A square with side $10^6$, with a corner square with side $10^{-3}$ cut off, is partitioned into $10$ rectangles. Prove that at least one of these rectangles has the ratio of the greater side to the smaller one at least $9$.

Samantha is about to celebrate her sweet 16th birthday. To celebrate, she chooses a five-digit positive integer of the form SWEET, in which the two E's represent the same digit but otherwise the digits are distinct. (The leading digit S can't be 0.) How many such integers are divisible by 16?

For any prime number $p>2,$ and an integer $a$ and $b,$ if $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{(p-1)^3}=\frac{a}{b},$ prove that $a$ is divisible by $p.$

Quadrilateral $ABCD$ is cyclic with $AB = CD = 6$. Given that $AC = BD = 8$ and $AD+3 = BC$, the area of $ABCD$ can be written in the form $\frac{p\sqrt{q}}{r}$, where $p, q$, and $ r$ are positive integers such that $p$ and $ r$ are relatively prime and that $q$ is square-free. Compute $p + q + r$.

A rhombus has vertices at $(0,0)$, $(6, 8)$, $(16, 8)$, and $(10, 0)$. A line with slope $m$ passes through the point $(3, 1)$ and splits the rhombus into $2$ regions of equal area. Find $m$.

Consider $2$ positive integers $a,b$ such that $a+2b=2020$. (a) Determine the largest possible value of the greatest common divisor of $a$ and $b$. (b) Determine the smallest possible value of the least common multiple of $a$ and $b$.

Let $A_1, A_2, . . . , A_{2021}$ be $2021$ points on the plane, no three collinear and $$\angle A_1A_2A_3 + \angle A_2A_3A_4 +... + \angle A_{2021}A_1A_2 = 360^o,$$ in which by the angle $\angle A_{i-1}A_iA_{i+1}$ we mean the one which is less than $180^o$ (assume that $A_{2022} =A_1$ and $A_0 = A_{2021}$). Prove that some of these angles will add up to $90^o$. [i]Proposed by Morteza Saghafian - Iran[/i]

Mykhailo chose three distinct real numbers \( a, b, c \) and wrote the following numbers on the board: \[ a + b, \quad b + c, \quad c + a, \quad ab, \quad bc, \quad ca. \]What is the minimum possible number of distinct numbers that can be written on the board? [i]Proposed by Anton Trygub[/i]

Let $D$ be a dodecahedron which can be inscribed in a sphere with radius $R$. Let $I$ be an icosahedron which can also be inscribed in a sphere of radius $R$. Which has the greater volume, and why? Note: A regular [i]polyhedron [/i] is a geometric solid, all of whose faces are congruent regular polygons, in which the same number of polygons meet at each vertex. A regular dodecahedron is a polyhedron with $12$ faces which are regular pentagons and a regular icosahedron is a polyhedron with $20$ faces which are equilateral triangles. A polyhedron is inscribed in a sphere if all of its vertices lie on the surface of the sphere. The illustration below shows a dodecahdron and an icosahedron, not necessarily to scale. [img]https://cdn.artofproblemsolving.com/attachments/7/5/9873b42aacf04bb5daa0fe70d4da3bf0b7be38.png[/img]

Consider the sequences with 2016 terms formed by the digits 1, 2, 3, and 4. Find the number of those sequences containing an even number of the digit 1.

Let $P$ be a point on the circumcircle of acute triangle $ABC$. Let $D,E,F$ be the reflections of $P$ in the $A$-midline, $B$-midline, and $C$-midline. Let $\omega$ be the circumcircle of the triangle formed by the perpendicular bisectors of $AD, BE, CF$. Show that the circumcircles of $\triangle ADP, \triangle BEP, \triangle CFP,$ and $\omega$ share a common point.

On a circle, points $A,B,C,D$ lie counterclockwise in this order. Let the orthocenters of $ABC,BCD,CDA,DAB$ be $H,I,J,K$ respectively. Let $HI=2$, $IJ=3$, $JK=4$, $KH=5$. Find the value of $13(BD)^2$.

Let $ABC$ be a triangle with circumcircle $\omega$ and $\ell$ a line without common points with $\omega$. Denote by $P$ the foot of the perpendicular from the center of $\omega$ to $\ell$. The side-lines $BC,CA,AB$ intersect $\ell$ at the points $X,Y,Z$ different from $P$. Prove that the circumcircles of the triangles $AXP$, $BYP$ and $CZP$ have a common point different from $P$ or are mutually tangent at $P$. [i]Proposed by Cosmin Pohoata, Romania[/i]

Let $S$ be the set of all ordered pairs $(x,y)$ of integer solutions to the equation $$6x^2+y^2+6x=3xy+6y+x^2y.$$ $S$ contains a unique ordered pair $(a,b)$ with a maximal value of $b$. Compute $a+b$. Proposed by Kenan Hasanaliyev (claserken)

Ten students are seated around a circular table. The teacher has a list of fifteen problems and each student is given six problems, in such a way that each problem is given exactly four times and any two students they have at most three problems in common. Prove that no matter how the teacher distributes the problems, there will always be two students sitting next to each other who have at least one problem in common.

Do there exist $1000 000$ distinct positive integers such that the sum of any collection of these numbers is never an exact square?

In a quadrilateral $ABCD,$ $AB=BC=DA/\sqrt{2},$ and $\angle ABC$ is a right angle. The midpoint of $BC$ is $E,$ the orthogonal projection of $C$ on $AD$ is $F,$ and the orthogonal projection of $B$ on $CD$ is $G.$ The second intersection point of circle $(BCF)$ (with center $H$) and line $BG$ is $K,$ and the second intersection point of circle $(BCF)$ and line $HK$ is $L.$ The intersection of lines $BL$ and $CF$ is $M.$ The center of the Feurbach circle of triangle $BFM$ is $N.$ Prove that $\angle BNE$ is a right angle. [i]Proposed by Zsombor Fehér, Cambridge[/i]

Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram. Prove that $GR=GS$. [i]Proposed by Hossein Karke Abadi, Iran[/i]

Each cell in a \( 2024 \times 2024 \) table contains the letter \( A \) or \( B \), with the number of \( A \)'s in each row being the same and the number of \( B \)'s in each column being the same. Alexandra and Boris play the following game, alternating turns, with Alexandra going first. On each turn, the player chooses a row or column and erases all the letters in it that have not yet been erased, as long as at least one letter is erased during the turn, and at the end of the turn, at least one letter remains in the table. The game ends when exactly one letter remains in the table. Alexandra wins the game if the letter is \( A \), and Boris wins if it is \( B \). What is the number of initial tables for which Alexandra has a winning strategy?

The four points of a circle are in the following order: $A, B, C, D$. Extensions of chord $AB$ beyond point $B$ and of chord $CD$ beyond point $C$ intersect at point $E$, with $\angle AED= 60^o$. If $\angle ABD =3 \angle BAC$ , prove that $AD$ is the diameter of the circle.

Let $ABCD$ be a rectangle with $AB=10$ and $BC=26$. Let $\omega_1$ be the circle with diameter $\overline{AB}$ and $\omega_2$ be the circle with diameter $\overline{CD}$. Suppose $\ell$ is a common internal tangent to $\omega_1$ and $\omega_2$ and that $\ell$ intersects $AD$ and $BC$ at $E$ and $F$ respectively. What is $EF$? [asy] size(10cm); draw((0,0)--(26,0)--(26,10)--(0,10)--cycle); draw((1,0)--(25,10)); draw(circle((0,5),5)); draw(circle((26,5),5)); dot((1,0)); dot((25,10)); label("$E$",(1,0),SE); label("$F$",(25,10),NW); label("$A$", (0,0), SW); label("$B$", (0,10), NW); label("$C$", (26,10), NE); label("$D$", (26,0), SE); dot((0,0)); dot((0,10)); dot((26,0)); dot((26,10)); [/asy] [i]Proposed by Nathan Xiong[/i]

A competition involving $n\ge 2$ players was held over $k$ days. In each day, the players received scores of $1,2,3,\ldots , n$ points with no players receiving the same score. At the end of the $k$ days, it was found that each player had exactly $26$ points in total. Determine all pairs $(n,k)$ for which this is possible.

Let $ABCD$ be a unit square. Points $E$ and $F$ are chosen on line segments $\overline{BC}$ and $\overline{CD}$, respectively, such that the area of $ABEF D$ is three times the area of triangle $\vartriangle ECF$. Compute the maximum possible area of triangle $\vartriangle AEF$.

Let $ ABCDE $ be a convex pentagon such that $ BC $ and $ DE $ are tangent to the circumcircle of $ ACD $. Prove that if the circumcircles of $ ABC $ and $ ADE $ intersect at the midpoint of $ CD $, then the circumcircles $ ABE $ and $ ACD $ are tangent to each other.

Let $a$ be a positive integer such that the number $a^n$ has an odd number of digits in the decimal representation for all $n > 0$. Prove that the number $a$ is an even power of $10$.