The Imomi archipelago consists of $n\geq 2$ islands. Between each pair of distinct islands is a unique ferry line that runs in both directions, and each ferry line is operated by one of $k$ companies. It is known that if any one of the $k$ companies closes all its ferry lines, then it becomes impossible for a traveller, no matter where the traveller starts at, to visit all the islands exactly once (in particular, not returning to the island the traveller started at). Determine the maximal possible value of $k$ in terms of $n$. [i]Anton Trygub, Ukraine[/i]

Let ${ABC}$ be a triangle inscribed in a circle. Point ${P}$ is the center of the arc ${BAC}$. The circle with the diameter ${CP}$ intersects the angle bisector of angle ${\angle BAC}$ at points ${K, L}$ ${(|AK| <|AL|)}$. Point ${M}$ is the reflection of ${L}$ with respect to line ${BC}$. Prove that the circumcircle of the triangle ${BKM}$ passes through the center of the segment ${BC}$ .

Find the smallest positive integer $n$ such that the $73$ fractions $\frac{19}{n+21}, \frac{20}{n+22},\frac{21}{n+23},...,\frac{91}{n+93}$ are all irreducible.

Suppose $a,b,c>0$ are integers such that \[abc-bc-ac-ab+a+b+c=2013.\] Find the number of possibilities for the ordered triple $(a,b,c)$.

For any integer $n+1,\ldots, 2n$ ($n$ is a natural number) consider its greatest odd divisor. Prove that the sum of all these divisors equals $n^2.$

[b]p1.[/b] Insert "$+$" signs between some of the digits in the following sequence to obtain correct equality: $$1\,\,\,\, 2\,\,\,\, 3\,\,\,\, 4\,\,\,\,5\,\,\,\, 6\,\,\,\, 7 = 100$$ [b]p2.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the big square $ABCD$ is $40$ cm. [img]https://cdn.artofproblemsolving.com/attachments/8/c/d54925cba07f63ec8578048f46e1e730cb8df3.png[/img] [b]p3.[/b] Jack made $3$ quarts of fruit drink from orange and apple juice. $\frac25$ of his drink is orange juice and the rest is apple juice. Nick prefers more orange juice in the drink. How much orange juice should he add to the drink to obtain a drink composed of $\frac35$ of orange juice? [b]p4.[/b] A train moving at $55$ miles per hour meets and is passed by a train moving moving in the opposite direction at $35$ miles per hour. A passenger in the first train sees that the second train takes $8$ seconds to pass him. How long is the second train? [b]p5.[/b] It is easy to arrange $16$ checkers in $10$ rows of $4$ checkers each, but harder to arrange $9$ checkers in $10$ rows of $3$ checkers each. Do both. [b]p6.[/b] Every human that lived on Earth exchanged some number of handshakes with other humans. Show that the number of people that made an odd number of handshakes is even. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Mr. Green receives a $ 10 \%$ raise every year. His salary after four such raises has gone up by what percent? \[ \textbf{(A)}\ \text{less than }40 \% \qquad \textbf{(B)}\ 40 \% \qquad \textbf{(C)}\ 44 \% \qquad \textbf{(D)}\ 45 \% \qquad \textbf{(E)}\ \text{More than }45 \% \]

The number $4567$ is tripled. The ones digit (units digit) in the resulting number is $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 1$

Let be $\triangle ABC$ .Let $A'$, $B'$, $C'$ be the foot of perpendiculars from $A$, $B$ and $C$ respectively. The points $E$ and $F$ are on the sides $CB'$ and $BC'$ respectively, such that $B'E\cdot C'F = BF\cdot CE$. Show that $AEA'F$ is cyclic.

The positive real numbers $a, b, c, d$ satisfy the equality $$\left(\frac{1}{a}+ \frac{1}{b}\right) \left(\frac{1}{c}+ \frac{1}{d}\right) + \frac{1}{ab}+ \frac{1}{cd} = \frac{6}{\sqrt{abcd}}$$ Find the value of the $$\frac{a^2+ac+c^2}{b^2-bd+d^2}$$

Find all positive integers $a$ such that for any positive integer $n\ge 5$ we have $2^n-n^2\mid a^n-n^a$.

A [i]kite[/i] is a quadrilateral whose diagonals are perpendicular. Let kite $ABCD$ be such that $\angle B = \angle D = 90^\circ$. Let $M$ and $N$ be the points of tangency of the incircle of $ABCD$ to $AB$ and $BC$ respectively. Let $\omega$ be the circle centered at $C$ and tangent to $AB$ and $AD$. Construct another kite $AB^\prime C^\prime D^\prime$ that is similar to $ABCD$ and whose incircle is $\omega$. Let $N^\prime$ be the point of tangency of $B^\prime C^\prime$ to $\omega$. If $MN^\prime \parallel AC$, then what is the ratio of $AB:BC$?

The area of the shaded region $\text{BEDC}$ in parallelogram $\text{ABCD}$ is [asy] unitsize(10); pair A,B,C,D,E; A=origin; B=(4,8); C=(14,8); D=(10,0); E=(4,0); draw(A--B--C--D--cycle); fill(B--E--D--C--cycle,gray); label("A",A,SW); label("B",B,NW); label("C",C,NE); label("D",D,SE); label("E",E,S); label("$10$",(9,8),N); label("$6$",(7,0),S); label("$8$",(4,4),W); draw((3,0)--(3,1)--(4,1)); [/asy] $\text{(A)}\ 24 \qquad \text{(B)}\ 48 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 80$

At a twins and triplets convention, there were $9$ sets of twins and $6$ sets of triplets, all from different families. Each twin shook hands with all the twins except his/her sibling and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and half the twins. How many handshakes took place? $ \textbf{(A)}\ 324 \qquad \textbf{(B)}\ 441 \qquad \textbf{(C)}\ 630 \qquad \textbf{(D)}\ 648 \qquad \textbf{(E)}\ 882$

An integer $n$ is said to be [i]good[/i] if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer. [i]Proposed by Hojoo Lee, Korea[/i]

Let a be a real number such that $0 < a < 1$, and let $n$ be a positive integer. Define the sequence $a_0, a_1, a_2, \ldots, a_n$ an recursively by \[a_0 = a, \quad a_{k+1} = a_k +\frac 1n a_k^2 \quad \text{ for } k = 0, 1, \ldots, n - 1.\] Prove that there exists a real number $A$, depending on $a$ but independent of $n$, such that \[0 < n(A - a_n) < A^3.\]

Let $ABC$ be a triangle, $I$ and $O$ be its incenter and circumcenter respectively. $A'$ is symmetric to $O$ with respect to line $AI$. Points $B'$ and $C'$ are defined similarly. Prove that the nine-point centers of triangles $ABC$ and $A'B'C'$ coincide.

Let $ABC$ be an equilateral triangle with side length $1$. Points $A_1$ and $A_2$ are chosen on side $BC$, points $B_1$ and $B_2$ are chosen on side $CA$, and points $C_1$ and $C_2$ are chosen on side $AB$ such that $BA_1<BA_2$, $CB_1<CB_2$, and $AC_1<AC_2$. Suppose that the three line segments $B_1C_2$, $C_1A_2$, $A_1B_2$ are concurrent, and the perimeters of triangles $AB_2C_1$, $BC_2A_1$, and $CA_2B_1$ are all equal. Find all possible values of this common perimeter. [i]Ankan Bhattacharya[/i]

The point $M$ is the midpoint of the side $BC$ of triangle $ABC$. A circle is passing through $B$, is tangent to the line $AM$ at $M$, and intersects the segment $AB$ secondary at the point $P$. Prove that the circle, passing through $A$, $P$, and the midpoint of the segment $AM$, is tangent to the line $AC$. [i](A. Voidelevich)[/i]

Let $n$ be a positive integer. Calculate the sum $\sum_{k=1}^n\ \ {\sum_{1\le i_1 < \ldots < i_k\le n}^{}{\frac {2^k}{(i_1 + 1)(i_2 + 1)\ldots (i_k + 1)}}}$

Find the number of distinguishable groupings into which you can place $3$ indistinguishable red balls and $3$ indistinguishable blue balls. Here the groupings $RR-BR-B-B$ and $B-RB-B-RR$ are indistinguishable because the groupings are merely rearranged, but $RRB-BR-B$ is distinguishable from $RBB-BR-R$.

Let $m,n$ are positive integers. a)Prove that $(m,n)=2\sum_{k=0}^{m-1}[\frac{kn}{m}]+m+n-mn$. b)If $m,n\geq 2$, prove that $\sum_{k=0}^{m-1}[\frac{kn}{m}]=\sum_{k=0}^{n-1}[\frac{km}{n}]$.

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $$\gcd(f(x),y)f(xy)=f(x)f(y)$$ for all positive integers $x, y$.

Determine the polynomial P(X) satisfying simoultaneously the conditions: a) The remainder obtained when dividing P(X) to the polynomial X^3 −2 is equal to the fourth power of quotient. b) P(−2) + P(2) = −34.

Alice and Bob play a game on a connected graph with $2n$ vertices, where $n\in \mathbb{N}$ and $n>1$.. Alice and Bob have tokens named A and B respectively. They alternate their turns with Alice going first. Alice gets to decide the starting positions of A and B. Every move, the player with the turn moves their token to an adjacent vertex. Bob's goal is to catch Alice, and Alice's goal is to prevent this. Note that positions of A, B are visible to both Alice and Bob at every moment. Provided they both play optimally, what is the maximum possible number of edges in the graph if Alice is able to evade Bob indefinitely? [i]Proposed by Shashank Ingalagavi and Vighnesh Sangle[/i]