Odysseus on his ten year return voyage to Ithaca sailed between two monsters. On one side, the creature Charybdis periodically sucked the oceans such that a whirlpool formed. On the opposing side the creature Scylla would lunge down from above and devour one sailor in each of her many mouths. Odysseus opted to sail near Scylla skirting Charybdis by 500 m. At this distance, the maximum drop in water level of the ocean was 0.2 m from between when Charybdis was draining the oceans and when he was not. At Charybdis' mouth the funnel of the whirlpool is 25 m wide. Assume that the oceans are perfectly calm and that there are no intermolecular attractions between water molecules. (a) How deep is Charybdis under water? (b) The boat with a crew of 40 men weighs 5,000 kg. Each crew member displaces 3 kg of water at a velocity of 5 m/s every stroke every second. If at some point Odysseus is traveling radially away from Charybdis, what is the closest his ship can be without being sucked in? Assume that Odysseus' vessel has an extremely shallow draft (low friction). [i]Problem proposed by Brian Yue[/i]

Positive integers $x_1,x_2,...,x_n$ ($n \in N_+$) satisfy $x_1^2 +x_2^2+...+x_n^2=111$, find the maximum possible value of $S =\frac{x_1 +x_2+...+x_n}{n}$.

Given real numbers $x$ and $y$, such that $$x^4 y^2 + y^4 + 2 x^3 y + 6 x^2 y + x^2 + 8 \le 0 .$$ Prove that $x \ge - \frac16$

A fair $20$-sided die has faces numbered $1$ through $20$. The die is rolled three times and the outcomes are recorded. If $a$ and $b$ are relatively prime integers such that $a/b$ is the probability that the three recorded outcomes can be the sides of a triangle with positive area, find $a+b$.

The road infrastructure in a country consists of an even number of direct roads, each of which is bidirectional. Moreover, for any two cities $X$ and $Y$, there is at most one direct road between the two of them and there exists a sequence $X = X_0, X_1, ..., X_{n - 1}, X_n = Y$ of cities such that for any $i = 0, ..., n - 1$, there exists a direct road between $X_i$ and $X_{i + 1}$. Prove that all direct roads in this country can be oriented (i.e. each road can become a one-way road) such that each city $X$ is the starting point for an even number of direct roads. Proposed by [i]Mirko Petrushevski[/i]

Given an interger $n\geq 2$, determine the maximum value the sum $\frac{a_1}{a_2}+\frac{a_2}{a_3}+...+\frac{a_{n-1}}{a_n}$ may achieve, and the points at which the maximum is achieved, as $a_1,a_2,...a_n$ run over all positive real numers subject to $a_k\geq a_1+a_2...+a_{k-1}$, for $k=2,...n$

Let $x,y$ be cartesian coordinates in the plane. $I$ denotes the line segment $1\leq x\leq 3 , y=1.$ For every point $P$ on $I$, let $P'$ denote the point that lies on the segment joining the origin to $P$ and such that the distance $P P'$ is equal to $1 \slash 100.$ As $P$ describes $I$, the point $P'$ describes a curve $C$. Which of $I$ and $C$ has greater length?

Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $ m$ and $ n$ be relatively prime positive integers such that $ \frac{m}{n}$ is the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. Find $ m\plus{}n$.

Let the sequence $\{a_n\}_{n \geq 1}$ be defined by \[ a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N} \] Prove that \[ a_n^{2025} >n^{2024} \] for all positive integers $n \geq 2$. $\textbf{Proposed by Prajit Adhikari, Nepal.}$

Let $ ABC $ a right triangle in $ A. $ Let $ D $ a point on the segment $ AC, $ and $ E,F $ the projections of $ A $ upon the lines $ BD, $ respectively, $ BC. $ Show that the quadrilateral $ CDEF $ is concyclic.

$M$ is an interior point of a rectangle $ABCD$ and $S$ is its area. Prove that $S \le AM \cdot CM + BM \cdot DM$. (I.J . Goldsheyd)

Let $n$ be a positive integer. The numbers $\{1, 2, ..., n^2\}$ are placed in an $n \times n$ grid, each exactly once. The grid is said to be [i]Muirhead-able[/i] if the sum of the entries in each column is the same, but for every $1 \le i,k \le n-1$, the sum of the first $k$ entries in column $i$ is at least the sum of the first $k$ entries in column $i+1$. For which $n$ can one construct a Muirhead-able array such that the entries in each column are decreasing? [i]Proposed by Evan Chen[/i]

In the Rainbow Nation there are two airways: Red Rockets and Blue Boeings. For any two cities in the Rainbow Nation it is possible to travel from the one to the other using either or both of the airways. It is known, however, that it is impossible to travel from Beanville to Mieliestad using only Red Rockets - not directly nor by travelling via other cities. Show that, using only Blue Boeings, one can travel from any city to any other city by stopping at at most one city along the way.

A triangle $XY Z$ and a circle $\omega$ of radius $2$ are given in a plane, such that $\omega$ intersects segment $\overline{XY}$ at the points $A$, $B$, segment $\overline{Y Z}$ at the points $C$, $D$, and segment $\overline{ZX}$ at the points $E$, $F$. Suppose that $XB > XA$, $Y D > Y C$, and $ZF > ZE$. In addition, $XA = 1$, $Y C = 2$, $ZE = 3$, and $AB = CD = EF$. Compute $AB$.

Let the sequence $a_1,a_2,\ldots,a_n,\ldots$ is defined by the conditions: $a_1=2$ and $a_{n+1}=a_n^2-a_n+1$ $(n=1,2,\ldots)$. Prove that: (a) $a_m$ and $a_n$ are relatively prime numbers when $m\ne n$. (b) $\lim_{n\to\infty}\sum_{k=1}^n\frac1{a_k}=1$ [i]I. Tonov[/i]

A circle $k$ with center $O$ and radius $r$ and a line $p$ which has no common points with $k$, are given. Let $E$ be the foot of the perpendicular from $O$ to $p$. Let $M$ be an arbitrary point on $p$, distinct from $E$. The tangents from the point $M$ to the circle $k$ are $MA$ and $MB$. If $H$ is the intersection of $AB$ and $OE$, then prove that $OH=\frac{r^2}{OE}$.

Let $O$ be the circumcenter and $H$ the orthocenter of an acute-angled triangle $ABC$ such that $BC>CA$. Let $F$ be the foot of the altitude $CH$ of triangle $ABC$. The perpendicular to the line $OF$ at the point $F$ intersects the line $AC$ at $P$. Prove that $\measuredangle FHP=\measuredangle BAC$.

Circles $\omega_1$ and $\omega_2$ have centres $O_1$ and $O_2$, respectively. These two circles intersect at points $X$ and $Y$. $AB$ is common tangent line of these two circles such that $A$ lies on $\omega_1$ and $B$ lies on $\omega_2$. Let tangents to $\omega_1$ and $\omega_2$ at $X$ intersect $O_1O_2$ at points $K$ and $L$, respectively. Suppose that line $BL$ intersects $\omega_2$ for the second time at $M$ and line $AK$ intersects $\omega_1$ for the second time at $N$. Prove that lines $AM, BN$ and $O_1O_2$ concur. [i]Proposed by Dominik Burek - Poland[/i]

Form an eight-digit number, using only the digits $1,2,3,4$ each twice, so that: there is one digit between the $1$'s, there are two digits between the $2$'s, there are three digits between the $3$'s, and there are four digits between the $4$'s.

Lev and Alex are racing on a number line. Alex is much faster, so he goes to sleep until Lev reaches $100$. Lev runs at $5$ integers per minute and Alex runs at $7$ integers per minute (in the same direction). How many minutes from the START of the race will it take Alex to catch up to Lev (who is still running after Alex wakes up)?

One-third of the students who attend Grant School are taking Algebra. One-quarter of the students taking Algebra are also on the track team. There are $15$ students on the track team who take Algebra. Find the number of students who attend Grant School.

Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$. [i]Proposed by North Korea[/i]

Find integer $ n$ with $ 8001 < n < 8200$ such that $ 2^n \minus{} 1$ divides $ 2^{k(n \minus{} 1)! \plus{} k^n} \minus{} 1$ for all integers $ k > n$.

The sum of the angles $A$ and $C$ of a convex quadrilateral $ABCD$ is less than $180^{\circ} .$ Prove that \[AB \cdot CD + AD \cdot BC < AC(AB + AD).\]

$39$ students participated in a math competition. The exam consisted of $6$ problems and each problem was worth $1$ point for a correct solution and $0$ points for an incorrect solution. For any $3$ students, there is at most $1$ problem that was not solved by any of the three. Let $B$ be the sum of all of the scores of the $39$ students. Find the smallest possible value of $B$.