Let $a, b, c$ be real numbers such that: $$\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}= 1$$ Determine all values ​​which the following expression can take : $$\frac{a^2}{b + c} + \frac{b^2}{c + a} + \frac{c^2}{a + b}.$$

Two operations are allowed: (i) to write two copies of number $1$; (ii) to replace any two identical numbers $n$ by $(n + 1)$ and $(n - 1)$. Find the minimal number of operations that required to produce the number $2005$ (at the beginning there are no numbers). [i](8 points)[/i]

Let $ 101$ marbles be numbered from $ 1$ to $ 101$. The marbles are divided over two baskets $ A$ and $ B$. The marble numbered $ 40$ is in basket $ A$. When this marble is removed from basket $ A$ and put in $ B$, the averages of the numbers $ A$ and $ B$ both increase by $ \frac{1}{4}$. How many marbles were there originally in basket $ A?$

Let $a,b,c$ be non-negative real numbers such that $a^2+b^2+c^2 \le 3$ then prove that; $$(a+b+c)(a+b+c-abc)\ge2(a^2b+b^2c+c^2a)$$

( "Sisyphian Labour" ) There are $1001$ steps going up a hill , with rocks on some of them {no more than 1 rock on each step ) . Sisyphus may pick up any rock and raise it one or more steps up to the nearest empty step . Then his opponent Aid rolls a rock (with an empty step directly below it) down one step . There are $500$ rocks, originally located on the first $500$ steps. Sisyphus and Aid move rocks in turn , Sisyphus making the first move . His goal is to place a rock on the top step. Can Aid stop him? ( S . Yeliseyev)

Let $n$ be a positive integer and $p > n+1$ a prime. Prove that $p$ divides the following sum $S = 1^n + 2^n +...+ (p - 1)^n$

A circle $k$ passes through the vertices $B, C$ of a scalene triangle $ABC$. $k$ meets the extensions of $AB, AC$ beyond $B, C$ at $P, Q$ respectively. Let $A_1$ is the foot the altitude drop from $A$ to $BC$. Suppose $A_1P=A_1Q$. Prove that $\widehat{PA_1Q}=2\widehat{BAC}$.

Define a sequence of real numbers $ a_1$, $ a_2$, $ a_3$, $ \dots$ by $ a_1 = 1$ and $ a_{n + 1}^3 = 99a_n^3$ for all $ n \ge 1$. Then $ a_{100}$ equals $ \textbf{(A)}\ 33^{33} \qquad \textbf{(B)}\ 33^{99} \qquad \textbf{(C)}\ 99^{33} \qquad \textbf{(D)}\ 99^{99} \qquad \textbf{(E)}\ \text{none of these}$

Show that the sequence $ a_n \equal{} \frac {(n \plus{} 1)^nn^{2 \minus{} n}}{7n^2 \plus{} 1}$ is strictly monotonically increasing, where $ n \equal{} 0,1,2, \dots$.

In how many ways can we fill the cells of a $4\times4$ grid such that each cell contains exactly one positive integer and the product of the numbers in each row and each column is $2020$?

A fair standard six-sided dice is tossed three times. Given that the sum of the first two tosses equal the third, what is the probability that at least one $2$ is tossed? $\displaystyle \textbf{(A)} \ \frac{1}{6} \qquad \textbf{(B)} \ \frac{91}{216} \qquad \textbf{(C)} \ \frac{1}{2} \qquad \textbf{(D)} \ \frac{8}{15} \qquad \textbf{(E)} \ \frac{7}{12}$

Consider $x_1 > 0, y_1 > 0, x_2 < 0, y_2 > 0, x_3 < 0, y_3 < 0, x_4 > 0, y_4 < 0.$ Suppose that for each $i = 1, ... ,4$ we have $ (x_i -a)^2 +(y_i -b)^2 \le c^2$. Prove that $a^2 + b^2 < c^2$. Restate this fact in the form of geometric result in plane geometry.

Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$. [i]Proposed by Jozsef Pelikan, Hungary[/i]

Let ABC be an acute-angled triangle. Let D be a point on the segment BC, I the incentre of ABC. The circumcircle of ABD meets BI at P and the circumcircle of ACD meets CI at Q. If the area of PID and the area of QID are equal, prove that PI*QD=QI*PD.

A man rows at a speed of $2$ mph in still water. He set out on a trip towards a spot $2$ miles downstream. He rowed with the current until he was halfway there, then turned back and rowed against the current for $15$ minutes. Then, he turned around again and rowed with the current until he reached his destination. The entire trip took him $70$ minutes. The speed of the current can be represented as $\frac{p}{q}$ mph where $p,q$ are relatively prime positive integers. Find $10p+q$.

In the isosceles triangle $ABC$, with $AB=AC$, $D$ is a point on the extension of $CA$ such that $DB$ is perpendicular to $BC$, $E$ is a point on the extension of $BC$ such that $CE=2BC$, and $F$ is a point on $ED$ such that $FC$ is parallel to $AB$. Prove that $FA$ is parallel to $BC$.

Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of the line segment $RT$. Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$. Line $AJ$ meets $\Omega$ again at $K$. Prove that the line $KT$ is tangent to $\Gamma$. [i]Proposed by Charles Leytem, Luxembourg[/i]

Let $S$ be the set of all numbers which are the sum of the squares of three consecutive integers. Then we can say that: $\textbf{(A) }\text{No member of }S\text{ is divisible by }2\qquad$ $\textbf{(B) }\text{No member of }S\text{ is divisible by }3\text{ but some member is divisible by }11\qquad$ $\textbf{(C) }\text{No member of }S\text{ is divisible by }3\text{ or }5\qquad$ $\textbf{(D) }\text{No member of }S\text{ is divisible by }3\text{ or }7\qquad$ $\textbf{(E) }\text{None of these}$

Given is a triangle $ABC$ with $\angle BAC=45$; $AD, BE, CF$ are altitudes and $EF \cap BC=X$. If $AX \parallel DE$, find the angles of the triangle.

There are three positive integers written on a blackboard every minute. You can pick two written numbers $a$ and $b$ and replace them with $a \cdot b$ and $|a-b|$. Prove that it is always possible to make two of the numbers zero.

Let $x$ be a real number. Suppose that there exist integers $a_0,a_1,\dots,a_n$, not all zero, such that \[\sum_{k=0}^n a_k\cos(kx)=\sum_{k=0}^na_k\sin(kx)=0.\] Characterize all possible values of $\cos x$. [i]Grisham Paimagam[/i]

Let $X= \{A_1, A_2, A_3, A_4\}$ be a set of four distinct points in the plane. Show that there exists a subset $Y$ of $X$ with the property that there is no (closed) disk $K$ such that $K\cap X = Y$.

Once upon a time there are $n$ pairs of princes and princesses who are in love with each other. One day a witch comes along and turns all the princes into frogs; the frogs can be distinguished by sight but the princesses cannot tell which frog corresponds to which prince. The witch tells the princesses that if any of them kisses the frog that corresponds to the prince very that she loves then that frog will immediately transform back into a prince. If each princess can stand kissing at most $k$ frogs, what is the maximum number of princes they can be sure to save? (The princesses may take turns kissing in any order, communicate with each other and vary their strategy for future kisses depending on information gained from past kisses.)

Determine all the two-digit numbers that satisfy the following condition: if we multiply their two digits, the result is equal to half the number. For example, $24$ does not satisfy the condition, because $2 \times 4 = 8$ and $8$ is not half of $24$.

In a math test, there are easy and hard questions. The easy questions worth 3 points and the hard questions worth D points.\\ If all the questions begin to worth 4 points, the total punctuation of the test increases 16 points.\\ Instead, if we exchange the questions scores, scoring D points for the easy questions and 3 for the hard ones, the total punctuation of the test is multiplied by $\frac{3}{2}$.\\ Knowing that the number of easy questions is 9 times bigger the number of hard questions, find the number of questions in this test.