Let $n{}$ be a positive integer. Find the number of noncongruent triangles with integer sidelengths and a perimeter of $2n$.

Consider the isosceles right triangle $ABC$ ($AB = AC$) and the points $M, P \in [AB]$ so that $AM = BP$. Let $D$ be the midpoint of the side $BC$ and $R, Q$ the intersections of the perpendicular from $A$ on$ CM$ with $CM$ and $BC$ respectively. Prove that a) $\angle AQC = \angle PQB$ b) $\angle DRQ = 45^o$

Let $n$ be an integer greater than or equal to $1$. Find, as a function of $n$, the smallest integer $k\ge 2$ such that, among any $k$ real numbers, there are necessarily two of which the difference, in absolute value, is either strictly less than $1 / n$, either strictly greater than $n$.

Let $a$, $b$ and $c$ be non-negative numbers such that $ab+ac+bc+abc=4.$ . Prove that: $$\sqrt{\frac{ab+ac+1}{a+2}}+\sqrt{\frac{ab+bc+1}{b+2}}+\sqrt{\frac{ac+bc+1}{c+2}}\leq3.$$ [hide="PS"]Dedicated to dear KhuongTrang :-D [/hide]

Let $k$ a positive integer number such that $\frac{k(k+1)}{3}$ is a perfect square. Show that $\frac{k}{3}$ and $k+1$ are both perfect squares.

Let a triangle $ABC$ inscribed in a circle $\Gamma$ with center $O$. Let $I$ the incenter of triangle $ABC$ and $D, E, F$ the contact points of the incircle with sides $BC, AC, AB$ of triangle $ABC$ respectively . Let also $S$ the foot of the perpendicular line from $D$ to the line $EF$.Prove that line $SI$ passes from the antidiametric point $N$ of $A$ in the circle $\Gamma$.( $AN$ is a diametre of the circle $\Gamma$).

Prove that in any polygon, there exist two sides whose radio is less than $2$.(Essentialy if $a_1\geq a_2\geq...\geq a_n$ are the sides of a polygon prove that there exist $i,j\in\{1,2,..,n\}$ so that $i<j$ and $\frac {a_i}{a_j}<2$).

A circle $k_1$ lies within a second circle $k_2$ and touches it at point $A$. A line through $A$ intersects $k_1$ again in $B$ and $k_2$ in $C$. The tangent to $k_1$ through $B$ intersects $k_2$ at points $D$ and $E$. The tangents at $k_1$ passing through $C$ intersects $k_1$ in points $F$ and $G$. Prove that $D, E, F$ and $G$ lie on a circle.

Given a nonzero natural number $n$, let $f_n$ be the function of the closed interval $[0, 1]$ in $R$ defined like this: $$f_n(x) = \begin{cases}n^2x, \,\,\, if \,\,\, 0 \le x < 1/n\\ 3/n, \,\,\,if \,\,\,1/n \le x \le 1 \end{cases}$$ a) Represent the function graphically. b) Calculate $A_n =\int_0^1 f_n(x) dx$. c) Find, if it exists, $\lim_{n\to \infty} A_n$ .

The vertices of a triangle with sides $a\ge b\ge c$ are centers of three circles, such that no two of the circles have common interior points and none contains any other vertex of the triangle. Determine the maximum possible total area of these three circles.

Which is the least positive integer $n$ for which it is possible to find a (non-degenerate) $n$-gon with sidelengths $1, 2,. . . , n$, and where all vertices have integer coordinates?

Let $ K$ be the figure bounded by the curve $ y\equal{}e^x$ and 3 lines $ x\equal{}0,\ x\equal{}1,\ y\equal{}0$ in the $ xy$ plane. (1) Find the volume of the solid formed by revolving $ K$ about the $ x$ axis. (2) Find the volume of the solid formed by revolving $ K$ about the $ y$ axis.

Solve in positive real numbers the following equation $x^{-y} + y^{-x} = 4 - x - y$.

A finite set $S$ of positive integers has the property that, for each $s \in S,$ and each positive integer divisor $d$ of $s$, there exists a unique element $t \in S$ satisfying $\text{gcd}(s, t) = d$. (The elements $s$ and $t$ could be equal.) Given this information, find all possible values for the number of elements of $S$.

Let $ABCD$ be a trapezoid where $AB$ is parallel to $CD.$ Let $P$ be the intersection of diagonal $AC$ and diagonal $BD.$ If the area of triangle $PAB$ is $16,$ and the area of triangle $PCD$ is $25,$ find the area of the trapezoid.

Let $m,k$ be positive integers, $k<m$ and $M$ a set with $m$ elements. Prove that the maximal number of subsets $A_1,A_2,\ldots ,A_p$ of $M$ for which $A_i\cap A_j$ has at most $k$ elements, for every $1\le i<j\le p$, equals \[ p_{max}=\binom{m}{0}+\binom{m}{1}+\binom{m}{2}+\ldots+\binom{m}{k+1}\]

Let $a$, $b$, $c$, $d$, $e$, $f$ and $g$ be seven distinct positive integers not bigger than $7$. Find all primes which can be expressed as $abcd+efg$

Let $A$ be a ring. $a)$ Show that the set $Z(A)=\{a\in A|ax=xa,\ \text{for all}\ x\in A\}$ is a subring of the ring $A$. $b)$ Prove that, if any commutative subring of $A$ is a field, then $A$ is a field.

Source: 2017 Canadian Open Math Challenge, Problem A2 ----- An equilateral triangle has sides of length $4$cm. At each vertex, a circle with radius $2$cm is drawn, as shown in the figure below. The total area of the shaded regions of the three circles is $a\cdot \pi \text{cm}^2$. Determine $a$. [center][asy] size(2.5cm); draw(circle((0,2sqrt(3)/3),1)); draw(circle((1,-sqrt(3)/3),1)); draw(circle((-1,-sqrt(3)/3),1)); draw((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle); fill(((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle),mediumgray); draw((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle); fill(((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle),mediumgray); draw((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle); fill(((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle),mediumgray); [/asy][/center]

$n$ students take a test with $m$ questions, where $m,n\ge 2$ are integers. The score given to every question is as such: for a certain question, if $x$ students fails to answer it correctly, then those who answer it correctly scores $x$ points, while those who answer it wrongly scores $0$. The score of a student is the sum of his scores for the $m$ questions. Arrange the scores in descending order $p_1\ge p_2\ge \ldots \ge p_n$. Find the maximum value of $p_1+p_n$.

Let $ p$ be an odd prime number. How many $ p$-element subsets $ A$ of $ \{1,2,\dots,2p\}$ are there, the sum of whose elements is divisible by $ p$?

A non-negative integer $p$ is a [i]3-choice[/i] if $\dfrac{k(k-1)(k-2)}{6}$ for some positive integer $k$. Let $p$ and $q$ be 3-choices with $p<q$. Show there is an integer $n$ such that $p \leq n^2 < q$.

Find all real solutions of the system of equations: \[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$

An accurate 12-hour analog clock has an hour hand, a minute hand, and a second hand that are aligned at 12:00 o’clock and make one revolution in 12 hours, 1 hour, and 1 minute, respectively. It is well known, and not difficult to prove, that there is no time when the three hands are equally spaced around the clock, with each separating angle $ \frac{2 \cdot \pi}{3}.$ Let $ f(t), g(t), h(t)$ be the respective absolute deviations of the separating angles from \frac{2 \cdot \pi}{3} at $ t$ hours after 12:00 o’clock. What is the minimum value of $ max\{f(t), g(t), h(t)\}?$

The roots of the polynomial $f(x) = x^8 +x^7 -x^5 -x^4 -x^3 +x+ 1 $ are all roots of unity. We say that a real number $r \in [0, 1)$ is nice if $e^{2i \pi r} = \cos 2\pi r + i \sin 2\pi r$ is a root of the polynomial $f$ and if $e^{2i \pi r}$ has positive imaginary part. Let $S$ be the sum of the values of nice real numbers $r$. If $S =\frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.