Luis' friends decided to play a prank on him in his geometry homework. They erased most of a triangle and, instead, drew an equivalent triangle with the sum of its three side lengths. Help Luis complete his homework by reconstructing the original triangle using only a straightedge and compass. Since Luis' method involves measurements, prove that his method results in a triangle longer than the sum of its three sides.

Let $E$ be the midpoint of the side $AB$ of the parallelogram $ABCD$. Let $F$ be the projection of $B$ on $AC$. Prove that the triangle $ABF$ is isosceles

If $b$ and $c$ are constants and \[(x + 2)(x + b) = x^2 + cx + 6,\] then $c$ is $ \textbf{(A)}\ -5\qquad\textbf{(B)}\ -3\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 5 $

Let $x$, $y$ and $z$ be positive real numbers such that $x+y+z = 1$. Prove the inequality: $$\frac{x^2}{1+y}+\frac{y^2}{1+z} +\frac{z^2}{1+x} \leq 1$$

For each positive integer $k$, let $a_k$ be the greatest integer not exceeding $\sqrt{k}$ and let $b_k$ be the greatest integer not exceeding $\sqrt[3]{k}$. Calculate $$\sum_{k=1}^{2003} (a_k-b_k).$$

In a circle four distinct points are fixed and each of them is assigned with a real number. Let those numbers be $x_1,x_2,x_3,x_4$ such that $x_1+x_2+x_3+x_4>0$. Now we define a game with these numbers: If one of them, i.e. $x_i$, is a negative number, the player makes a move by adding the number $x_i$ to his neighbors and changes the sign of the chosen number. The game ends when all the numbers are negative. Prove that this game ends in a finite number of steps.

What is the area of the shaded figure shown below? [asy] size(200); defaultpen(linewidth(0.4)+fontsize(12)); pen s = linewidth(0.8)+fontsize(8); pair O,X,Y; O = origin; X = (6,0); Y = (0,5); fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2)); for (int i=1; i<7; ++i) { draw((i,0)--(i,5), gray+dashed); label("${"+string(i)+"}$", (i,0), 2*S); if (i<6) { draw((0,i)--(6,i), gray+dashed); label("${"+string(i)+"}$", (0,i), 2*W); } } label("$0$", O, 2*SW); draw(O--X+(0.15,0), EndArrow); draw(O--Y+(0,0.15), EndArrow); draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5); [/asy]

Alice, Bob, and Chris each roll $4$ dice. Each only knows the result of their own roll. Alice claims that there are at least $5$ multiples of $3$ among the dice rolled. Bob has $1$ six and no threes, and knows that Alice wouldn’t claim such a thing unless he had at least $2$ multiples of $3$. Bob can call Alice a liar, or claim that there are at least $6$ multiples of $3$, but Chris says that he will immediately call Bob a liar if he makes this claim. Bob wins if he calls Alice a liar and there aren't at least $5$ multiples of $3$, or if he claims there are at least $6$ multiples of $3$, and there are. What is the probability that Bob loses no matter what he does?

For any function $f:\mathbb{N}\to\mathbb{N}$ we define $P(n)=f(1)f(2)...f(n)$ . Find all functions $f:\mathbb{N}\to\mathbb{N}$ st for each $a,b$ : $$P(a)+P(b) | a! + b!$$

All the vertices of a convex pentagon are on lattice points. Prove that the area of the pentagon is at least $\frac{5}{2}$. [i]Bogdan Enescu[/i]

a) Does there exist a real number $x$ such that $x+\sqrt{3}$ and $x^2+\sqrt{3}$ are both rationals? b) Does there exist a real number $y$ such that $y+\sqrt{3}$ and $y^3+\sqrt{3}$ are both rationals?

The three sides of the quadrilateral are equal, the angles between them are equal, respectively $90^o$ and $150^o$. Find the smallest angle of this quadrilateral in degrees.

Circle $C_1$ has center $O$ and radius $OA$, and circle $C_2$ has diameter $OA$. $AB$ is a chord of circle $C_1$ and $BD$ may be constructed with $D$ on $OA$ such that $BD$ and $OA$ are perpendicular. Let $C$ be the point where $C_2$ and $BD$ intersect. If $AC = 1$, find $AB$.

Let triangle$ABC(AB<AC)$ with incenter $I$ circumscribed in $\odot O$. Let $M,N$ be midpoint of arc $\widehat{BAC}$ and $\widehat{BC}$, respectively. $D$ lies on $\odot O$ so that $AD//BC$, and $E$ is tangency point of $A$-excircle of $\bigtriangleup ABC$. Point $F$ is in $\bigtriangleup ABC$ so that $FI//BC$ and $\angle BAF=\angle EAC$. Extend $NF$ to meet $\odot O$ at $G$, and extend $AG$ to meet line $IF$ at L. Let line $AF$ and $DI$ meet at $K$. Proof that $ML\bot NK$.

Let $N$ be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1's than 0's. Find the remainder when $N$ is divided by 1000.

Let $a_1, a_2,...,a_{2018}$ be a sequence of numbers such that all its elements are elements of a set $\{-1,1\}$. Sum $$S=\sum \limits_{1 \leq i < j \leq 2018} a_i a_j$$ can be negative and can also be positive. Find the minimal value of this sum

The solution to inequality $\left|\frac{1}{\log_{\frac{1}{2}}x}+2\right|>\frac{3}{2}$ is________(express answer with a set).

Consider the sequence of numbers $\left[n+\sqrt{2n}+\frac12\right]$, where $[x]$ denotes the greatest integer not exceeding $x$. If the missing integers in the sequence are $n_1<n_2<n_3<\ldots$ find $n_{12}$

Prove that the sum of the squares of any three pairwise different positive odd integers can be represented as the sum of the squares of six (not necessarily different) positive integers.

Given an $m\times n$ grid with unit squares coloured either black or white, a black square in the grid is [i]stranded [/i]if there is some square to its left in the same row that is white and there is some square above it in the same column that is white. Find a closed formula for the number of $2\times n$ grids with no stranded black square. Note that $n$ is any natural number and the formula must be in terms of $n$ with no other variables.

On a sphere with radius $1$, a point $ P $ is given. Three mutually perpendicular the rays emanating from the point $ P $ intersect the sphere at the points $ A $, $ B $ and $ C $. Prove that all such possible $ ABC $ planes pass through fixed point, and find the maximum possible area of the triangle $ ABC $

Let $n$ points be given inside a rectangle $R$ such that no two of them lie on a line parallel to one of the sides of $R$. The rectangle $R$ is to be dissected into smaller rectangles with sides parallel to the sides of $R$ in such a way that none of these rectangles contains any of the given points in its interior. Prove that we have to dissect $R$ into at least $n + 1$ smaller rectangles. [i]Proposed by Serbia[/i]

Find all pairs of integers $m, n \geq 2$ such that $$n\mid 1+m^{3^n}+m^{2\cdot 3^n}.$$

Let two ants stand on the perimeter of a regular $2019$-gon of unit side length. One of them stands on a vertex and the other one is on the midpoint of the opposite side. They start walking along the perimeter at the same speed counterclockwise. The locus of their midpoints traces out a figure $P$ in the plane with $N$ corners. Let the area enclosed by the convex hull of $P$ be $\tfrac{A}{B}\tfrac{\sin^m\left(\tfrac{\pi}{4038}\right)}{\tan\left(\tfrac{\pi}{2019}\right)}$, where $A$ and $B$ are coprime positive integers, and $m$ is the smallest possible positive integer such that this formula holds. Find $A+B+m+N$. [i]Note:[/i] The [i]convex hull[/i] of a figure $P$ is the convex polygon of smallest area which contains $P$.

Let $a$ be a positive real number and let $b= \sqrt[3] {a+ \sqrt {a^{2}+1}} + \sqrt[3] {a- \sqrt {a^{2}+1}}$. Prove that $b$ is a positive integer if, and only if, $a$ is a positive integer of the form $\frac{1}{2} n(n^{2}+3)$, for some positive integer $n$.