Determine the smallest possible value of the expression $$\frac{ab+1}{a+b}+\frac{bc+1}{b+c}+\frac{ca+1}{c+a}$$ where $a,b,c \in \mathbb{R}$ satisfy $a+b+c = -1$ and $abc \leqslant -3$

Find the unique integer $\overline{CA7DB}$ with nonzero digits so that $\overline{ABCD} \cdot 3 = \overline{CA7DB}.$

[i]Reduced name[/i] of a natural number $A$ with $n$ digits ($n \ge 2$) a number of $n-1$ digits obtained by deleting one of the digits of $A$: For example, the [i]reduced names[/i] of $1024$ is $124$, $104$ and $120$. Determine how many seven-digit numbers cannot be written as the sum of one natural numbers $A$ and a [i]reduced name[/i] of $A$.

Prove that if the function $ ax^2 + bx + c $ takes an integer value for every integer value of the variable $ x $, then $ 2a $, $ a + b $, $ c $ are integers and vice versa.

Suppose that $S\subseteq \mathbb Z$ has the following property: if $a,b\in S$, then $a+b\in S$. Further, we know that $S$ has at least one negative element and one positive element. Is the following statement true? There exists an integer $d$ such that for every $x\in \mathbb Z$, $x\in S$ if and only if $d|x$. [i]proposed by Mahyar Sefidgaran[/i]

[asy] draw(unitsquare);draw((0,0)--(.25,sqrt(3)/4)--(.5,0)); label("Z",(0,1),NW);label("Y",(1,1),NE);label("A",(0,0),SW);label("X",(1,0),SE);label("B",(.5,0),S);label("P",(.25,sqrt(3)/4),N); //Credit to Zimbalono for the diagram[/asy] Equilateral triangle $ABP$ (see figure) with side $AB$ of length $2$ inches is placed inside square $AXYZ$ with side of length $4$ inches so that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square until $P$ returns to its original position. The length of the path in inches traversed by vertex $P$ is equal to $\textbf{(A) }20\pi/3\qquad\textbf{(B) }32\pi/3\qquad\textbf{(C) }12\pi\qquad\textbf{(D) }40\pi/3\qquad \textbf{(E) }15\pi$

Define an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \le a<b<c<d \le 10$, and $a+d>b+c$. How many ordered quadruples are there?

Prove that there exist infinitely many positive integers $ n$ such that $ p \equal{} nr,$ where $ p$ and $ r$ are respectively the semiperimeter and the inradius of a triangle with integer side lengths.

In a triangle $ABC$, let $A'$ be a point on the segment $BC$, $B'$ be a point on the segment $CA$ and $C'$ a point on the segment $AB$ such that $$ \frac{AB'}{B'C}= \frac{BC'}{C'A} =\frac{CA'}{A'B}=k,$$ where $k$ is a positive constant. Let $\triangle$ be the triangle formed by the interesctions of $AA'$, $BB'$ and $CC'$. Prove that the areas of $\triangle $ and $ABC$ are in the ratio $$\frac{(k-1)^{2}}{k^2 +k+1}.$$

Given an $ABC$ acute triangle with $O$ the center of the circumscribed circle. Suppose that $\omega$ is a circle that is tangent to the line $AO$ at point $A$ and also tangent to the line $BC$. Prove that $\omega$ is also tangent to the circumcircle of the triangle $BOC$.

Let $X= a1b9$ and $Y ab = 51ab$ be two positive integers where $a$ and $b$ are digits. $X$ is known to be multiple of a positive two-digit number $n$ and $Y$ is the next multiple of that number $n$. Find the number $n$ and the digits $a$ and $b$. Justify why there are no other possibilities.

There are $n$ values of $x$ in the interval $0<x<2\pi$ where $f(x)=\sin(7\pi\cdot\sin(5x))=0$. For $t$ of these $n$ values of $x$, the graph of $y=f(x)$ is tangent to the $x$-axis. Find $n+t$.

[b]7.[/b] Consider four real numbers $t_{1},t_{2},t_{3},t_{4}$ such that each is less than the sum of the others. Show that there exists a tetrahedron whose faces have areas $t_{1},t_{2}, t_{3}$ and $t_{4},$ respectively. [b](G. 9)[/b]

Find all strictly ascending functions $f$ such that for all $x\in \mathbb R$, \[f(1-x)=1-f(f(x)).\]

Let $H$ be the orthocentre of $\triangle ABC$ and let $P$ be a point on the circumcircle of $\triangle ABC$, distinct from $A,B,C$. Let $E$ and $F$ be the feet of altitudes from $H$ onto $AC$ and $AB$ respectively. Let $PAQB$ and $PARC$ be parallelograms. Suppose $QA$ meets $RH$ at $X$ and $RA$ meets $QH$ at $Y$. Prove that $XE$ is parallel to $YF$.

Ali wants to move from point $A$ to point $B$. He cannot walk inside the black areas but he is free to move in any direction inside the white areas (not only the grid lines but the whole plane). Help Ali to find the shortest path between $A$ and $B$. Only draw the path and write its length. [img]https://1.bp.blogspot.com/-nZrxJLfIAp8/W1RyCdnhl3I/AAAAAAAAIzQ/NM3t5EtJWMcWQS0ig0IghSo54DQUBH5hwCK4BGAYYCw/s1600/igo%2B2016.el1.png[/img] by Morteza Saghafian

A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$. How many edges does the remaining solid have? $\textbf{(A) }36\qquad \textbf{(B) }60\qquad \textbf{(C) }72\qquad \textbf{(D) }84\qquad \textbf{(E) }108\qquad$

Let $p \ge 5$ be a prime number. For a positive integer $k$, let $R(k)$ be the remainder when $k$ is divided by $p$, with $0 \le R(k) \le p-1$. Determine all positive integers $a < p$ such that, for every $m = 1, 2, \cdots, p-1$, $$ m + R(ma) > a. $$

Let $ABCD$ be a trapezium in which $AB //CD$ and $AD \perp AB$. Suppose $ABCD$ has an incircle which touches $AB$ at $Q$ and $CD$ at $P$. Given that $PC = 36$ and $QB = 49$, find $PQ$.

In a given country, all inhabitants are knights or knaves. A knight never lies; a knave always lies. We meet three persons, $A, B$, and $C$. Person $A$ says, “If $C$ is a knight, $B$ is a knave.” Person $C$ says, “$A$ and I are different; one is a knight and the other is a knave.” Who are the knights, and who are the knaves ?

A sequence $(u_{n})$ is defined by \[ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots \] Prove that for any positive integer $n$ we have \[ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} \](where [x] denotes the smallest integer $\leq$ x)$.$

Let $ I $ be the incenter of a triangle $ ABC $ with $ AB \neq AC $, and let $ M $ be the midpoint of the arc $ BAC $ of the circumcircle of the triangle. The perpendicular line to $ AI $ passing through $ I $ intersects line $ BC $ at point $ D $. The line $ MI $ intersects the circumcircle of triangle $ BIC $ at point $ N $. Prove that line $ DN $ is tangent to the circumcircle of triangle $ BIC $.

For given positive integer $a$, find every $(x_1, x_2, …, x_{2002})$ that satisfies the following: (1) $x_1 \geq x_2 \geq … \geq x_{2002} \geq 0$ (2) $0< x_1+x_2+…+x_{2003}<a+1$ (3) $ x^2_1+x^2_2+…+x^2_{2003}+9=a^2$

Values for $A,B,C,$ and $D$ are to be selected from $\{1, 2, 3, 4, 5, 6\}$ without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves are listed does not matter; for example, the choices $A=3, B=2, C=4, D=1$ is considered the same as the choices $A=4, B=1, C=3, D=2.$) $\textbf{(A) }30 \qquad \textbf{(B) }60 \qquad \textbf{(C) }90 \qquad \textbf{(D) }180 \qquad \textbf{(E) }360$

[b]i)[/b] Let $0<a<b$.Prove that amongst all triangles having base $a$ and perimeter $a+b$ the triangle having two sides(other than the base) equal to $\frac {b}{2}$ has the maximum area. [b]ii)[/b]Using $i)$ or otherwise, prove that amongst all quadrilateral having give perimeter the square has the maximum area.