If $T(n)$ is the numbers of triangles with integers sizes(not congruent with each other) with it´s perimeter is equal to $n$, prove that: $$T(2012)<T(2015)$$ $$T(2013)=T(2016)$$

Let $S$ be a set of integers such that [list][*] there exist $a, b \in S$ with $\gcd(a, b)=\gcd(a-2,b-2)=1$, [*] if $x,y\in S$, then $x^2 -y\in S$.[/list] Prove that $S=\mathbb{Z}$.

Suppose that $a,b,c$ are real numbers such that $a < b < c$ and $a^3-3a+1=b^3-3b+1=c^3-3c+1=0$. Then $\frac1{a^2+b}+\frac1{b^2+c}+\frac1{c^2+a}$ can be written as $\frac pq$ for relatively prime positive integers $p$ and $q$. Find $100p+q$. [i]Proposed by Michael Ren[/i]

A floor is tiled with equilateral triangles of side length $1$, as shown. If you drop a needle of length $2$ somewhere on the floor , what is the largest number of triangles it could end up intersecting? (Only count the triangles whose interiors are met by the needle --- touching along edges or at corners doesn't qualify.) [img]https://cdn.artofproblemsolving.com/attachments/5/6/e7555c22ffe890b46a3ebdbda2169d23e43700.png[/img]

Let $a$ and $b$ be positive integers such that the number $b^2 + (b +1)^2 +...+ (b + a)^2-3$ is multiple of $5$ and $a + b$ is odd. Calculate the digit of the units of the number $a + b$ written in decimal notation.

In the plane points $A,B,C$ are marked in blue and points $P,Q$ are marked in red (no 3 marked points lie on a line, and no 4 marked points lie on a circle). A circle is called [b]separating[/b] if all points of one color are inside it, and all points of the other color are outside of it. Denote by $O$ the circumcenter of $ABC$ and by $R$ the circumradius of $ABC$. Prove that [b]exactly one[/b] of the following holds: [list] [*] There exists a separating circle; [*] There exists a point $X$ on the segment $PQ$ which also lies inside the triangle $ABC$, for which $PX\cdot XQ = R^2-OX^2$.

Let $k$ be an even positive integer and define a sequence $<x_n>$ by \[ x_1= 1 , x_{n+1} = k^{x_n} +1. \] Show that $x_n ^2$ divides $x_{n-1}x_{n+1}$ for each $n \geq 2.$

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

Let $ABCD$ be a unit square. Let $Q_1$ be the midpoint of $\overline{CD}$. For $i=1,2,\dots,$ let $P_i$ be the intersection of $\overline{AQ_i}$ and $\overline{BD}$, and let $Q_{i+1}$ be the foot of the perpendicular from $P_i$ to $\overline{CD}$. What is $$\sum_{i=1}^{\infty} \text{Area of } \triangle DQ_i P_i \, ?$$ $\textbf{(A)}\ \frac{1}{6} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{1}{3} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ 1$

On Halloween 31 children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order. "Are you a truth-teller?" The principal gave a piece of candy to each of the 22 children who answered yes. "Are you an alternater?" The principal gave a piece of candy to each of the 15 children who answered yes. "Are you a liar?" The principal gave a piece of candy to each of the 9 children who answered yes. How many pieces of candy in all did the principal give to the children who always tell the truth? $\textbf{(A) }7\qquad\textbf{(B) }12\qquad\textbf{(C) }21\qquad\textbf{(D) }27\qquad\textbf{(E) }31$

Let $x$ be in the interval $\left(0, \frac{\pi}{2}\right)$ such that $\sin x - \cos x = \frac12$ . Then $\sin^3 x + \cos^3 x = \frac{m\sqrt{p}}{n}$ , where $m, n$, and $p$ are relatively prime positive integers, and $p$ is not divisible by the square of any prime. Find $m + n + p$.

Alice and Bob are playing a guessing game. Bob is thinking of a number n of the form $2^a3^b$, where a and b are positive integers between $ 1$ and $2020$, inclusive. Each turn, Alice guess a number m, and Bob will tell her either $\gcd (m, n)$ or $lcm (m, n)$ (letting her know that he is saying that $gcd$ or $lcm$), as well as whether any of the respective powers match up in their prime factorization. In particular, if $m = n$, Bob will let Alice know this, and the game is over. Determine the smallest number $k$ so that Alice is always able to find $n$ within $k$ guesses, regardless of Bob’s number or choice of revealing either the $lcm$, or the $gcd$ .

Let $p(x)$ be a polynomial of degree strictly less than $100$ and such that it does not have $(x^3-x)$ as a factor. If $$\frac{d^{100}}{dx^{100}}\bigg(\frac{p(x)}{x^3-x}\bigg)=\frac{f(x)}{g(x)}$$ for some polynomials $f(x)$ and $g(x)$ then find the smallest possible degree of $f(x)$.

$a$, $b$, $c$ are real numbers. Show that $\min((a-b)^2,(b-c)^2,(c-a)^2)\leq \frac{a^2+b^2+c^2}{2}$

From a point $ M $ on the surface of a sphere, three mutually perpendicular chords $ MA $, $ MB $, $ MC $ are drawn. Prove that the segment joining the point $ M $ with the center of the sphere intersects the plane of the triangle $ ABC $ at the center of gravity of this triangle.

Let $ f(x)$ be a $ n \minus{}$degree polynomial all of whose coefficients are equal to $ \pm 1$, and having $ x \equal{} 1$ as its $ m$ multiple root. If $ m\ge 2^k (k\ge 2,k\in N)$, then $ n\ge 2^{k \plus{} 1} \minus{} 1.$

Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.

Given a set $P$ of $n>100$ points on the plane such that no three of them are collinear, and a set $S$ of $20n$ distinct segments, each joining a pair of points from $P$. Prove that there exists a line not passing through a point from $P$ and intersecting at least $200$ segments from $S$.

A line through the vertex $A{}$ of the triangle $ABC{}$ which doesn't coincide with $AB{}$ or $AC{}$ intersectes the altitudes from $B{}$ and $C{}$ at $D{}$ and $E{}$ respectively. Let $F{}$ be the reflection of $D{}$ in $AB{}$ and $G{}$ be the reflection of $E{}$ in $AC{}.$ Prove that the circles $ABF{}$ and $ACG{}$ are tangent.

Define a positive integer $n$ to be [i]almost-cubic [/i] if it becomes a perfect cube upon concatenating the digit $5$. For example, $12$ is almost-cubic because $125 = 5^3$. Find the remainder when the sum of all almost-cubic $n < 10^8$ is divided by $1000$.

Is it possible to color each square on a $9\times 9$ board so that each $2\times 3$ or $3\times 2$ block contains exactly $2$ black squares? If so, what is/are the possible total number(s) of black squares?

There are $n \ge 4$ cities on a round lake, between which $n -4$ people travel and one green drivers operate. Each ferry connects two non-adjacent cities, and itself do not cross two driving routes so that collisions can be avoided. In order to better adapt the transport routes to the needs of the passengers, the following change can be done: A new route can be assigned to any driver. The routes of the remaining drives must not cross and also must not be changed at the same time. Let Santa Marta and Cape Town be two non-adjacent cities. Show that you have finitely many route changes so that the Green Driver will operate between Santa Marta and Cape Town after these changes. Note: At no time may two trips between the same cities or one drive between two neighboring cities. [hide=original wording]An einem runden See liegen $n >= 4$ Stadte, zwischen denen $n - 4$ Personenfahren und eine grune Autofahre verkehren. Jede Fahre verbindet zwei nicht benachbarte Stadte, wobei sich keine zwei Fahrenrouten uberkreuzen, damit Kollisionen vermieden werden konnen. Um die Transportrouten besser den Bedurfnissen der Passagiere anzupassen, kann folgende Anderung vorgenommen werden: Einer beliebigen Fahre kann eine neue Route zugeordnet werden. Dabei durfen die Routen der restlichen Fahren nicht uberkreuzt und auch nicht gleichzeitig verandert werden. Seien Santa Marta und Kapstadt zwei nicht benachbarte Stadte. Zeige, dass man endlich viele Routenanderungen vornehmen kann, sodass die grune Autofahre nach diesen Anderungen zwischen Santa Marta und Kapstadt verkehrt. Bemerkung: Zu keinem Zeitpunkt durfen zwei Fahren zwischen denselben Stadten oder eine Fahre zwischen zwei benachbarten Stadten verkehren.[/hide]

[b]p1.[/b] The length of the side $AB$ of the trapezoid with bases $AD$ and $BC$ is equal to the sum of lengths $|AD|+|BC|$. Prove that bisectors of angles $A$ and $B$ do intersect at a point of the side $CD$. [b]p2.[/b] Polynomials $P(x) = x^4 + ax^3 + bx^2 + cx + 1$ and $Q(x) = x^4 + cx^3 + bx^2 + ax + 1$ have two common roots. Find these common roots of both polynomials. [b]p3.[/b] A girl has a box with $1000$ candies. Outside the box there is an infinite number of chocolates and muffins. A girl may replace: $\bullet$ two candies in the box with one chocolate bar, $\bullet$ two muffins in the box with one chocolate bar, $\bullet$ two chocolate bars in the box with one candy and one muffin, $\bullet$ one candy and one chocolate bar in the box with one muffin, $\bullet$ one muffin and one chocolate bar in the box with one candy. Is it possible that after some time it remains only one object in the box? [b]p4.[/b] There are $9$ straight lines drawn in the plane. Some of them are parallel some of them intersect each other. No three lines do intersect at one point. Is it possible to have exactly $17$ intersection points? [b]p5.[/b] It is known that $x$ is a real number such that $x+\frac{1}{x}$ is an integer. Prove that $x^n+\frac{1}{x^n}$ is an integer for any positive integer $n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be an acute triangle, $H$ its orthocentre, $D$ a point on the side $[BC]$, and $P$ a point such that $ADPH$ is a parallelogram. Show that $\angle BPC > \angle BAC$.

Hi everybody! I've an interesting problem! Can you solve it? Prove [b]Erdös-Ginzburg-Ziv Theorem[/b]: [i]"Among any $2n-1$ integers, there are some $n$ whose sum is divisible by $n$."[/i]