Let $n$ be a positive integer. What is the smallest value of $m$ with $m > n$ such that the set $M = \{n, n + 1, ..., m\}$ can be partitioned into subsets so that in each subset, there is a number which equals to the sum of all other numbers of this subset?

The sequence $ (x_n)$ of real numbers is defined by $ x_1\equal{}\frac{29}{10}$ and $ x_{n\plus{}1}\equal{}\frac{x_n}{\sqrt{x_n^2\minus{}1}}\plus{}\sqrt{3}$ for all $ n\ge 1$. Find a real number $ a$ (if exists) such that $ x_{2k\minus{}1}>a>x_{2k}$.

Consider the sequence of rational numbers defined by $x_1=\frac{4}{3}$ and $x_{n+1}=\frac{x_n^2}{x_n^2-x_n+1}$ , $n\geq 1$. Show that the numerator of the lowest term expression of each sum $\sum_{k=1}^{n}x_k$ is a perfect square. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Given an $ a \times b$ rectangle with $ a > b > 0,$ determine the minimum side of a square that covers the rectangle. (A square covers the rectangle if each point in the rectangle lies inside the square.)

How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits? $ \textbf{(A)}\ 41\qquad \textbf{(B)}\ 42\qquad \textbf{(C)}\ 43\qquad \textbf{(D)}\ 44\qquad \textbf{(E)}\ 45$

Five men, $ A$, $ B$, $ C$, $ D$, $ E$ are wearing caps of black or white colour without each knowing the colour of his cap. It is known that a man wearing black cap always speaks the truth while the ones wearing white always tell lies. If they make the following statements, find the colour worn by each of them: $ A$ : I see three black caps and one white cap. $ B$ : I see four white caps $ C$ : I see one black cap and three white caps $ D$ : I see your four black caps.

Let $ABC$ be a triangle and $D$ is a point inside of $\triangle ABC$. The point $A'$ is the midpoint of the arc $BDC$, in the circle which passes by $B,C,D$. Analogously define $B'$ and $C'$, being the midpoints of the arc $ADC$ and $ADB$ respectively. Prove that the four points $D,A',B',C'$ are concyclic.

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for each $x \in \mathbb{R}^+$, there is exactly one $y \in \mathbb{R}^+$ satisfying $$xf(y)+yf(x) \leq 2$$

Compute the sum of all positive integers $N$ for which there exists a unique ordered triple of non-negative integers $(a,b,c)$ such that $2a+3b+5c=200$ and $a+b+c=N$. [i]Proposed by Kyle Lee[/i]

Let $n \in \mathbb{N}_{\geq 2}$ and $a_1,a_2, \dots , a_n \in (1,\infty).$ Prove that $f:[0,\infty) \to \mathbb{R}$ with $$f(x)=(a_1a_2...a_n)^x-a_1^x-a_2^x-...-a_n^x$$ is a strictly increasing function.

A circle $k$ touches a larger circle $K$ from inside in a point $P$. Let $Q$ be point on $k$ different from $P$. The line tangent to $k$ at $Q$ intersects $K$ in $A$ and $B$. Show that the line $PQ$ bisects $\angle APB$.

If $\left|x\right|-x+y=42$ and $x+\left|y\right|+y=24$, then what is the value of $x+y$? Express your answer in simplest terms. $\text{(A) }-4\qquad\text{(B) }\frac{26}{5}\qquad\text{(C) }6\qquad\text{(D) }10\qquad\text{(E) }18$

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

Let $S$ be a point inside $\angle pOq$, and let $k$ be a circle which contains $S$ and touches the legs $Op$ and $Oq$ in points $P$ and $Q$ respectively. Straight line $s$ parallel to $Op$ from $S$ intersects $Oq$ in a point $R$. Let $T$ be the intersection point of the ray $PS$ and circumscribed circle of $\vartriangle SQR$ and $T \ne S$. Prove that $OT // SQ$ and $OT$ is a tangent of the circumscribed circle of $\vartriangle SQR$.

Let $P,Q,R$ be three points on a circle $k_1$ with $|PQ|=|PR|$ and $|PQ|>|QR|$. Let $k_2$ be the circle with center in $P$ that goes through $Q$ and $R$. The circle with center $Q$ through $R$ intersects $k_1$ in another point $X\ne R$ and intersects $k_2$ in another point $Y\ne R$. The two points $X$ and $R$ lie on different sides of the line through $PQ$. Show that the three points $P$, $X$, $Y$ lie on a common line.

Let $S$ be the set of ordered pairs $(x,y)$ of positive integers for which $x+y\le 20$. Evaluate \[\sum_{(x, y) \in S} (-1)^{x+y}xy.\] [i]Proposed by Andrew Wen[/i]

The Fibonacci numbers are defined recursively by $F_0=0, F_1=1,$ and $F_i=F_{i-1}+F_{i-2}$ for $i \ge 2.$ Given $15$ wooden blocks of weights $F_2, F_3, \ldots, F_{16},$ compute the number of ways to paint each block either red or blue such that the total weight of the red blocks equals the total weight of the blue blocks.

Let $ABC$ be a acute-angled triangle with centroid $G$, the angle bisector of $\angle ABC$ intersects $AC$ in $D$. Let $P$ and $Q$ be points in $BD$ where $\angle PBA = \angle PAB$ and $\angle QBC = \angle QCB$. Let $M$ be the midpoint of $QP$, let $N$ be a point in the line $GM$ such that $GN = 2GM$(where $G$ is the segment $MN$), prove that: $\angle ANC + \angle ABC = 180$

In acute triangle $ ABC,$ points $ P,Q$ lie on its sidelines $ AB,AC,$ respectively. The circumcircle of triangle $ ABC$ intersects of triangle $ APQ$ at $ X$ (different from $ A$). Let $ Y$ be the reflection of $ X$ in line $ PQ.$ Given $ PX>PB.$ Prove that $ S_{\bigtriangleup XPQ}>S_{\bigtriangleup YBC}.$ Where $ S_{\bigtriangleup XYZ}$ denotes the area of triangle $ XYZ.$

Let $ABCD$ be a rectangle. Point $E$ lies on side $AB$, and point $F$ lies on segment $CE$. Prove that if triangles $ADE$ and $CDF$ have equal areas, then triangles $BCE$ and $DEF$ also have equal areas.

Given that $ a_1\equal{}1$, $ a_2\equal{}5$, $ \displaystyle a_{n\plus{}1} \equal{} \frac{a_n \cdot a_{n\minus{}1}}{\sqrt{a_n^2 \plus{} a_{n\minus{}1}^2 \plus{} 1}}$. Find a expression of the general term of $ \{ a_n \}$.

For which non-negative integers $ a<2007$ the congruence $ x^2\plus{}a \equiv 0 \mod 2007$ has got exactly two different non-negative integer solutions? That means, that there exist exactly two different non-negative integers $ u$ and $ v$ less than $ 2007$, such that $ u^2\plus{}a$ and $ v^2\plus{}a$ are both divisible by $ 2007$.

Let $S = \{(x,y) | x = 1, 2, \ldots, 1993, y = 1, 2, 3, 4\}$. If $T \subset S$ and there aren't any squares in $T.$ Find the maximum possible value of $|T|.$ The squares in T use points in S as vertices.

At a gala banquet, $12n + 6$ chairs, where $n \in N$, are equally arranged around a large round table. A seating will be called a proper seating of rank $n$ if a gathering of $6n + 3$ married couples sit around this table such that each seated person also has exactly one sibling (brother/sister) of the opposite gender present (siblings cannot be married to each other) and each man is seated closer to his wife than his sister. Among all proper seats of rank n find the maximum possible number of women seated closer to their brother than their husband. (The maximum is taken not only across all possible seating arrangements for a given gathering, but also across all possible gatherings.)

Source: 2018 Canadian Open Math Challenge Part C Problem 3 ----- Consider a convex quadrilateral $ABCD$. Let rays $BA$ and $CD$ intersect at $E$, rays $DA$ and $CB$ intersect at $F$, and the diagonals $AC$ and $BD$ intersect at $G$. It is given that the triangles $DBF$ and $DBE$ have the same area. $\text{(a)}$ Prove that $EF$ and $BD$ are parallel. $\text{(b)}$ Prove that $G$ is the midpoint of $BD$. $\text{(c)}$ Given that the area of triangle $ABD$ is 4 and the area of triangle $CBD$ is 6, [color=transparent](C.)[/color]compute the area of triangle $EFG$.