The diagram below shows a $1\times2\times10$ duct with $2\times2\times2$ cubes attached to each end. The resulting object is empty, but the entire surface is solid sheet metal. A spider walks along the inside of the duct between the two marked corners. There are positive integers $m$ and $n$ so that the shortest path the spider could take has length $\sqrt{m}+\sqrt{n}$. Find $m + n$. [asy] size(150); defaultpen(linewidth(1)); draw(origin--(43,0)--(61,20)--(18,20)--cycle--(0,-43)--(43,-43)--(43,0)^^(43,-43)--(61,-23)--(61,20)); draw((43,-43)--(133,57)--(90,57)--extension((90,57),(0,-43),(61,20),(18,20))); draw((0,-43)--(0,-65)--(43,-65)--(43,-43)^^(43,-65)--(133,35)--(133,57)); draw((133,35)--(133,5)--(119.5,-10)--(119.5,20)^^(119.5,-10)--extension((119.5,-10),(100,-10),(43,-65),(133,35))); dot(origin^^(133,5)); [/asy]

We call a positive integer $\textit{cheesy}$ if we can obtain the average of the digits in its decimal representation by putting a decimal separator after the leftmost digit. Prove that there are only finitely many $\textit{cheesy}$ numbers.

Find all pairs of positive integers $(m,n)$ for which there exist relatively prime integers $a$ and $b$ greater than $1$ such that $$\frac{a^m+b^m}{a^n+b^n}$$ is an integer.

Karakade has three flash drives of each of the six capacities $1, 2, 4, 8, 16, 32$ gigabytes. She gives each of her $6$ servants three flash drives of different capacities. Prove that either there are two capacities where each servant has at most one of the two capacities, or all servants have flash drives with different sums of capacities.

If $ {|x|}<{1}$ and ${|y|}<1$ then prove that $|\frac{x-y}{1-xy}|<1$

Let $a, b$ and $c$ be pairwise different natural numbers. Prove $\frac{a^3 + b^3 + c^3}{3} \ge abc + a + b + c$. When does equality holds? (Karl Czakler)

Let $S$ be the incenter of the triangle $ABC$ and let the line $AS$ intersect the circumcircle of triangle $ABC$ at point $D$ ($D\ne A$). Prove that the segments $BD, CD$ and $SD$ are of equal length.

Rhombus $ABCD$ has side length $ 1$. The size of $\angle A$ (in degrees) is randomly selected from all real numbers between $0$ and $90$. Find the expected value of the area of $ABCD$.

For every four-digit number, we take the product of its four digits. Then we add all of these products together . What is the result? ( G Galperin)

A right rectangular prism is inscribed within a sphere. The total area of all the faces [of] the prism is $88$, and the total length of all its edges is $48$. What is the surface area of the sphere? $\text{(A) }40\pi\qquad\text{(B) }32\pi\sqrt{2}\qquad\text{(C) }48\pi\qquad\text{(D) }32\pi\sqrt{3}\qquad\text{(E) }56\pi$

There are $n \geq 2$ coins numbered from $1$ to $n$. These coins are placed around a circle, not necesarily in order. In each turn, if we are on the coin numbered $i$, we will jump to the one $i$ places from it, always in a clockwise order, beginning with coin number 1. For an example, see the figure below. Find all values of $n$ for which there exists an arrangement of the coins in which every coin will be visited.

A pair of positive integers $a$ and $b$ is such that their greatest common divisor is $5$ and their least common multiple is $55$. Find the smallest possible value of $a + b$.

Rodrigo has a very large sheet of graph paper. First he draws a line segment connecting point $(0,4)$ to point $(2,0)$ and colors the $4$ cells whose interiors intersect the segment, as shown below. Next Rodrigo draws a line segment connecting point $(2000,3000)$ to point $(5000,8000)$. How many cells will he color this time? [asy] filldraw((0,4)--(1,4)--(1,3)--(0,3)--cycle, gray(.75), gray(.5)+linewidth(1)); filldraw((0,3)--(1,3)--(1,2)--(0,2)--cycle, gray(.75), gray(.5)+linewidth(1)); filldraw((1,2)--(2,2)--(2,1)--(1,1)--cycle, gray(.75), gray(.5)+linewidth(1)); filldraw((1,1)--(2,1)--(2,0)--(1,0)--cycle, gray(.75), gray(.5)+linewidth(1)); draw((-1,5)--(-1,-1),gray(.9)); draw((0,5)--(0,-1),gray(.9)); draw((1,5)--(1,-1),gray(.9)); draw((2,5)--(2,-1),gray(.9)); draw((3,5)--(3,-1),gray(.9)); draw((4,5)--(4,-1),gray(.9)); draw((5,5)--(5,-1),gray(.9)); draw((-1,5)--(5, 5),gray(.9)); draw((-1,4)--(5,4),gray(.9)); draw((-1,3)--(5,3),gray(.9)); draw((-1,2)--(5,2),gray(.9)); draw((-1,1)--(5,1),gray(.9)); draw((-1,0)--(5,0),gray(.9)); draw((-1,-1)--(5,-1),gray(.9)); dot((0,4)); label("$(0,4)$",(0,4),NW); dot((2,0)); label("$(2,0)$",(2,0),SE); draw((0,4)--(2,0)); draw((-1,0) -- (5,0), arrow=Arrow); draw((0,-1) -- (0,5), arrow=Arrow); [/asy] $\textbf{(A) }6000\qquad\textbf{(B) }6500\qquad\textbf{(C) }7000\qquad\textbf{(D) }7500\qquad\textbf{(E) }8000$

If Bobby’s age is increased by $6$, it’s a number with an integral (positive) square root. If his age is decreased by $6$, it’s that square root. How old is Bobby?

Given $n \in\mathbb{N}$, find all continuous functions $f : \mathbb{R}\to \mathbb{R}$ such that for all $x\in\mathbb{R},$ \[\sum_{k=0}^{n}\binom{n}{k}f(x^{2^{k}})=0. \]

Consider the set $A$ containing all positive integers whose decimal expansion contains no $0$, and whose sum $S(N)$ of the digits divides $N$. (a) Prove that there exist infinitely many elements in $A$ whose decimal expansion contains each digit the same number of times as each other digit. (b) Explain that for each positive integer $k$ there exist an element in $A$ having exactly $k$ digits.

A digital calendar displays the date: day, month, and year, with $2$ digits for the day, $2$ digits for the month, and $2$ digits for the year. For example, $01-01-01$ is January $1$, $2001$ and $05-25-23$ is May $25$, $2023$. In front of the calendar is a mirror. The digits of the calendar are as in the figure [img]https://cdn.artofproblemsolving.com/attachments/c/5/a08a4e34071fff4d33b95b23690254f55b33e1.gif[/img] If $0, 1, 2, 5$, and $8$ are reflected, respectively, in $0, 1, 5, 2$, and $8$, and the other digits lose meaning when reflected, determine how many days of the century, when reflected in the mirror, also correspond to a date.

Show that for any function $ f: (0,\plus{}\infty)\to (0,\plus{}\infty)$ there exist real numbers $ x>0$ and $ y>0$ such that: $ f(x\plus{}y)<yf(f(x)).$ [b]Dan Schwarz[/b]

Milos arranged the numbers $1$ through $49$ into the cells of a $7\times7$ board. Djordje wants to guess the arrangement of the numbers. He can choose a square covering some cells of the board and ask Milos which numbers are found inside that square. At least, how many questions does Djordje need so as to be able to guess the arrangement of the numbers?

In a field, $2023$ friends are standing in such a way that all distances between them are distinct. Each of them fires a water pistol at the friend that stands closest. Prove that at least one person does not get wet.

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

Let $n$ be a posititve integer and let $M$ the set of all all integer cordinates $(a,b,c)$ such that $0 \le a,b,c \le n$. A frog needs to go from the point $(0,0,0)$ to the point $(n,n,n)$ with the following rules: $\cdot$ The frog can jump only in points of $M$ $\cdot$ The frog can't jump more than $1$ time over the same point. $\cdot$ In each jump the frog can go from $(x,y,z)$ to $(x+1,y,z)$, $(x,y+1,z)$, $(x,y,z+1)$ or $(x,y,z-1)$ In how many ways the Frog can make his target?

For any given real $a, b, c$ solve the following system of equations: $$\left\{\begin{array}{l}ax^3+by=cz^5,\\az^3+bx=cy^5,\\ay^3+bz=cx^5.\end{array}\right.$$ [i]Proposed by Oleksiy Masalitin, Bogdan Rublov[/i]

Alka finds a number $n$ written on a board that ends in $5.$ She performs a sequence of operations with the number on the board. At each step, she decides to carry out one of the following two operations: $1.$ Erase the written number $m$ and write it´s cube $m^3$. $2.$ Erase the written number $m$ and write the product $2023m$. Alka performs each operation an even number of times in some order and at least once, she finally obtains the number $r$. If the tens digit of $r$ is an odd number, find all possible values that the tens digit of $n^3$ could have had.

Find all the natural numbers $a,b,c$ such that: 1) $a^2+1$ and $b^2+1$ are primes 2) $(a^2+1)(b^2+1)=(c^2+1)$