Prove that if $x, y, z$ are real numbers such that $x^2+y^2+z^2 = 2$, then \[x + y + z \leq xyz + 2.\]

Let $a,b,a_2,\ldots,a_{n-2}$ be real numbers with $ab\ne0$ such that all the roots of the equation $$ax^n-ax^{n-1}+a_2x^{n-2}+\ldots+a_{n-2}x^2-n^2bx+b=0$$are positive and real. Prove that these roots are all equal.

In the television program “Field of Miracles,” the presenter played the prize as follows. The player was shown three boxes, one of which contained a prize. The player pointed to one of the boxes, after which the leader opened one of the other two remaining boxes, which turned out to be empty. After this, the player could either insist on the original choice, or change it and choose the third box. In what case does his chance of winning increase? (There are three possible answers: both boxes are equal, it is better to keep the original choice, it is better to change it. Try to justify your answer.)

Let $ a$ be a real number. Prove that if the equation $ x^2\minus{}ax\plus{}a\equal{}0$ has two real roots $ x_1$ and $ x_2$, then: $ x_1^2\plus{}x_2^2 \ge 2(x_1\plus{}x_2).$

Let $ a, b, c$ be positive integers for which $ abc \equal{} 1$. Prove that $ \sum \frac{1}{b(a\plus{}b)} \ge \frac{3}{2}$.

A trader owns horses of $3$ races, and exacly $b_j$ of each race (for $j=1,2,3$). He want to leave these horses heritage to his $3$ sons. He knowns that the boy $i$ for horse $j$ (for $i,j=1,2,3$) would pay $a_{ij}$ golds, such that for distinct $i,j$ holds holds $a_{ii}> a_{ij}$ and $a_{jj} >a_{ij}$. Prove that there exists a natural number $n$ such that whenever it holds $\min\{b_1,b_2,b_3\}>n$, trader can give the horses to their sons such that after getting the horses each son values his horses more than the other brother is getting, individually.

Let $\vartriangle ABC$ be an equilateral triangle with side $1$. $1011$ points $P_1$, $P_2$, $P_3$, $...$, $P_{1011}$ on the side $AC$ and $1011$ points $Q_1$, $Q_2$, $Q_3$, $...$ ,$ Q_{1011}$ on side AB (see figure) in such a way as to generate $2023$ triangles of equal area. Find the length of the segment $AP_{1011}$. [img]https://cdn.artofproblemsolving.com/attachments/f/6/fea495c16a0b626e0c3882df66d66011a1a3af.png[/img] PS. Harder version of [url=https://artofproblemsolving.com/community/c4h3323135p30741470]2023 Chile NMO L1 P3[/url]

Let $a_0, a_1,...$ be a sequence of non-negative integers and $b_0, b_1,... $ be a sequence of non-negative integers defined by the following rule: $b_i=gcd(a_i, a_{i+1})$ for every $i=>0$ Is it possible every positive integer to occur exactly once in the sequence $b_0, b_1,... $

Let $ABCD$ be a convex quadrilateral and let $P$ be the intersection of the diagonals $AC$ and $BD$. The radii of the circles inscribed in the triangles $\vartriangle ABP$, $\vartriangle BCP$, $\vartriangle CDP$ and $\vartriangle DAP$ are the same. Prove that $ABCD$ is a rhombus,

The circumference of the circle with center $O$ is divided into 12 equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$? [asy] size(230); defaultpen(linewidth(0.65)); pair O=origin; pair[] circum = new pair[12]; string[] let = {"$A$","$B$","$C$","$D$","$E$","$F$","$G$","$H$","$I$","$J$","$K$","$L$"}; draw(unitcircle); for(int i=0;i<=11;i=i+1) { circum[i]=dir(120-30*i); dot(circum[i],linewidth(2.5)); label(let[i],circum[i],2*dir(circum[i])); } draw(O--circum[4]--circum[0]--circum[6]--circum[8]--cycle); label("$x$",circum[0],2.75*(dir(circum[0]--circum[4])+dir(circum[0]--circum[6]))); label("$y$",circum[6],1.75*(dir(circum[6]--circum[0])+dir(circum[6]--circum[8]))); label("$O$",O,dir(60)); [/asy] $\textbf{(A) }75\qquad\textbf{(B) }80\qquad\textbf{(C) }90\qquad\textbf{(D) }120\qquad \textbf{(E) }150$

A rising number, such as $ 34689$, is a positive integer each digit of which is larger than each of the digits to its left. There are $ \dbinom{9}{5} \equal{} 126$ five-digit rising numbers. When these numbers are arranged from smallest to largest, the $ 97$th number in the list does not contain the digit $ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$

Let $P(x)$ be a polynomial with real coefficients. Prove that not all roots of $x^3P(x)+1$ are real.

Let $f: \mathbb{R}\to \mathbb{R}$ be a real function. Prove or disprove each of the following statements. (a) If f is continuous and range(f)=$\mathbb{R}$ then f is monotonic (b) If f is monotonic and range(f)=$\mathbb{R}$ then f is continuous (c) If f is monotonic and f is continuous then range(f)=$\mathbb{R}$

Jane has $4$ pears, $5$ bananas, $3$ lemons, $1$ orange, and $6$ apples. If she uses one of each fruit to make a fruit smoothie, what is the total number of fruits that she has left?

Given two points $P,Q$ of the plane, denote $P+Q$ the midpoint of (possibly degenerate) segment $PQ$ and $P\cdot Q$ the image of $P$ in rotation around the origin $Q$ under $+90^\circ.$ a) Are these operations commutative? b) Given two distinct points $A,B$ the equation \[Y\cdot X=(A\cdot X)+B\] defines a map $X\mapsto Y.$ Determine what the mapping is. c) Construct all fixed points of the map from b).

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.

Prove that for any $ n$, there is a subset $ \{a_1,\dots,a_n\}$ of $ \mathbb N$ such that for each subset $ S$ of $ \{1,\dots,n\}$, $ \sum_{i\in S}a_i$ has the same set of prime divisors.

Two fair octahedral dice, each with the numbers $1$ through $8$ on their faces, are rolled. Let $N$ be the remainder when the product of the numbers showing on the two dice is divided by $8$. Find the expected value of $N$.

Let $\omega$ be the circumcircle of triangle $ABC$ with $AB<AC$. Let $I$ be its incenter and let $M$ be the midpoint of $BC$. The foot of the perpendicular from $M$ to $AI$ is $H$. The lines $MH, BI, AB$ form a triangle $T_b$ and the lines $MH, CI, AC$ form a triangle $T_c$. The circumcircle of $T_b$ meets $\omega$ at $B'$ and the circumcircle of $T_c$ meets $\omega$ at $C'$. Prove that $B', H, C'$ are collinear.

Determine the largest constant $K\geq 0$ such that $$\frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2$$ holds for all positive real numbers $a,b,c$ such that $ab+bc+ca=abc$. [i]Proposed by Orif Ibrogimov (Czech Technical University of Prague).[/i]

$a, b, c$ are real, and the sum of any two is greater than the third. Show that $\frac{2(a + b + c)(a^2 + b^2 + c^2)}{3} > a^3 + b^3 + c^3 + abc$.

Let $A=\{-2562,-2561,...,2561,2562\}$. Prove that for any bijection (1-1, onto function) $f:A\to A$, $$\sum_{k=1}^{2562}\left\lvert f(k)-f(-k)\right\rvert\text{ is maximized if and only if } f(k)f(-k)<0\text{ for any } k=1,2,...,2562.$$

Let $S$ be the set of positive integers not divisible by $p^4$ for all primes $p$. Anastasia and Bananastasia play a game. At the beginning, Anastasia writes down the positive integer $N$ on the board. Then the players take moves in turn; Bananastasia moves first. On any move of his, Bananastasia replaces the number $n$ on the blackboard with a number of the form $n-a$, where $a\in S$ is a positive integer. On any move of hers, Anastasia replaces the number $n$ on the blackboard with a number of the form $n^k$, where $k$ is a positive integer. Bananastasia wins if the number on the board becomes zero. Compute the second-smallest possible value of $N$ for which Anastasia can prevent Bananastasia from winning. [i]Proposed by Brandon Wang and Vincent Huang[/i]

In a convex quadrilateral $ABCD$, ${\angle}ABD=65^\circ$,${\angle}CBD=35^\circ$, ${\angle}ADC=130^\circ$ and $BC=AB$.Find the angles of $ABCD$.

A solid pyramid $TABCD$, with a quadrilateral base $ABCD$ is to be coloured on each of the five faces such that no two faces with a common edge will have the same colour. If five different colours are available, what is the number of ways to colour the pyramid?