A group of $25$ friends were discussing a large positive integer. "It can be divided by $1$," said the first friend. "It can be divided by $2$," said the second friend. "And by $3$," said the third friend. "And by $4$," added the fourth friend. This continued until everyone had made such a comment. If exactly $2$ friends were incorrect, and those two friends said consecutive numbers, what was the least possible integer they were discussing?

Determine whether there exist an odd positive integer $n$ and $n\times n$ matrices $A$ and $B$ with integer entries, that satisfy the following conditions: [list=1] [*]$\det (B)=1$;[/*] [*]$AB=BA$;[/*] [*]$A^4+4A^2B^2+16B^4=2019I$.[/*] [/list] (Here $I$ denotes the $n\times n$ identity matrix.) [i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan[/i]

Compute the sum of the real solutions to $\lfloor x \rfloor \{x\} = 2020x$. Here, $\lfloor x \rfloor$ is defined as the greatest integer less than or equal to $x$, and$ \{x\} = x -\lfloor x \rfloor$.

In the right triangle $ABC$ ($m ( \angle A) = 90^o$) $D$ is the foot of the altitude from $A$. The bisectors of the angles $ABD$ and $ADB$ intersect in $I_1$ and the bisectors of the angles $ACD$ and $ADC$ in $I_2$. Find the angles of the triangle if the sum of distances from $I_1$ and $I_2$ to $AD$ is equal to $\frac14$ of the length of $BC$.

A cube $ABCDA_1B_1C_1D_1$ is given. Find the locus of points $E$ on the face $A_1B_1C_1D_1$ for which there exists a line intersecting the lines $AB$, $A_1D_1$, $B_1D$, and $EC$.

$\definecolor{A}{RGB}{190,0,60}\color{A}\fbox{A1.}$ Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $$\definecolor{A}{RGB}{80,0,200}\color{A} x^4+y^4+z^4\ge f(xy)+f(yz)+f(zx)\ge xyz(x+y+z)$$holds for all $a,b,c\in\mathbb{R}$. [i]Proposed by [/i][b][color=#FFFF00]usjl[/color][/b]. [color=#B6D7A8]#1733[/color]

For every $n$ non-negative integer let $S(n)$ denote a subset of the positive integers, for which $i$ is an element of $S(n)$ if and only if the $i$-th digit (from the right) in the base two representation of $n$ is a digit $1$. Two players, $A$ and $B$ play the following game: first, $A$ chooses a positive integer $k$, then $B$ chooses a positive integer $n$ for which $2^n\geqslant k$. Let $X$ denote the set of integers $\{ 0,1,\dotsc ,2^n-1\}$, let $Y$ denote the set of integers $\{ 0,1,\dotsc ,2^{n+1}-1\}$. The game consists of $k$ rounds, and in each round player $A$ chooses an element of set $X$ or $Y$, then player $B$ chooses an element from the other set. For $1\leqslant i\leqslant k$ let $x_i$ denote the element chosen from set $X$, let $y_i$ denote the element chosen from set $Y$. Player $B$ wins the game, if for every $1\leqslant i\leqslant k$ and $1\leqslant j\leqslant k$, $x_i<x_j$ if and only if $y_i<y_j$ and $S(x_i)\subset S(x_j)$ if and only if $S(y_i)\subset S(y_j)$. Which player has a winning strategy? [i]Proposed by Levente Bodnár, Cambridge[/i]

For a positive integer $n$, prove that $$\sum_{n \le p \le n^4} \frac{1}{p} < 4$$ where the sum is taken across primes $p$ in the range $[n, n^4]$ [i]N. Filonov[/i]

Based on a city's rules, the buildings of a street may not have more than $9$ stories. Moreover, if the number of stories of two buildings is the same, no matter how far they are from each other, there must be a building with a higher number of stories between them. What is the maximum number of buildings that can be built on one side of a street in this city?

Let $a,b,c&gt;0$ and $a+b+c+abc=4$. Prove that $$\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{c+a}}+\frac{c}{\sqrt{a+b} }\ge \frac{1}{\sqrt{2}}(a+b+c).$$ [i](Zhuge Liang)[/i]

$ABC$ is any triangle. Tangent at $C$ to circumcircle ($O$) of $ABC$ meets $AB$ at $M$. Line perpendicular to $OM$ at $M$ intersects $BC$ at $P$ and $AC$ at $Q$. P.T. $MP=MQ$.

A trapezoid has side lengths $24$, $25$, $26$, and $27$ in some order. Find its area.

$A$ is a set of integers which is closed under addition and contains both positive and negative numbers. Show that the difference of any two elements of $A$ also belongs to $A$.

Given integers \( m, n \geq 2 \), the points \( A_1, A_2, \dots, A_n \) are chosen independently and uniformly at random on a circle of circumference \( 1 \). That is, for each \( i = 1, \dots, n \), for any \( x \in (0,1) \), and for any arc \( \mathcal{C} \) of length \( x \) on the circle, we have $\mathbb{P}(A_i \in \mathcal{C}) = x$. What is the probability that there exists an arc of length \( \frac{1}{m} \) on the circle that contains all the points \( A_1, A_2, \dots, A_n \)?

There exists a unique circle that is both tangent to the parabola $y=x^2$ at two points and tangent to the curve $x=\sqrt{\tfrac{y^3}{1-y}}.$ Compute the radius of this circle.

Let $x, y, z$ be the real numbers that satisfies the following. $(x-y)^2+(y-z)^2+(z-x)^2=8, x^3+y^3+z^3=1$ Find the minimum value of $x^4+y^4+z^4$.

A tree has 10 pounds of apples at dawn. Every afternoon, a bird comes and eats x pounds of apples. Overnight, the amount of food on the tree increases by 10%. What is the maximum value of x such that the bird can sustain itself indefinitely on the tree without the tree running out of food?

Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p = n/729$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.

Determine the greatest positive integer \(n\) for which there exists a sequence of distinct positive integers \(s_1\), \(s_2\), \(\ldots\), \(s_n\) satisfying \[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\] [i]Proposed by Holden Mui[/i]

Let $G$ be a finite group and $f: G \to G$ a map, such that $f (xy) = f (x) f (y)$ for all $x, y \in G$ and $f (x) = x^{-1}$ for more than $\frac34$ of all $x \in G$ is fulfilled. Show that $f (x) =x^{-1}$ even holds for all $x \in G$ holds.

The following figure shows a [i]walk[/i] of length 6: [asy] unitsize(20); for (int x = -5; x <= 5; ++x) for (int y = 0; y <= 5; ++y) dot((x, y)); label("$O$", (0, 0), S); draw((0, 0) -- (1, 0) -- (1, 1) -- (0, 1) -- (-1, 1) -- (-1, 2) -- (-1, 3)); [/asy] This walk has three interesting properties: [list] [*] It starts at the origin, labelled $O$. [*] Each step is 1 unit north, east, or west. There are no south steps. [*] The walk never comes back to a point it has been to.[/list] Let's call a walk with these three properties a [i]northern walk[/i]. There are 3 northern walks of length 1 and 7 northern walks of length 2. How many northern walks of length 6 are there?

In triangle $ABC$, $\angle BAC = 94^{\circ},\ \angle ACB = 39^{\circ}$. Prove that \[ BC^2 = AC^2 + AC\cdot AB\].

Let $ABCD$ be a square with side length $1$. If point $E$ is on $BC$, point $F$ is on $DC$, and triangle $AEF$ is equilateral, compute the side length of triangle $AEF$. (Note: if your answer has a square root inside a square root, you have not fully simplified your answer.)

The polynomial $x^8 +x^7$ is written on a blackboard. In a move, Peter can erase the polynomial $P(x)$ and write down $(x+1)P(x)$ or its derivative $P'(x).$ After a while, the linear polynomial $ax+b$ with $a\ne 0$ is written on the board. Prove that $a-b$ is divisible by $49.$

Suppose $\mathcal{E}_1 \neq \mathcal{E}_2$ are two intersecting ellipses with a common focus $X$; let the common external tangents of $\mathcal{E}_1$ and $\mathcal{E}_2$ intersect at a point $Y$. Further suppose that $X_1$ and $X_2$ are the other foci of $\mathcal{E}_1$ and $\mathcal{E}_2$, respectively, such that $X_1\in \mathcal{E}_2$ and $X_2\in \mathcal{E}_1$. If $X_1X_2=8, XX_2=7$, and $XX_1=9$, what is $XY^2$?