Let $n\ge5$ be an integer. A trapezoid with base lengths of $1$ and $r$ is tiled by $n$ (not necessarily congruent) equilateral triangles. In terms of $n$, find the maximum possible value of $r$. [i]Linus Tang[/i]

Charlotte the cat is placed at the origin of the coordinate plane, such that the positive $x$ direction is pointing east, and the positive $y$ direction is pointing north. Then, every single lattice square (a unit square with vertices all on lattice points) in the first quadrant whose southwest vertex is a lattice point with both odd coordinates is completely removed. Charlotte can traverse the coordinate plane by drawing a segment between two valid vertices, given that they do not intersect a lattice square that has not been removed. Let $P_1$ and $P_2$ denote the distances of the first and second shortest paths Charlotte can take to $(5,7),$ respectively. Find $P_1-P_2.$ [i]Tiebreaker #4[/i]

Prove that among any $18$ consecutive positive integers not exceeding $2005$ there is at least one divisible by the sum of its digits.

Let $a,b,c,d$ be positive real numbers with $a+b+c+d=2$. Prove the following inequality: $$\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right).$$ [i]Proposed by Mohammad Jafari[/i]

A regular tetrahedron of height $h$ has a tetrahedron of height $xh$ cut off by a plane parallel to the base. When the remaining frustrum is placed on one of its slant faces on a horizontal plane, it is just on the point of falling over. (In other words, when the remaining frustrum is placed on one of its slant faces on a horizontal plane, the projection of the center of gravity G of the frustrum is a point of the minor base of this slant face.) Show that $x$ is a root of the equation $x^3 + x^2 + x = 2$.

The numbers $1, 2, \dots ,N$ are arranged in a circle where $N \geq  2$. If each number shares a common digit with each of its neighbours in decimal representation, what is the least possible value of $N$? $ \textbf{a)}\ 18 \qquad\textbf{b)}\ 19 \qquad\textbf{c)}\ 28 \qquad\textbf{d)}\ 29 \qquad\textbf{e)}\ \text{None of above} $

Sketch the graph of the derivative and the graph of the integral of a function given by a free-hand graph.

A dot-trapezium consists of several rows of dots such that each row contains one more dot than the row immediately above (apart from the top row). For example here is a dot-trapezium consisting of $15$ dots, having $3$ rows and $4$ dots in the top row. [asy] //wonderfully scuffed asymptote code , please don't laugh at me. constructed from the diagram at https://www.mathsolympiad.org.nz/competitions/nzmo/problems/nzmo1_2024.pdf //top row dot((.05,.1)); dot((-.05,.1)); dot((-.15,.1)); dot((.15,.1)); //middle row dot((0,0)); dot((.1,0)); dot((-.1,0)); dot((.2,0)); dot((-.2,0)); //bottom row dot((.05,-.1)); dot((-.05,-.1)); dot((-.15,-.1)); dot((.15,-.1)); dot((.25,-.1)); dot((-.25,-.1)); [/asy] A positive integer $n$ is called a trapezium-number if there exists a dot-trapezium consisting of exactly $n$ dots, with at least two rows and at least two dots in the top row. How many trapezium-numbers are there less than $100$?

Let $\triangle ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. The incircle touches sides $BC,CA,$ and $AB$ at $D,E,$ and $F$ respectively, and $A'$ is the reflection of $A$ over $O$. The circumcircles of $ABC$ and $A'EF$ meet at $G$, and the circumcircles of $AMG$ and $A'EF$ meet at a point $H\neq G$, where $M$ is the midpoint of $EF$. Prove that if $GH$ and $EF$ meet at $T$, then $DT\perp EF$. [i]Proposed by Ankit Bisain[/i]

Let $ P$ be a polyhedron whose all edges are congruent and tangent to a sphere. Suppose that one of the facesof $ P$ has an odd number of sides. Prove that all vertices of $ P$ lie on a single sphere.

The vertices of a convex $1991$-gon are enumerated with integers from $1$ to $1991$. Each side and diagonal of the $1991$-gon is colored either red or blue. Prove that, for an arbitrary renumeration of vertices, one can find integers $k$ and $l$ such that the segment connecting the vertices numbered $k$ and $l$ before the renumeration has the same color as the segment connecting the vertices numbered $k$ and $l$ after the renumeration.

Let $A=(a_{ij})$ be a real $n\times n$ matrix such that $A^n\ne0$ and $a_{ij}a_{ji}\le0$ for all $i,j$. Prove that there exist two nonreal numbers among eigenvalues of $A$.

Find all non-negative real numbers $a$ such that the equation \[ \sqrt[3]{1+x}+\sqrt[3]{1-x}=a \] has at least one real solution $x$ with $0 \leq x \leq 1$. For all such $a$, what is $x$?

Let $p\in \mathbb P,p>3$. Calcute: a)$S=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{2k^2}{p}\right]-2 \cdot \left[\frac{k^2}{p}\right]$ if $ p\equiv 1 \mod 4$ b) $T=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{k^2}{p}\right]$ if $p\equiv 1 \mod 8$

An arrangement of 12 real numbers in a row is called [i]good[/i] if for any four consecutive numbers the arithmetic mean of the first and last numbers is equal to the product of the two middle numbers. How many good arrangements are there in which the first and last numbers are 1, and the second number is the same as the third?

There is a frog in every vertex of a regular 2n-gon with circumcircle($n \geq 2$). At certain time, all frogs jump to the neighborhood vertices simultaneously (There can be more than one frog in one vertex). We call it as $\textsl{a way of jump}$. It turns out that there is $\textsl{a way of jump}$ with respect to 2n-gon, such that the line connecting any two distinct vertice having frogs on it after the jump, does not pass through the circumcentre of the 2n-gon. Find all possible values of $n$.

Set $A$ consists of $7$ consecutive positive integers less than $2011$, while set $B$ consists of $11$ consecutive positive integers. If the sum of the numbers in $A$ is equal to the sum of the numbers in $B$ , what is the maximum possible element that $A$ could contain?

Prove that $x^3+y^3+z^3+t^3=1999$ has infinitely many soln. over $\mathbb{Z}$.

Let $ABC$ be a triangle with $E$ being the centre of its Euler circle. Through $E$, construct the lines $PS, MQ, NR$ parallel to $BC,CA,AB$ ($R,Q$ are on the line $BC, N, P$ on the line $AC,M, S$ on the line $AB$). Prove that the four Euler lines of triangles $ABC,AMN,BSR,CPQ$ are concurrent. Nguyễn Văn Linh

Let $x$, $y$ be two positive integers such that $\displaystyle 3x^2+x=4y^2+y$. Prove that $x-y$ is a perfect square.

A natural number $n$ is called [i]perfect [/i] if it is equal to the sum of all its natural divisors other than $n$. For example, the number $6$ is perfect because $6 = 1 + 2 + 3$. Find all even perfect numbers that can be given as the sum of two cubes positive integers.

What is the last two digits of the decimal representation of $9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}$? $ \textbf{(A)}\ 81 \qquad\textbf{(B)}\ 61 \qquad\textbf{(C)}\ 41 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 01 $

Zach rolls five tetrahedral dice, each of whose faces are labeled $1, 2, 3$, and $4$. Compute the probability that the sum of the values of the faces that the dice land on is divisible by $3$.

Higher Secondary P3 Let $ABCDEF$ be a regular hexagon with $AB=7$. $M$ is the midpoint of $DE$. $AC$ and $BF$ intersect at $P$, $AC$ and $BM$ intersect at $Q$, $AM$ and $BF$ intersect at $R$. Find the value of $[APB]+[BQC]+[ARF]-[PQMR]$. Here $[X]$ denotes the area of polygon $X$.

Prove that if $ a > 0 $, $ b > 0 $, $ c > 0 $, then $$ \frac{a}{b + c} + \frac{b}{c+ a} + \frac{c}{a+b} \geq \frac{3}{2}.$$