Let $d(n)$ be the number of positive divisors of a positive integer $n$. Let $\mathbb{N}$ be the set of all positive integers. Say that a bijection $F$ from $\mathbb{N}$ to $\mathbb{N}$ is [i]divisor-friendly[/i] if $d(F(mn)) = d(F(m)) d(F(n))$ for all positive integers $m$ and $n$. (Note: A bijection is a one-to-one, onto function.) Does there exist a divisor-friendly bijection? Prove or disprove.

The triangles $ABG$ and $AEF$ are in the same plane. Between them the following conditions hold: (a) $E$ is the mid-point of $AB$; (b) points $A,G$ and $F$ are on the same line; (c) there is a point $C$ at which $BG$ and $EF$ intersect; (d) $|CE|=1$ and $|AC|=|AE|=|FG|$. Show that if $|AG|=x$, then $|AB|=x^3$.

Let $ABCD$ be convex quadrilateral, such that exist $M,N$ inside $ABCD$ for which $\angle NAD= \angle MAB; \angle NBC= \angle MBA; \angle MCB=\angle NCD; \angle NDA=\angle MDC$ Prove, that $S_{ABM}+S_{ABN}+S_{CDM}+S_{CDN}=S_{BCM}+S_{BCN}+S_{ADM}+S_{ADN}$, where $S_{XYZ}$-area of triangle $XYZ$

Does there exist a triangle with length $a,b,c$ such that: [b]a)[/b] $\begin{cases} a+b+c=6 \\ a^2+b^2+c^2=13 \\ a^3+b^3+c^3=28 \end{cases}$ [b]b)[/b] $\begin{cases} a+b+c=6 \\ a^2+b^2+c^2=13 \\ a^3+b^3+c^3=30 \end{cases}$

Let $n\ge 3$ be a positive integer. Prove that a polynomial of the form \[x^n+a_{n-3}x^{n-3}+a_{n-4}x^{n-4}+\ldots +a_1x+a_0,\] where at least one of the real coefficients $a_0,a_1,\ldots ,a_{n-3}$ is nonzero, cannot have all real roots.

For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied: [list][*] $a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$, [*] If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$[/list] Determine $N(n)$ for all $n\geq 2$. [i]Proposed by Dan Schwarz, Romania[/i]

A city is laid out with a rectangular grid of roads with 10 streets numbered from 1 to 10 running east-west and 16 avenues numbered from 1 to 16 running northsouth. All streets end at First and Sixteenth Avenues, and all avenues end at First and Tenth Streets. A rectangular city park is bounded on the north and south by Sixth Street and Eighth Street, and bounded on the east and west by Fourth Avenue and Twelfth Avenue. Although there are no breaks in the roads that bound the park, no road goes through the park. The city paints a crosswalk from every street corner across any adjacent road. Thus, where two roads cross such as at Second Street and Second Avenue, there are four crosswalks painted, while at corners such as First Street and First Avenue, there are only two crosswalks painted. How many crosswalks are there painted on the roads of this city?

In an acute triangle $ABC$, points $A_1, B_1, C_1$ are the midpoints of the sides $BC, AC, AB$, respectively. It is known that $AA_1 = d(A_1, AB) + d(A_1, AC)$, $BB1 = d(B_1, AB) + d(A_1, BC)$, $CC_1 = d(C_1, AC) + d(C_1, BC)$, where $d(X, Y Z)$ denotes the distance from point $X$ to the line $YZ$. Prove, that triangle $ABC$ is equilateral.

For what positive integers $n\geq 4$ does there exist a set $S$ of $n$ points on the plane, not all collinear, such that for any three non-collinear points $A,B,C$ in $S$, either the incenter, $A$-excenter, $B$-excenter, or $C$-excenter of triangle $ABC$ is also contained in $S$?

Let $ABCD$ be a convex quadrilateral with $AC=20$, $BC=12$ and $BD=17$. If $\angle{CAB}=80^{\circ}$ and $\angle{DBA}=70^{\circ}$, then find the area of $ABCD$. [i]2017 CCA Math Bonanza Team Round #7[/i]

Consider the assertion that for each positive integer $n\geq2$, the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of $4$. Either prove the assertion or find (with proof) a counterexample.

Real numbers $a_1,a_2,...,a_n$ ($n \ge 3$) satisfy the conditions $a_1 = a_n = 0$ and $$a_{k-1}+a_{k+1} \ge 2a_k$$ for $k = 2$,$3$$,...,$$n -1$. Prove that none of the numbers $a_1$,$...$,$a_n$ is positive.

A lattice point in the Carthesian plane is a point with both coordinates integers. A rectangle minor (respectively a square minor) is a set of lattice points lying inside or on the boundaries of a rectangle (respectively square) with vertices lattice points. We assign to each lattice point a real number, such that the sum of all the numbers in any square minor is less than $1$ in absolute value. Prove that the sum of all the numbers in any rectangle minor is less than $4$ in absolute value.

There are $2000$ apples , contained in several baskets. One can remove baskets and /or remove apples from the baskets. Prove that it is possible to then have an equal number of apples in each of the remaining baskets, with the total number of apples being not less than $100$ . (A. Razborov)

Find all strictly monotonic functions $f: \mathbb{R}^+\rightarrow\mathbb{R}^+$ for which $f\left(\frac{x^2}{f(x)}\right)=x$ for all $x$.

Let $ABCD$ be a convex quadrilateral such that $AC \perp BD$. (a) Prove that $AB^2 + CD^2 = BC^2 + DA^2$. (b) Let $PQRS$ be a convex quadrilateral such that $PQ = AB$, $QR = BC$, $RS = CD$ and $SP = DA$. Prove that $PR \perp QS$.

Let $f\colon \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a map such that the image of every compact set is compact, and the image of every connected set is connected. Prove that $f$ is continuous.

Let $F=\dfrac{6x^2+16x+3m}6$ be the square of an expression which is linear in $x$. Then $m$ has a particular value between: $\textbf{(A) }3\text{ and }4\qquad \textbf{(B) }4\text{ and }5\qquad \textbf{(C) }5\text{ and }6\qquad$ $\textbf{(D) }-4\text{ and }-3\qquad \textbf{(E) }-6\text{ and }-5$

In the game of rock-paper-scissors-lizard-Spock, rock defeats scissors and lizard, paper defeats rock and Spock, scissors defeats paper and lizard, lizard defeats paper and Spock, and Spock defeats rock and scissors, as shown in the below diagram. As before, if two players choose the same move, then there is a draw. If three people each play a game of rock-paper-scissors-lizard-Spock at the same time by choosing one of the five moves at random, what is the probability that one player beats the other two? [img]https://cdn.artofproblemsolving.com/attachments/6/0/3129da5998a2e872673e34351f786ffd47e1a1.png[/img]

Prove that among any $12$ consecutive positive integers there is at least one which is smaller than the sum of its proper divisors. (The proper divisors of a positive integer n are all positive integers other than $1$ and $n$ which divide $n$. For example, the proper divisors of $14$ are $2$ and $7$.)

Find all functions $f : \mathbb N \to \mathbb N$ satisfying, for all $x \in \mathbb N$, \[ f(2f(x)) = x + 1998 . \]

$ABCD$ is a tetrahedron. Let $K$ be the center of the incircle of $CBD$. Let $M$ be the center of the incircle of $ABD$. Let $L$ be the gravycenter of $DAC$. Let $N$ be the gravycenter of $BAC$. Suppose $AK$, $BL$, $CM$, $DN$ have one common point. Is $ABCD$ necessarily regular?

Circle inscribed in square $ABCD$ , is tangent to sides $AB$ and $CD$ at points $M$ and $K$ respectively. Line $BK$ intersects this circle at the point $L, X$ is the midpoint of $KL$. Find the angle $\angle MXK $.

Let $\triangle{ABC}$ be a triangle with $AB = 10$ and $AC = 11$. Let $I$ be the center of the inscribed circle of $\triangle{ABC}$. If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$, then $BC$ can be expressed in the form $\frac{\sqrt{a}-b}{c}$ where $a$, $b$, and $c$ are positive integers. Find $a+b+c$. [color=#00f]Note that this problem is null because a diagram is impossible.[/color] [i]Proposed by Andy Xu[/i]

Paula the painter had just enough paint for $ 30$ identically sized rooms. Unfortunately, on the way to work, three cans of paint fell of her truck, so she had only enough paint for $ 25$ rooms. How many cans of paint did she use for the $ 25$ rooms? $ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 25$