Consider any sequence of real numbers $a_0$, $a_1$, $\cdots$. If, for all pairs of nonnegative integers $(m, s)$, there exists some integer $n \in [m+1, m+2024(s+1)]$ satisfying $a_m+a_{m+1}+\cdots+a_{m+s}=a_n+a_{n+1}+\cdots+a_{n+s}$, say that this sequence has [i]repeating sums[/i]. Is a sequence with repeating sums always eventually periodic?

Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $\frac {1} {2}$. Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.

Let $\Omega$ be the circumcircle of cyclic quadrilateral $ABCD$. Consider such pairs of points $P$, $Q$ of diagonal $AC$ that the rays $BP$ and $BQ$ are symmetric with respect the bisector of angle $B$. Find the locus of circumcenters of triangles $PDQ$.

Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1) \] is a divisor of $abc-1.$

A sequence of seven digits is randomly chosen in a weekly lottery. Every digit can be any of the digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9.$ What is the probability of having at most five different digits in the sequence?

Find integer solutions of $x^3+y^3-2xy+x+y+2=0$

Consider $\vartriangle ABC$ such that $CA + AB = 3BC$. Let the incircle $\omega$ touch segments $\overline{CA}$ and $\overline{AB}$ at $E$ and $F$, respectively, and define $P$ and $Q$ such that segments $\overline{P E}$ and $\overline{QF}$ are diameters of $\omega$. Define the function $D$ of a point $K$ to be the sum of the distances from $K$ to $P$ and $Q$ (i.e. $D(K) = KP + KQ$). Let $W, X, Y$ , and $Z$ be points chosen on lines $\overleftrightarrow {BC}$, $\overleftrightarrow {CE}$, $\overleftrightarrow {EF}$, and $\overleftrightarrow {F B}$, respectively. Given that $BC =\sqrt{133}$ and the inradius of $\vartriangle ABC$ is $\sqrt{14}$, compute the minimum value of $D(W) + D(X) + D(Y ) + D(Z)$.

Given the rays $ OP$ and $OQ$.Inside the smaller angle $POQ$ selected points $M$ and $N$, such that $\angle POM=\angle QON $ and $\angle POM<\angle PON $ The circle, which concern the rays $OP$ and $ON$, intersects the second circle, which concern the rays $OM$ and $OQ$ at the points $B$ and $C$. Prove that$\angle POC=\angle QOB $

A sequence of integers fsng is defined as follows: fix integers $a$, $b$, $c$, and $d$, then set $s_1 = a$, $s_2 = b$, and $$s_n = cs_{n-1} + ds_{n-2}$$ for all $n \ge 3$. Create a second sequence $\{t_n\}$ by defining each $t_n$ to be the remainder when $s_n$ is divided by $2018$ (so we always have $0 \le t_n \le 2017$). Let $N = (2018^2)!$. Prove that $t_N = t_{2N}$ regardless of the choices of $a$, $b$, $c$, and $d$.

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.