Let $ P$ be a point in the plane of a triangle $ ABC$. Lines $ AP,BP,CP$ respectively meet lines $ BC,CA,AB$ at points $ A',B',C'$. Points $ A'',B'',C''$ are symmetric to $ A,B,C$ with respect to $ A',B',C',$ respectively. Show that: $ S_{A''B''C''}\equal{}3S_{ABC}\plus{}4S_{A'B'C'}$.

Let $ a_1$, $ \ldots$, $ a_k$, $ a_{k\plus{}1}$, $ \ldots$, $ a_n$ be $ n$ positive numbers ($ k<n$). Suppose that the values of $ a_{k\plus{}1}$, $ a_{k\plus{}2}$, $ \ldots$, $ a_n$ are fixed. Choose the values of $ a_1$, $ a_2$, $ \ldots$, $ a_k$ that minimize the sum $ \sum_{i, j, i\neq j}\frac{a_i}{a_j}$

How many triples $(p,q,n)$ are there such that $1/p+2013/q = n/5$ where $p$, $q$ are prime numbers and $n$ is a positive integer? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 4 $

Let $ABC$ be a triangle such that $AB<AC$. Let $D$ be a point on the segment $BC$ such that $BD<CD$. The angle bisectors of $\angle ADB$ and $\angle ADC$ meet the segments $AB$ and $AC$ at $E$ and $F$ respectively. Let $\omega$ be the circumcircle of $AEF$ and $M$ be the midpoint of $EF$. The ray $AD$ meets $\omega$ at $X$ and the line through $X$ parallel to $EF$ meets $\omega$ again at $Y$. If $YM$ meets $\omega$ at $T$, show that $AT$, $EF$ and $BC$ are concurrent. [i]Authored by Nikola Velov[/i]

$f$ is a function twice differentiable on $[0,1]$ and such that $f''$ is continuous. We suppose that : $f(1)-1=f(0)=f'(1)=f'(0)=0$. Prove that there exists $x_0$ on $[0,1]$ such that $|f''(x_0)| \geq 4$

1. Let $f(x)=x^2+bx+c$, M = {x | |f(x)|<1}. Prove $|M|\leq 2\sqrt{2}$ (|...| = length of interval(s))

A positive integer is called [i]monotone[/i] if has at least two digits and all its digits are nonzero and appear in a strictly increasing or strictly decreasing order. (a) Compute the sum of all monotone five-digit numbers. (b) Find the number of final zeros in the least common multiple of all monotone numbers (with any number of digits).

Can the equation $x^3 + ax^2 + bx + c = 0$ have only negative roots , if we know that $a+2b+4c=- \frac12 $?

Given a simple, connected graph with $n$ vertices and $m$ edges. Prove that one can find at least $m$ ways separating the set of vertices into two parts, such that the induced subgraphs on both parts are connected.

Show that for all primes $p$, \[\sum^{p-1}_{k=1}\left \lfloor \frac{k^{3}}{p}\right \rfloor =\frac{(p+1)(p-1)(p-2)}{4}.\]

A line through the origin passes through the curve whose equation is $5y=2x^2-9x+10$ at two points whose $x-$coordinates add up to $77.$ Find the slope of the line.

A circle $\omega$ passes through the two vertices $B$ and $C$ of a triangle $ABC.$ Furthermore, $\omega$ intersects segment $AC$ in $D\ne C$ and segment $AB$ in $E\ne B.$ On the ray from $B$ through $D$ lies a point $K$ such that $|BK| = |AC|,$ and on the ray from $C$ through $E$ lies a point $L$ such that $|CL| = |AB|.$ Show that the circumcentre $O$ of triangle $AKL$ lies on $\omega$.

Solve for $g : \mathbb{Z}^+ \to \mathbb{Z}^+$ such that $$g(m + n) = g(m) + mn(m + n) + g(n)$$ Show that $g(n)$ is of the form $\sum_{i=0}^{d} {c_i n^i}$ \\ and find necessary and sufficient conditions on $d$ and $c_0, c_1, \cdots , c_d$

Let $ f(x) \equal{} \prod^n_{k\equal{}1} (x \minus{} a_k) \minus{} 2,$ where $ n \geq 3$ and $ a_1, a_2, \ldots,$ an are distinct integers. Suppose that $ f(x) \equal{} g(x)h(x),$ where $ g(x), h(x)$ are both nonconstant polynomials with integer coefficients. Prove that $ n \equal{} 3.$

Find all triples of positive integers $x, y, z$ satisfying \[\frac{1}{x} +\frac{1}{y} + \frac{1}{z} = \frac{4}{5} .\]

What is the smallest positive integer that consists base 10 of each of the ten digits, each used exactly once, and is divisible by each of the digits $2$ through $9$?

Suppose that $a, b, c$ and $d$ are four different integers. Explain why $$(a - b)(a - c)(a - d)(b - c)(b -d)(c - d)$$ must be a multiple of $12$.

Find the greatest positive integer $k$ such that the following inequality holds for all $a,b,c\in\mathbb{R}^+$ satisfying $abc=1$ \[ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{k}{a+b+c+1}\geqslant 3+\frac{k}{4} \]

Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$

In a ABCD cyclic quadrilateral 4 points K, L ,M, N are taken on AB , BC , CD and DA , respectively such that KLMN is a parallelogram. Lines AD, BC and KM have a common point. And also lines AB, DC and NL have a common point. Prove that KLMN is rhombus.

[b]Q.[/b] Find all polynomials $P: \mathbb{R \times R}\to\mathbb{R\times R}$ with real coefficients, such that $$P(x,y) = P(x+y,x-y), \ \forall\ x,y \in \mathbb{R}.$$ [i]Proposed by TuZo[/i]

Let $1\leq a_1 < a_2 <... < a_{2n} \leq 4n-2$ be integers, such that their sum is even. Prove that for all sufficiently large n, there exist $\varepsilon_1 , ..., \varepsilon_{2n} = \pm1$ such that $$\sum\varepsilon_i = \sum\varepsilon_i a_i = 0$$

From a group of boys and girls, $15$ girls leave. There are then left two boys for each girl. After this $45$ boys leave. There are then $5$ girls for each boy. The number of girls in the beginning was: $\textbf{(A)}\ 40 \qquad \textbf{(B)}\ 43 \qquad \textbf{(C)}\ 29 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ \text{None of these}$

Calculate the minimum value of $ \text{tr} (A^tA) , $ where $ A $ in the cases where is a matrix of pairwise distinct nonnegative integers and: [b]a)[/b] $ \det A\equiv 1\pmod 2 $ [b]b)[/b] $ \det A=0 $ [i]Vlad Mihaly[/i]

[b]p1.[/b] What is the maximum possible value of $m$ such that there exist $m$ integers $a_1, a_2, ..., a_m$ where all the decimal representations of $a_1!, a_2!, ..., a_m!$ end with the same amount of zeros? [b]p2.[/b] Let $f : R \to R$ be a function such that $f(x) + f(y^2) = f(x^2 + y)$, for all $x, y \in R$. Find the sum of all possible $f(-2017)$. [b]p3. [/b] What is the sum of prime factors of $1000027$? [b]p4.[/b] Let $$\frac{1}{2!} +\frac{2}{3!} + ... +\frac{2016}{2017!} =\frac{n}{m},$$ where $n, m$ are relatively prime. Find $(m - n)$. [b]p5.[/b] Determine the number of ordered pairs of real numbers $(x, y)$ such that $\sqrt[3]{3 - x^3 - y^3} =\sqrt{2 - x^2 - y^2}$ [b]p6.[/b] Triangle $\vartriangle ABC$ has $\angle B = 120^o$, $AB = 1$. Find the largest real number $x$ such that $CA - CB > x$ for all possible triangles $\vartriangle ABC$. [b]p7. [/b]Jung and Remy are playing a game with an unfair coin. The coin has a probability of $p$ where its outcome is heads. Each round, Jung and Remy take turns to flip the coin, starting with Jung in round $ 1$. Whoever gets heads first wins the game. Given that Jung has the probability of $8/15$ , what is the value of $p$? [b]p8.[/b] Consider a circle with $7$ equally spaced points marked on it. Each point is $ 1$ unit distance away from its neighbors and labelled $0,1,2,...,6$ in that order counterclockwise. Feng is to jump around the circle, starting at the point $0$ and making six jumps counterclockwise with distinct lengths $a_1, a_2, ..., a_6$ in a way such that he will land on all other six nonzero points afterwards. Let $s$ denote the maximum value of $a_i$. What is the minimum possible value of $s$? [b]p9. [/b]Justin has a $4 \times 4 \times 4$ colorless cube that is made of $64$ unit-cubes. He then colors $m$ unit-cubes such that none of them belong to the same column or row of the original cube. What is the largest possible value of $m$? [b]p10.[/b] Yikai wants to know Liang’s secret code which is a $6$-digit integer $x$. Furthermore, let $d(n)$ denote the digital sum of a positive integer $n$. For instance, $d(14) = 5$ and $d(3) = 3$. It is given that $$x + d(x) + d(d(x)) + d(d(d(x))) = 999868.$$ Please find $x$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].