There are $n$ cities in a country, where $n>1$. There are railroads connecting some of the cities so that you can travel between any two cities through a series of railroads (railroads run in both direction.) In addition, in this country, it is impossible to travel from a city, through a series of distinct cities, and return back to the original city. We define the [b]degree[/b] of a city as the number of cities directly connected to it by a single segment of railroad. For a city $A$ that is directly connected to $x$ cities, with $y$ of those cities having a smaller degree than city $A$, the [b]significance[/b] of city $A$ is defined as $\frac{y}{x}$. Find the smallest positive real number $t$ so that, for any $n>1$, the sum of the significance of all cities is less than $tn$, no matter how the railroads are paved. [i]Proposed by houkai[/i]

In convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at point $O$ at angle $90^{\circ}$. Let $K$, $L$, $M$ and $N$ be orthogonal projections of point $O$ to sides $AB$, $BC$, $CD$ and $DA$ of quadrilateral $ABCD$. Prove that $KLMN$ is cyclic

For each positive integer $n$, find all triples $a,b,c$ of real numbers for which \[\begin{cases}a=b^n+c^n\\ b=c^n+a^n\\ c=a^n+b^n\end{cases}\]

Let $AB$ and $CD$ be two diameters of the circle $C$. For an arbitrary point $P$ on $C$, let $R$ and $S$ be the feet of the perpendiculars from $P$ to $AB$ and $CD$, respectively. Show that the length of $RS$ is independent of the choice of $P$.

If $n$ is composite, prove that $\phi(n) \le n- \sqrt{n}$.

The positive numbers $a,b,c,A,B,C$ satisfy a condition $$a + A = b + B = c + C = k$$ Prove that $$aB + bC + cA \le k^2$$

Let $ N$ be the number of ways to write $ 2010$ in the form \[2010 \equal{} a_3 \cdot 10^3 \plus{} a_2 \cdot 10^2 \plus{} a_1 \cdot 10 \plus{} a_0,\] where the $ a_i$'s are integers, and $ 0 \le a_i \le 99$. An example of such a representation is $ 1\cdot10^3 \plus{} 3\cdot10^2 \plus{} 67\cdot10^1 \plus{} 40\cdot10^0$. Find $ N$.

Prove that \[ \sum_{n=1}^\infty {1\over n^n} = \int_0^1 x^{-x}\,dx. \]

Let $ABCD$ be a convex quadrilateral with area $202, AB = 4,$ and $\angle A = \angle B = 90^\circ$ such that there is exactly one point $E$ on line $CD$ satisfying $\angle AEB = 90^\circ.$ Compute the perimeter of $ABCD.$

Find the greatest $a$ for which there is $b$ such that the system $$\begin{cases} y=x^4+a \\ x=\dfrac{1}{y^4}+b \end{cases}$$ has exactly two solutions.

Gandalf (the wizard) and Bilbo (the assistant) are presenting a magic trick to Nitzan (the audience). While Gandalf leaves the room, Nitzan chooses a number $1\leq x\leq 2^{2022}$ and shows it to Bilbo. Now bilbo writes on the board a long row of $N$ digits, each of which is $0$ or $1$. After this Nitzan can, if he wishes, switch the order of two consecutive digits in the row, but only once. Then Gandalf returns to the room, looks at the row, and guesses the number $x$. Can Bilbo and Gandalf come up with a strategy that allows Gandalf to guess $x$ correctly no matter how Nitzan acts, if [b]a)[/b] $N=2500$? [b]b)[/b] $N=2030$? [b]c)[/b] $N=2040$?

Let $\triangle ABC$ be an acute triangle, and let $I_B, I_C,$ and $O$ denote its $B$-excenter, $C$-excenter, and circumcenter, respectively. Points $E$ and $Y$ are selected on $\overline{AC}$ such that $\angle ABY=\angle CBY$ and $\overline{BE}\perp\overline{AC}$. Similarly, points $F$ and $Z$ are selected on $\overline{AB}$ such that $\angle ACZ=\angle BCZ$ and $\overline{CF}\perp\overline{AB}$. Lines $\overleftrightarrow{I_BF}$ and $\overleftrightarrow{I_CE}$ meet at $P$. Prove that $\overline{PO}$ and $\overline{YZ}$ are perpendicular. [i]Proposed by Evan Chen and Telv Cohl[/i]

Let $ABCD$ be a trapezoid with $AB // CD$, $2|AB| = |CD|$ and $BD \perp BC$. Let $M$ be the midpoint of $CD$ and let $E$ be the intersection $BC$ and $AD$. Let $O$ be the intersection of $AM$ and $BD$. Let $N$ be the intersection of $OE$ and $AB$. (a) Prove that $ABMD$ is a rhombus. (b) Prove that the line $DN$ passes through the midpoint of the line segment $BE$.

An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent unit squares (forming a 1-by-2 or 2-by-1 rectangle), and flip their colors: black becomes white and white becomes black. We call this operation a [i]step[/i]. If $C$ is the original coloring, let $S(C)$ be the least number of steps required to make all the unit squares black. Find with proof the greatest possible value of $S(C)$.

Circle centred at point $O$ intersects sides $AC, AB$ of triangle $\triangle ABC$ at points $B_1$ and $C_1$ respectively and passes through points $B,C$. It is known that lines $AO, CC_1, BB_1 $ are concurrent. Prove that $\triangle ABC$ is isosceles.

The value(s) of $y$ for which the following pair of equations \[x^2+y^2-16=0\text{ and }x^2-3y+12=0\] may have a real common solution, are $\textbf{(A) }4\text{ only}\qquad\textbf{(B) }-7,~4\qquad\textbf{(C) }0,~4\qquad\textbf{(D) }\text{no }y\qquad \textbf{(E) }\text{all }y$

Let $x,y$ be the positive integers such that $3x^2 +x = 4y^2 + y$. Prove that $x - y$ is a perfect (square).

Is it possible for any finite group of people to choose a positive integer $N$ and assign a positive integer to each person in the group such that the product of two persons' number is divisible by $N$ if and only if they are friends?

It is known that some \(d\) distinct divisors of a positive integer number \(n\) form an arithmetic progression. Prove that the number \(n\) has at least \(2d - 2\) divisors. [i]Proposed by Anton Trygub[/i]

Let $n \in \mathbb{N}$ be such that $gcd(n, 6) = 1$. Let $a_1 < a_2 < \cdots < a_n$ and $b_1 < b_2 < \cdots < b_n$ be two collection of positive integers such that $a_j + a_k + a_l = b_j + b_k + b_l$ for all integers $1 \le j < k < l \le n$. Prove that $a_j = b_j$ for all $1 \le j \le n$.

Solve in natural numbers the system of equations $3x^2+6y^2+5z^2=1997$ and $3x+6y+5z=161$ .

What is the value of $$(\sqrt{2}-1)^4+\frac{1}{(\sqrt{2}-1)^4}?$$ $\textbf{(A) }32-16\sqrt{2}\qquad\textbf{(B) }30\qquad\textbf{(C) }34\qquad\textbf{(D) }15+15\sqrt{2}\qquad\textbf{(E) }16+16\sqrt{2}$

Determine all real numbers $ a$ such that there is a function $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ x\plus{}f(y)\equal{}af(y\plus{}f(x))\] for all real numbers $ x$ and $ y$. [i]Hery Susanto, Malang[/i]

Compute $$\lim_{x\rightarrow0}\frac{\sqrt[5]{\cos x}-\sqrt[3]{\cos x}}{x^2}.$$

Let $\{a_n\}_{n \geq 1}$ be a sequence in which $a_1=1$ and $a_2=2$ and \[a_{n+1}=1+a_1a_2a_3 \cdots a_{n-1}+(a_1a_2a_3 \cdots a_{n-1} )^2 \qquad \forall n \geq 2.\] Prove that \[\lim_{n \to \infty} \biggl( \frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots + \frac{1}{a_n} \biggr) =2\]