The positive reals $ x,y$ and $ z$ are satisfying the relation $ x \plus{} y \plus{} z\geq 1$. Prove that: $ \frac {x\sqrt {x}}{y \plus{} z} \plus{} \frac {y\sqrt {y}}{z \plus{} x} \plus{} \frac {z\sqrt {z}}{x \plus{} y}\geq \frac {\sqrt {3}}{2}$ [i]Proposer[/i]:[b] Baltag Valeriu[/b]

Let $a,b,c$ be positive real numbers such that \[ a+b+c\geq abc. \] Prove that at least two of the inequalities \[ \frac{2}{a}+\frac{3}{b}+\frac{6}{c}\geq6,\;\;\;\;\;\frac{2}{b}+\frac{3}{c}+\frac{6}{a}\geq6,\;\;\;\;\;\frac{2}{c}+\frac{3}{a}+\frac{6}{b}\geq6 \] are true.

Find all the functions $f: \mathbb{R} \to\mathbb{R}$ such that \[f(x-f(y))=f(f(y))+xf(y)+f(x)-1\] for all $x,y \in \mathbb{R} $.

For each positive integer $n$ (written with no leading zeros), let $t(n)$ equal the number formed by reversing the digits of $n$. For example, $t(461) = 164$ and $t(560) = 65$. For how many three-digit positive integers $m$ is $m + t(t(m))$ odd? [i]Proposed by David Altizio[/i]

In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?

Joe generates 4 independent random numbers in $(0,1)$ according to the uniform distribution. He shows one the numbers to Bill, who has to guess whether the number shown is one of the extremal numbers (that is, the smallest or the greatest) of the four numbers or not. Can Joe have a deterministic strategy such that no matter what Bill's method is, the probability of the right guess of Bill is at most $\frac12$?

Homer goes on the $100$-Donut Diet. A $100$-Donut Diet Plan specifies how many of $100$ total donuts Homer will eat each day. The diet requires that the number of donuts he eats does not increase from one day to the next. For example, one $5$-day Donut Diet Plan is $40$, $25$, $25$, $8$, $2$. Are there more $100$-Donut Diet Plans with an odd number of days or plans where Homer eats an odd number of donuts on the first day?

The numbers from 1 to 360 are partitioned into 9 subsets of consecutive integers and the sums of the numbers in each subset are arranged in the cells of a $3 \times 3$ square. Is it possible that the square turns out to be a magic square? Remark: A magic square is a square in which the sums of the numbers in each row, in each column and in both diagonals are all equal.

Given $12$ complex numbers $z_1,...,z_{12}$ st for each $1 \leq i \leq 12$: $$|z_i|=2 , |z_i - z_{i+1}| \geq 1$$ prove that : $$\sum_{1 \leq i \leq 12} \frac{1}{|z_i\overline{z_{i+1}}+1|^2} \geq \frac{1}{2}$$

10. Let $p=47$ be a prime. Call a function $f$ defined on the integers [i]lit[/i] if $f(x)$ is an integer from 1 to $p$ inclusive and $f(x+p)=f(x)$ for all integers $x$. How many [i]lit[/i] functions $g$ are there such that for all integers $x$, $p$ divides $g(x^2)-g(x)-x^8+x$? [i]Proposed by Monkey_king1[/i]

Ten distinct positive real numbers are given and the sum of each pair is written (So 45 sums). Between these sums there are 5 equal numbers. If we calculate product of each pair, find the biggest number $k$ such that there may be $k$ equal numbers between them.

When planning a trip from Temuco to the extreme north of the country, a truck driver notices that to cross the Atacama desert you must cross a distance of $800$ km between two stations consecutive service. Your truck can only store $50$ liters of benzene, and has a yield of $10$ km per liter. The trucker can leave gasoline stored in cans on the side of the road in different points along the way. For example, with an initial total charge of $50$ liters you can travel $100$ km, leave $30$ liters stored at the point you reached, and return to the starting point (with zero load) to refuel. The trucker decides to start the trip and arrives at the first service station with a zero load of fuel. a) Can the trucker cross the desert if at this service station the total supply is $140$ liters? b) Can the trucker cross the desert if the total supply of gasoline at the station is $180$ liters?

Let $ABC$ be an acute triangle with $AB<AC<BC$ and let $D$ be a point on it's extension of $BC$ towards $C$. Circle $c_1$, with center $A$ and radius $AD$, intersects lines $AC,AB$ and $CB$ at points $E,F$, and $G$ respectively. Circumscribed circle $c_2$ of triangle $AFG$ intersects again lines $FE,BC,GE$ and $DF$ at points $J,H,H' $ and $J'$ respectively. Circumscribed circle $c_3$ of triangle $ADE$ intersects again lines $FE,BC,GE$ and $DF$ at points $I,K,K' $ and $I' $ respectively. Prove that the quadrilaterals $HIJK$ and $H'I'J'K '$ are cyclic and the centers of their circumscribed circles coincide. by Evangelos Psychas, Greece

Prove that the inequality $x^2+y^2+1\ge 2(xy-x+y)$ is satisfied by any $x$, $y$ real numbers. Indicate when the equality is satisfied.

What is the greatest common divisor of the set of numbers \[\{{16}^{n}+10n-1 \; \vert \; n=1,2,\cdots \}?\]

In an increasing infinite sequence of positive integers, every term starting from the $2002$-th term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.

A sequence of real numbers $ \{a_{n}\}$ is defineds by $ a_{0}\neq 0,1$, $ a_1\equal{}1\minus{}a_0$,$ a_{n\plus{}1}\equal{}1\minus{}a_n(1\minus{}a_n)$, $ n\equal{}1,2,...$. Prove that for any positive integer $ n$, we have $ a_{0}a_{1}...a_{n}(\frac{1}{a_0}\plus{}\frac{1}{a_1}\plus{}...\plus{}\frac{1}{a_n})\equal{}1$

If $x, 2x+2, 3x+3, \dots$ are in geometric progression, the fourth term is: ${{ \textbf{(A)}\ -27 \qquad\textbf{(B)}\ -13\frac{1}{2} \qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\frac{1}{2} }\qquad\textbf{(E)}\ 27 } $

Define $ x\otimes y \equal{} x^3 \minus{} y$. What is $ h\otimes (h\otimes h)$? $ \textbf{(A) } \minus{} h\qquad \textbf{(B) } 0\qquad \textbf{(C) } h\qquad \textbf{(D) } 2h\qquad \textbf{(E) } h^3$

Let $ V$ be a bounded, closed, convex set in $ \mathbb{R}^n$, and denote by $ r$ the radius of its circumscribed sphere (that is, the radius of the smallest sphere that contains $ V$). Show that $ r$ is the only real number with the following property: for any finite number of points in $ V$, there exists a point in $ V$ such that the arithmetic mean of its distances from the other points is equal to $ r$. [i]Gy. Szekeres[/i]

[i]a)[/i] Let $M$ be the set of the lengths of the edges of an octahedron whose sides are congruent quadrangles. Prove that $M$ has at most three elements. [i]b)[/i] Let an octahedron whose sides are congruent quadrangles be given. Prove that each of these quadrangles has two equal sides meeting at a common vertex.

A positive integer is called [i]Tiranian[/i] if it can be written as $x^2+6xy+y^2$, where $x$ and $y$ are (not necessarily distinct) positive integers. The integer $36^{2023}$ is written as the sum of $k$ Tiranian integers. What is the smallest possible value of $k$? [i]Proposed by Miroslav Marinov, Bulgaria[/i]

Let $a_{1},a_{2},...,a_{n}$ be positive real number $(n \geq 2)$,not all equal,such that $\sum_{k=1}^n a_{k}^{-2n}=1$,prove that: $\sum_{k=1}^n a_{k}^{2n}-n^2.\sum_{1 \leq i<j \leq n}(\frac{a_{i}}{a_{j}}-\frac{a_{j}}{a_{i}})^2 >n^2$

At certain store, a package of 3 apples and 12 oranges costs 5 dollars, and a package of 20 apples and 5 oranges costs 13 dollars. Given that apples and oranges can only be bought in these two packages, what is the minimum nonzero amount of dollars that must be spent to have an equal number of apples and oranges? [i]Ray Li[/i]

[color=darkred]Let $g:\mathbb{R}\to\mathbb{R}$ be a continuous and strictly decreasing function with $g(\mathbb{R})=(-\infty,0)$ . Prove that there are no continuous functions $f:\mathbb{R}\to\mathbb{R}$ with the property that there exists a natural number $k\ge 2$ so that : $\underbrace{f\circ f\circ\ldots\circ f}_{k\text{ times}}=g$ . [/color]