ be a natural number. Prove that $$\sum_{k\in S}\left\lfloor\sqrt{\frac nk}\right\rfloor=n.$$