Prove that there are infinitely many positive integers $k$ such that $k(k+1)(k+2)(k+3)$ has no prime divisor of the form $8t+5.$

Let $n$ be a positive integer. Each cell on an $n \times n$ board will be filled with a positive integer less than or equal to $2n-1$ such that for each index $i$ with $1 \leq i \leq n$, the $2n-1$ cells in the $i^{\text{th}}$ row or $i^{\text{th}}$ collumn contain distinct integers. (a) Is this filling possible for $n=4$? (b) Is this filling possible for $n=5$?

Liam writes the number $0$ on a board, then performs a series of turns. Each turn, he chooses a nonzero integer so that for every nonzero integer $N,$ he chooses $N$ with $3^{- |N|}$ probability. He adds his chosen integer $N$ to the last number written on the board, yielding a new number. He writes the new number on the board and uses it for the next turn. Liam repeats the process until either $8$ or $9$ is written on the board, at which point he stops. Given that Liam eventually stopped, find the probability the last number he wrote on the board was $9.$

The difference of fractions $\frac{2024}{2023} - \frac{2023}{2024}$ was represented as an irreducible fraction $\frac{p}{q}$. Find the value of $p$.

Let $ f(x) \equal{} x^8 \plus{} 4x^6 \plus{} 2x^4 \plus{} 28x^2 \plus{} 1.$ Let $ p > 3$ be a prime and suppose there exists an integer $ z$ such that $ p$ divides $ f(z).$ Prove that there exist integers $ z_1, z_2, \ldots, z_8$ such that if \[ g(x) \equal{} (x \minus{} z_1)(x \minus{} z_2) \cdot \ldots \cdot (x \minus{} z_8),\] then all coefficients of $ f(x) \minus{} g(x)$ are divisible by $ p.$

Let $ 0 \leq x \leq 2\pi$. Prove the inequality $ \sqrt {\frac {\sin^{2}x}{1 + \cos^{2}x}} + \sqrt {\frac {\cos^{2}x}{1 + \sin^{2}x}}\geq 1 $

A subsum of $n$ real numbers $a_1,\dots,a_n$ is a sum of elements of a subset of the set $\{a_1,\dots,a_n\}$. In other words a subsum is $\epsilon_1a_1+\dots+\epsilon_na_n$ in which for each $1\leq i \leq n$ ,$\epsilon_i$ is either $0$ or $1$. Years ago, there was a valuable list containing $n$ real not necessarily distinct numbers and their $2^n-1$ subsums. Some mysterious creatures from planet Tarator has stolen the list, but we still have the subsums. (a) Prove that we can recover the numbers uniquely if all of the subsums are positive. (b) Prove that we can recover the numbers uniquely if all of the subsums are non-zero. (c) Prove that there's an example of the subsums for $n=1392$ such that we can not recover the numbers uniquely. Note: If a subsum is sum of element of two different subsets, it appears twice. Time allowed for this question was 75 minutes.

Let $ABCD$ be a circumscribed quadrilateral, assume $ABCD$ is not a kite. Denote the circumcenters of triangle $ABC,BCD,CDA,DAB$ by $O_D,O_A,O_B,O_C$ respectively. a. Prove that $O_AO_BO_CO_D$ is circumscribed. b. Let the angle bisector of $\angle BAD$ intersect the angle bisector of $\angle O_BO_AO_D$ in $X$. Similarly define the points $Y,Z,W$. Denote the incenters of $ABCD, O_AO_BO_CO_D$ by $I,J$ respectively. Express the angles $\angle ZYJ,\angle XYI$ in terms of angles of quadrilateral $ABCD$.

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [b]D1.[/b] A Giant Hopper is $200$ meters away from you. It can hop $50$ meters. How many hops would it take for it to reach you? [b]D2.[/b] A rope of length $6$ is used to form the edges of an equilateral triangle (a triangle with equal side lengths). What is the length of one of these edges? [b]D3 / Z1.[/b] Point $E$ is on side $AB$ of rectangle $ABCD$. Find the area of triangle $ECD$ divided by the area of rectangle $ABCD$. [b]D4 / Z2.[/b] Garb and Grunt have two rectangular pastures of area $30$. Garb notices that his has a side length of $3$, while Grunt’s has a side length of $5$. What’s the positive difference between the perimeters of their pastures? [b]D5.[/b] Let points $A$ and $B$ be on a circle with radius $6$ and center $O$. If $\angle AOB = 90^o$, find the area of triangle $AOB$. [b]D6 / Z3.[/b] A scalene triangle (the $3$ side lengths are all different) has integer angle measures (in degrees). What is the largest possible difference between two angles in the triangle? [b]D7.[/b] Square $ABCD$ has side length $6$. If triangle $ABE$ has area $9$, find the sum of all possible values of the distance from $E$ to line $CD$. [b]D8 / Z4.[/b] Let point $E$ be on side $\overline{AB}$ of square $ABCD$ with side length $2$. Given $DE = BC+BE$, find $BE$. [b]Z5.[/b] The two diagonals of rectangle $ABCD$ meet at point $E$. If $\angle AEB = 2\angle BEC$, and $BC = 1$, find the area of rectangle $ABCD$. [b]Z6.[/b] In $\vartriangle ABC$, let $D$ be the foot of the altitude from $A$ to $BC$. Additionally, let $X$ be the intersection of the angle bisector of $\angle ACB$ and $AD$. If $BD = AC = 2AX = 6$, find the area of $ABC$. [b]Z7.[/b] Let $\vartriangle ABC$ have $\angle ABC = 40^o$. Let $D$ and $E$ be on $\overline{AB}$ and $\overline{AC}$ respectively such that DE is parallel to $\overline{BC}$, and the circle passing through points $D$, $E$, and $C$ is tangent to $\overline{AB}$. If the center of the circle is $O$, find $\angle DOE$. [b]Z8.[/b] Consider $\vartriangle ABC$ with $AB = 3$, $BC = 4$, and $AC = 5$. Let $D$ be a point of $AC$ other than $A$ for which $BD = 3$, and $E$ be a point on $BC$ such that $\angle BDE = 90^o$. Find $EC$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In Survev.io, Calvin observes that he has exactly twice as much blue ammo as red ammo. After firing one blue bullet and $9$ red bullets, he remarks that the amount of blue ammo he has is divisible by $5$ and the amount of red ammo he has is divisible by $7$. Find the least amount of red ammo he could have started with.

Which of the two numbers is bigger : $\sqrt{1997}+2\sqrt{1999} + 2\sqrt{2001} + \sqrt{2003}$ or $2\sqrt{1998} +2\sqrt{2000}+2\sqrt{2002}$ ?

Let triangle $ABC$ have $\angle BAC = 45^{\circ}$ and circumcircle $\Gamma$ and let $M$ be the intersection of the angle bisector of $\angle BAC$ with $\Gamma$. Let $\Omega$ be the circle tangent to segments $\overline{AB}$ and $\overline{AC}$ and internally tangent to $\Gamma$ at point $T$. Given that $\angle TMA = 45^{\circ}$ and that $TM = \sqrt{100 - 50 \sqrt{2}}$, the length of $BC$ can be written as $a \sqrt{b}$, where $b$ is not divisible by the square of any prime. Find $a + b$.

Niki has $15$ dollars more than twice as much money as her sister Amy. If Niki gives Amy $30$ dollars, then Niki will have hals as much money as her sister. How many dollars does Niki have?

From a point $P$ outside of a circle with centre $O$, tangent segments $\overline{PA}$ and $\overline{PB}$ are drawn. If $\frac1{\left|\overline{OA}\right|^2}+\frac1{\left|\overline{PA}\right|^2}=\frac1{16}$, then $\left|\overline{AB}\right|=$? $\textbf{(A)}~4$ $\textbf{(B)}~6$ $\textbf{(C)}~8$ $\textbf{(D)}~10$

From a square of side length $1$, four identical triangles are removed, one at each corner, leaving a regular octagon. What is the area of the octagon?

In a solid sphere of radius $R$ the density $\rho$ is a function of $r$, the distance from the center of the sphere. If the magnitude of the gravitational force of attraction due to the sphere at any point inside the sphere is $k r^2$, where $k$ is a constant, find $\rho$ as a function of $r.$ Find also the magnitude of the force of attraction at a point outside the sphere at a distance $r$ from the center.

After years at sail, you and your crew have found the island that houses the great treasure of Scottybeard, the greatest pirate to ever sail the high seas. The island takes the shape of a unit square, and the treasure (which we treat as a single point) could be buried under any point on the island. To assist you in finding his treasure, Scotty has left a peculiar instrument. To use this instrument, you may draw any directed line (possibly one that never hits the island!), and the instrument will tell you whether the treasure lies to the "left" or the "right" of the line.* [asy] draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((1,0.5)--(0.8,0.42),arrow=Arrow()); draw((0.8,0.42)--(0.6,0.34),arrow=Arrow()); draw((0.6,0.34)--(0.4,0.26),arrow=Arrow()); draw((0.4,0.26)--(0.2,0.18),arrow=Arrow()); draw((0.2,0.18)--(0,0.1)); label("``Right''", (0.5,0.55)); label("``Left''", (0.8,0.2)); [/asy] However, Scotty also left a trap! If the instrument ever reports ``left'' three times in a row or ``right'' three times in a row, the island will suddenly sink into the sea, submerging the treasure forever and drowning you and your crew! You want to avoid this at all costs. To minimize the amount of energy spent digging, you would like to narrow down the set of possible locations of the treasure to be as small as possible. However, Scotty left one last trick; you can only use the instrument 12 times before it breaks! Devise an algorithm to use the instrument no more than 12 times that can never result in the island sinking and narrows the worst-case space of possible locations of the treasure to have as small an area as possible. * [size=75]Where "left" or "right" is taken with respect to an observer walking along the line in its designated direction. There is also a probability zero chance the treasure is precisely on the line; this won't affect anything, but for the sake of clarity let's say the instrument reports "left" in this case.[/size] [b]Scoring:[/b] An algorithm that achieves a worst-case area of $K$ will be awarded: [list] [*] 1 point for any $K<1$ [*] 10 points for $K=\tfrac 14$ [*] 20 points for $\tfrac 1{128}<K<\tfrac 14$ [*] 30 points for $K=\tfrac 1{128}$ [*] 50 points for $K_{\text{min}}<K<\tfrac 1 {128}$ [*] 75 points for $K=K_{\text{min}}$ [*] 100 points for $K=K_{\text{min}}$, with a proof that this is optimal [/list] (where $K_{\text{min}}$ is the smallest possible worst-case area, which we are not disclosing to avoid giving anything away) [i]Proposed by Connor Gordon[/i]

There are $32$ students in the class with $10$ interesting group. Each group contains exactly $16$ students. For each couple of students, the square of the number of the groups which are only involved by just one of the two students is defined as their $interests-disparity$. Define $S$ as the sum of the $interests-disparity$ of all the couples, $\binom{32}{2}\left ( =\: 496 \right )$ ones in total. Determine the minimal possible value of $S$.

Find all the triples of integers $ (a, b,c)$ such that: \[ \begin{array}{ccc}a\plus{}b\plus{}c &\equal{}& 24\\ a^{2}\plus{}b^{2}\plus{}c^{2}&\equal{}& 210\\ abc &\equal{}& 440\end{array}\]

Outside the parallelogram $ABCD$ on its sides $AB$ and $BC$ are constructed equilateral triangles $ABK$, and $BCM$. Prove that the triangle $KMD$ is equilateral.

Show that if all lateral edges of a pentagonal pyramid are of equal length and all the angles between neighboring lateral faces are equal, then the pyramid is regular.

Cris has the equation $-2x^2 + bx + c = 0$, and Cristian increases the coefficients of the Cris equation by $1$, obtaining the equation $-x^2 + (b + 1) x + (c + 1) = 0$. Mariloli notices that the real solutions of the Cristian's equation are the squares of the real solutions of the Cris equation. Find all possible values that can take the coefficients $b$ and $c$.

Let $n\in\mathbb{Z}$, $n\geq 2$. Find all functions $f:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$ such that $$f(x_1+\dots +x_n)^2=\sum_{i=1}^nf(x_i) ^2+ 2\sum_{i<j}f(x_ix_j),$$ for all $x_1,\dots ,x_n\in\mathbb{R}_{>0}$. [i]Proposed by Andrei Vila[/i]

Let $\alpha < 0 < \beta$ and consider the polynomial $f(x) = x(x-\alpha)(x-\beta)$. Let $S$ be the set of real numbers $s$ such that $f(x) - s$ has three different real roots. For $s\in S$, let $p(x)$ the product of the smallest and largest root of $f(x)-s$. Determine the smallest possible value that $p(s)$ for $s\in S$.

Find all triples $(x,y,z)$ of positive real numbers such that $$ \begin{cases} 2x^2+y^3=3 \\ 3y^2+z^3=4 \\ 4z^2+x^3=5 \\ \end{cases} $$ [i]M. Zorka[/i]