When two distinct digits are randomly chosen in $N=123456789$ and their places are swapped, one gets a new number $N'$ (for example, if 2 and 4 are swapped, then $N'=143256789$). The expected value of $N'$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute the remainder when $m+n$ is divided by $10^6$. [i]Proposed by Yannick Yao[/i]

Let $n\ge 3$, $d$ be positive integers. For an integer $x$, denote $r(x)$ be the remainder of $x$ when divided by $n$ such that $0\le r(x)\le n-1$. Let $c$ be a positive integer with $1<c<n$ and $\gcd(c,n)=1$, and suppose $a_1, \cdots, a_d$ are positive integers with $a_1+\cdots+a_d\le n-1$. \\ (a) Prove that if $n<2d$, then $\displaystyle\sum_{i=1}^d r(ca_i)\ge n.$ \\ (b) For each $n$, find the smallest $d$ such that $\displaystyle\sum_{i=1}^d r(ca_i)\ge n$ always holds. [i]Proposed by Yeoh Zi Song & Anzo Teh Zhao Yang[/i]

In triangle $ABC$, $AB = 52$, $BC = 56$, $CA = 60$. Let $D$ be the foot of the altitude from $A$ and $E$ be the intersection of the internal angle bisector of $\angle BAC$ with $BC$. Find $DE$.

A set of $n\ge 9$ points is given on the plane. For any 9 it points can be selected from all circles so that all these points end up on selected circles. Prove that all n points lie on two circles

A positive integer is [b]superb[/b] if it is the least common multiple of $1,2,\ldots, n$ for some positive integer $n$. Find all superb $x,y,z$ such that $x+y=z$. [i] Proposed by usjl[/i]

$ABCD$ is a rectangle with $AB = CD = 2$. A circle centered at $O$ is tangent to $BC$, $CD$, and $AD$ (and hence has radius $1$). Another circle, centered at $P$, is tangent to circle $O$ at point $T$ and is also tangent to $AB$ and $BC$. If line $AT$ is tangent to both circles at $T$, find the radius of circle $P$.

The lengths of the sides of a rectangle are given to be odd integers. Prove that there does not exist a point within that rectangle that has integer distances to each of its four vertices.

Let $n$ be a positive integer and let $a_1$, $a_2$,..., $a_n$ be positive integers from set $\{1, 2,..., n\}$ such that every number from this set occurs exactly once. Is it possible that numbers $a_1$, $a_1 + a_2 ,..., a_1 + a_2 + ... + a_n$ all have different remainders upon division by $n$, if: $a)$ $n=7$ $b)$ $n=8$

In $ \bigtriangleup ABC $, $ AB = 86 $, and $ AC = 97 $. A circle with center $ A $ and radius $ AB $ intersects $ \overline{BC} $ at points $ B $ and $ X $. Moreover $ \overline{BX} $ and $ \overline{CX} $ have integer lengths. What is $ BC $? $ \textbf{(A)} \ 11 \qquad \textbf{(B)} \ 28 \qquad \textbf{(C)} \ 33 \qquad \textbf{(D)} \ 61 \qquad \textbf{(E)} \ 72 $

Allen and Alan play a game. A nonconstant polynomial $P(x,y)$ with real coefficients and a positive integer $d$ greater than the degree of $P$ are known to both Allen and Alan. Alan thinks of a polynomial $Q(x,y)$ with real coefficients and degree at most $d$ and keeps it secret. Allen can make queries of the form $(s,t)$, where $s$ and $t$ are real numbers such that $P(s,t)\neq0$. Alan must respond with the value $Q(s,t)$. Allen's goal is to determine whether $P$ divides $Q$. Find (in terms of $P$ and $d$) the smallest positive integer, $g$, such that Allen can always achieve this goal making no more than $g$ queries. [i]Linus Tang[/i]

A bagel is a loop of $2a+2b+4$ unit squares which can be obtained by cutting a concentric $a\times b$ hole out of an $(a +2)\times (b+2)$ rectangle, for some positive integers a and b. (The side of length a of the hole is parallel to the side of length $a+2$ of the rectangle.) Consider an infinite grid of unit square cells. For each even integer $n \ge 8$, a bakery of order $n$ is a finite set of cells $ S$ such that, for every $n$-cell bagel $B$ in the grid, there exists a congruent copy of $B$ all of whose cells are in $S$. (The copy can be translated and rotated.) We denote by $f(n)$ the smallest possible number of cells in a bakery of order $ n$. Find a real number $\alpha$ such that, for all sufficiently large even integers $n \ge 8$, we have $$\frac{1}{100}<\frac{f (n)}{n^ {\alpha}}<100$$ [i]Proposed by Nikolai Beluhov[/i]

A segment $AB$ is given in (Euclidean) plane. Consider all triangles $XYZ$ such, that $X$ is an inner point of $AB$, triangles $XBY$ and $XZA$ are similar (in this order of vertices), and points $A, B, Y, Z$ lie on a circle in this order. Find the locus of midpoints of all such segments $YZ$. (Day 1, 2nd problem authors: Michal Rolínek, Jaroslav Švrček)

Let $x,y,z$ be the non-negative real numbers satisfying $xy+yz+zx\leq 1$. Prove that: $$1-xy-yz-zx\leq (6-2\sqrt{6})(1-\min\{x,y,z\}).$$

For which integers $n \geq 3$ does there exist a regular $n$-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system?

Let $n$ be an integer with $n \ge 2$. Show that $n$ does not divide $2^{n}-1$.

A set $A$ is endowed with a binary operation $*$ satisfying the following four conditions: (1) If $a, b, c$ are elements of $A$, then $a * (b * c) = (a * b) * c$ , (2) If $a, b, c$ are elements of $A$ such that $a * c = b *c$, then $a = b$ , (3) There exists an element $e$ of $A$ such that $a * e = a$ for all $a$ in $A$, and (4) If a and b are distinct elements of $A-\{e\}$, then $a^3 * b = b^3 * a^2$, where $x^k = x * x^{k-1}$ for all integers $k \ge 2$ and all $x$ in $A$. Determine the largest cardinality $A$ may have. proposed by Bojan Basic, Serbia

Find the least positive integer $m$ such that $105| 9^{(p^2)} -29^p +m$ for all prime numbers $p > 3$.

In the fictional country of Mahishmati, there are $50$ cities, including a capital city. Some pairs of cities are connected by two-way flights. Given a city $A$, an ordered list of cities $C_1,\ldots, C_{50}$ is called an [i]antitour[/i] from $A$ if [list] [*] every city (including $A$) appears in the list exactly once, and [*] for each $k\in \{1,2,\ldots, 50\}$, it is impossible to go from $A$ to $C_k$ by a sequence of exactly $k$ (not necessarily distinct) flights. [/list] Baahubali notices that there is an antitour from $A$ for any city $A$. Further, he can take a sequence of flights, starting from the capital and passing through each city exactly once. Find the least possible total number of antitours from the capital city. [i]Proposed by Sutanay Bhattacharya[/i]

Let $ n$ be a non-negative integer, which ends written in decimal notation on exactly $ k$ zeros, but which is bigger than $ 10^k$. For a $ n$ is only $ k\equal{}k(n)\geq2$ known. In how many different ways (as a function of $ k\equal{}k(n)\geq2$) can $ n$ be written as difference of two squares of non-negative integers at least?

Fix a triangle $ABC$. Let $\Gamma_1$ the circle through $B$, tangent to edge in $A$. Let $\Gamma_2$ the circle through C tangent to edge $AB$ in $A$. The second intersection of $\Gamma_1$ and $\Gamma_2$ is denoted by $D$. The line $AD$ has second intersection $E$ with the circumcircle of $\vartriangle ABC$. Show that $D$ is the midpoint of the segment $AE$.

Find the real-numbered solution to the equation below and demonstrate that it is unique. \[\dfrac{36}{\sqrt{x}}+\dfrac{9}{\sqrt{y}}=42-9\sqrt{x}-\sqrt{y}\]

Positive integers $a, b,$ and $c$ have the property that $a^b, b^c,$ and $c^a$ end in $4, 2,$ and $9,$ respectively. Compute the minimum possible value of $a+b+c.$

Let $P $ be the intersection of $AC $ and $BD $ in isosceles trapezoid  $ABCD $ ($AB\parallel CD$ , $BC=AD $) . The circumcircle of triangle $ABP $ inersects $BC $ for the second time at $X $. Point $Y $ lies on $AX $ such that $DY\parallel BC $. Prove that $\hat {YDA} =2.\hat {YCA} $.

To write all consecutive natural numbers from $1ab$ to $ab2$ inclusive, $1ab1$ digits have been used. Determine how many more digits are needed to write the natural numbers up to $aab$ inclusive. Give all chances. ($a$ and $b$ represent digits)

What is $10! - 7! \cdot 6!$? $ \textbf{(A) }-120 \qquad \textbf{(B) }0 \qquad \textbf{(C) }120 \qquad \textbf{(D) }600 \qquad \textbf{(E) }720 \qquad $