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?

A four-digit integer in base $10$ is [i]friendly[/i] if its digits are four consecutive digits in any order. A four-digit integer is [i]shy[/i] if there exist two adjacent digits in its representation that differ by $1.$ Compute the number of four-digit integers that are both friendly and shy.

Find all positive integer solutions $(x, y, z)$ of the equation $1 + 2^x \cdot 7^y=z^2$.

Let the sum of positive reals $a,b,c$ be equal to 1. Prove an inequality \[ \sqrt{{ab}\over {c+ab}}+\sqrt{{bc}\over {a+bc}}+\sqrt{{ac}\over {b+ac}}\le 3/2 \]. [b]proposed by Fedor Petrov[/b]

We call a tiling of an $m\times$ n rectangle with arabos (see figure below) [i]regular[/i] if there is no sub-rectangle which is tiled with arabos. Prove that if for some $m$ and $n$ there exists a [i]regular[/i] tiling of the $m\times n$ rectangle then there exists a [i]regular[/i] tiling also for the $2m \times 2n$ rectangle. [img]https://cdn.artofproblemsolving.com/attachments/1/1/2ab41cc5107a21760392253ed52d9e4ecb22d1.png[/img]

Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^2$