Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$f(xf(x + y)) = yf(x) + 1$$ holds for all $x, y \in \mathbb{R}^{+}$. [i]Proposed by Nikola Velov, North Macedonia[/i]

Six members of the team of Fatalia for the International Mathematical Olympiad are selected from $13$ candidates. At the TST the candidates got $a_1,a_2, \ldots, a_{13}$ points with $a_i \neq a_j$ if $i \neq j$. The team leader has already $6$ candidates and now wants to see them and nobody other in the team. With that end in view he constructs a polynomial $P(x)$ and finds the creative potential of each candidate by the formula $c_i = P(a_i)$. For what minimum $n$ can he always find a polynomial $P(x)$ of degree not exceeding $n$ such that the creative potential of all $6$ candidates is strictly more than that of the $7$ others? [i]Proposed by F. Petrov, K. Sukhov[/i]

Let $a$ and $b$ be the roots of the polynomial $x^2+2020x+c$. Given that $\frac{a}{b}+\frac{b}{a}=98$, compute $\sqrt c$.

Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that $$|f(A)\cap f(B)|=|A\cap B|$$ whenever $A$ and $B$ are two distinct subsets of $X$. [i] (Sergiu Novac)[/i]

The set $C$ of all complex numbers $z$ satisfying $(z +1)^2 = az$ for some $a \in [-10,3]$ is the union of two curves intersecting at a single point in the complex plane. If the sum of the lengths of these two curves is $\ell,$ find $\lfloor \ell \rfloor.$

Find all natural $n$ such that for some different natural $a, b, c$ and $d$ among numbers $$\frac{(a-c)(b-d)}{(b-c)(a-d)} , \frac{(b-c)(a-d)}{(a-c)(b-d)} , \frac{(a-b)(d-c)}{(a-d)(b-c)} , \frac{(a-c)(b-d)}{(a-b)(c-d)} ,$$ there are at least two numbers equal to $n$.

Let $a,b$ be two constants within the open interval $(0,\frac{1}{2})$. Find all continous functions $f(x)$ such that \[f(f(x))=af(x)+bx\] holds for all real $x$. This is much harder than the problems we had in the 1st TST...

A sequence $a_1,a_2,\ldots$ of positive integers satisfies the following properties.[list][*]$a_1 = 1$ [*]$a_{3n+1} = 2a_n + 1$ [*]$a_{n+1}\ge a_n$ [*]$a_{2001} = 200$[/list]Find the value of $a_{1000}$. [i]Note[/i]. In the original statement of the problem, there was an extra condition:[list][*]every positive integer appears at least once in the sequence.[/list]However, with this extra condition, there is no solution, i.e., no such sequence exists. (Try to prove it.) The problem as written above does have a solution.

Let $x,y,a,b$ be positive real numbers such that $x\not= y$, $x\not= 2y$, $y\not= 2x$, $a\not=3b$ and $\frac{2x-y}{2y-x}=\frac{a+3b}{a-3b}$. Prove that $\frac{x^2+y^2}{x^2-y^2}\ge 1$.

Let \( p_1, p_2, \cdots, p_{2025} \) be real numbers. For \( 1 \leq i \leq 2025 \), let \[\{a_n^{(i)}\}_{n \geq 0}\] be an infinite real sequence satisfying \[a_0^{(i)} = 0.\] It is known that: (1) \[a_1^{(1)}, a_1^{(2)}, \cdots, a_1^{(2025)}\] are not all zero. (2) For any integer \( n \geq 0 \) and any \( 1 \leq i \leq 2025 \), the following holds: \[p_i \cdot a_n^{(i+1)} = a_{n-1}^{(i)} + a_n^{(i)} + a_{n+1}^{(i)},\] where the sequence \[\{a_n^{(2026)}\}\] satisfies \[a_n^{(2026)} = a_n^{(1)}, \, n = 0, 1, 2, \cdots.\] Prove that there exists a positive real number \( r \) such that for infinitely many positive integers \( n \), \[\max \left\{ |a_n^{(1)}|, |a_n^{(2)}|, \cdots, |a_n^{(2025)}|\right\} \geq r.\]

Let $\, a,b \,$ be odd positive integers. Define the sequence $\, (f_n ) \,$ by putting $\, f_1 = a,$ $f_2 = b, \,$ and by letting $\, f_n \,$ for $\, n \geq 3 \,$ be the greatest odd divisor of $\, f_{n-1} + f_{n-2}$. Show that $\, f_n \,$ is constant for $\, n \,$ sufficiently large and determine the eventual value as a function of $\, a \,$ and $\, b$.

Let $a,b,c$ be three real numbers such that $1 \geq a \geq b \geq c \geq 0$. prove that if $\lambda$ is a root of the cubic equation $x^3 + ax^2 + bx + c = 0$ (real or complex), then $| \lambda | \leq 1.$

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [b]D1.[/b] The product of two positive integers is $5$. What is their sum? [b]D2.[/b] Gavin is $4$ feet tall. He walks $5$ feet before falling forward onto a cushion. How many feet is the top of Gavin’s head from his starting point? [b]D3.[/b] How many times must Nathan roll a fair $6$-sided die until he can guarantee that the sum of his rolls is greater than $6$? [b]D4 / Z1.[/b] What percent of the first $20$ positive integers are divisible by $3$? [b]D5.[/b] Let $a$ be a positive integer such that $a^2 + 2a + 1 = 36$. Find $a$. [b]D6 / Z2.[/b] It is said that a sheet of printer paper can only be folded in half $7$ times. A sheet of paper is $8.5$ inches by $11$ inches. What is the ratio of the paper’s area after it has been folded in half $7$ times to its original area? [b]D7 / Z3.[/b] Boba has an integer. They multiply the number by $8$, which results in a two digit integer. Bubbles multiplies the same original number by 9 and gets a three digit integer. What was the original number? [b]D8.[/b] The average number of letters in the first names of students in your class of $24$ is $7$. If your teacher, whose first name is Blair, is also included, what is the new class average? [b]D9 / Z4.[/b] For how many integers $x$ is $9x^2$ greater than $x^4$? [b]D10 / Z5.[/b] How many two digit numbers are the product of two distinct prime numbers ending in the same digit? [b]D11 / Z6.[/b] A triangle’s area is twice its perimeter. Each side length of the triangle is doubled,and the new triangle has area $60$. What is the perimeter of the new triangle? [b]D12 / Z7.[/b] Let $F$ be a point inside regular pentagon $ABCDE$ such that $\vartriangle FDC$ is equilateral. Find $\angle BEF$. [b]D13 / Z8.[/b] Carl, Max, Zach, and Amelia sit in a row with $5$ seats. If Amelia insists on sitting next to the empty seat, how many ways can they be seated? [b]D14 / Z9.[/b] The numbers $1, 2, ..., 29, 30$ are written on a whiteboard. Gumbo circles a bunch of numbers such that for any two numbers he circles, the greatest common divisor of the two numbers is the same as the greatest common divisor of all the numbers he circled. Gabi then does the same. After this, what is the least possible number of uncircled numbers? [b]D15 / Z10.[/b] Via has a bag of veggie straws, which come in three colors: yellow, orange, and green. The bag contains $8$ veggie straws of each color. If she eats $22$ veggie straws without considering their color, what is the probability she eats all of the yellow veggie straws? [b]Z11.[/b] We call a string of letters [i]purple[/i] if it is in the form $CVCCCV$ , where $C$s are placeholders for (not necessarily distinct) consonants and $V$s are placeholders for (not necessarily distinct) vowels. If $n$ is the number of purple strings, what is the remainder when $n$ is divided by $35$? The letter $y$ is counted as a vowel. [b]Z12.[/b] Let $a, b, c$, and d be integers such that $a+b+c+d = 0$ and $(a+b)(c+d)(ab+cd) = 28$. Find $abcd$. [b]Z13.[/b] Griffith is playing cards. A $13$-card hand with Aces of all $4$ suits is known as a godhand. If Griffith and $3$ other players are dealt $13$-card hands from a standard $52$-card deck, then the probability that Griffith is dealt a godhand can be expressed in simplest form as $\frac{a}{b}$. Find $a$. [b]Z14.[/b] For some positive integer $m$, the quadratic $x^2 + 202200x + 2022m$ has two (not necessarily distinct) integer roots. How many possible values of $m$ are there? [b]Z15.[/b] Triangle $ABC$ with altitudes of length $5$, $6$, and $7$ is similar to triangle $DEF$. If $\vartriangle DEF$ has integer side lengths, find the least possible value of its perimeter. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a,b\in\mathbb{R},~a>1,~b>0.$ Find the least possible value for $\alpha$ such that :$$(a+b)^x\geq a^x+b,~(\forall)x\geq\alpha.$$

Find all functions $ f: \mathbb{R}^{ \plus{} }\to\mathbb{R}^{ \plus{} }$ satisfying $ f\left(x \plus{} f\left(y\right)\right) \equal{} f\left(x \plus{} y\right) \plus{} f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ \plus{} }$ denotes the set of all positive reals. [i]Proposed by Paisan Nakmahachalasint, Thailand[/i]

Answer the following questions : $\textbf{(a)}$ Find all real solutions of the equation $$\Big(x^2-2x\Big)^{x^2+x-6}=1$$ Explain why your solutions are the only solutions. $\textbf{(b)}$ The following expression is a rational number. Find its value. $$\sqrt[3]{6\sqrt{3}+10} -\sqrt[3]{6\sqrt{3}-10}$$

Let $m$ and $n$ be two positive integers greater than $1$. Prove that there are $m$ positive integers $N_1$ , $\ldots$ , $N_m$ (some of them may be equal) such that \[\sqrt{m}=\sum_{i=1}^m{(\sqrt{N_i}-\sqrt{N_i-1})^{\frac{1}{n}}.}\]

Find the largest $c$ such that for any $\lambda \ge 1$ there is an a that satisfies the inequality $$\sin a + \sin (a\lambda ) \ge c.$$

Let $f : \mathbb R \to \mathbb R$ be a function such that $f(1)=1$ and \[f(x+y)=f(x)+f(y)\] And for all $x \in \mathbb R / \{0\}$ we have $f\left( \frac 1x \right) = \frac{1}{f(x)}.$ Find all such functions $f.$

Let $P(X)$ be a nonconstant polynomial with real coefficients such that for every rational number $q{}$ the equation $P(X)=q$ has no irrational solutions. Show that $P(X)$ is a first degree polynomial.

Let f be a rational function (i.e. the quotient of two real polynomials) and suppose that $f(n)$ is an integer for infinitely many integers n. Prove that f is a polynomial.

Find all real polynomials $P(x, y)$ such that $P(x+y, x-y) = 2P(x, y)$, for all $x, y$ in $R$.

A given infinite geometric series with first term $a \ne 0$ and common ratio $2r$ sums to a value that is $6$ times the sum of an infinite geometric series with first term $2a$ and common ratio $r$. Then $r = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Solve the system of equations in integers: \[ ab + 1 = (c+1)(d+1), \quad cd + 1 = (a-1)(b-1). \]

[b]p1.[/b] How many perfect squares are in the set: $\{1, 2, 4, 9, 10, 16, 17, 25, 36, 49\}$? [b]p2.[/b] If $a \spadesuit b = a^b - ab - 5$, what is the value of $2 \spadesuit 11$? [b]p3.[/b] Joe can catch $20$ fish in $5$ hours. Jill can catch $35$ fish in $7$ hours. If they work together, and the number of days it takes them to catch $900$ fish is represented by $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, what is $m + n$? Assume that they work at a constant rate without taking breaks and that there are an infinite number of fish to catch. [b]p4.[/b] What is the units digit of $187^{10}$? [b]p5.[/b] What is the largest number of regions we can create by drawing $4$ lines in a plane? [b]p6.[/b] A regular hexagon is inscribed in a circle. If the area of the circle is $2025\pi$, given that the area of the hexagon can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, $c$ where $gcd(a, c) = 1$ and $b$ is not divisible by the square of any number other than $1$, find $a + b + c$. [b]p7.[/b] Find the number of trailing zeroes in the product $3! \cdot 5! \cdot 719!$. [b]p8.[/b] How many ordered triples $(x, y, z)$ of odd positive integers satisfy $x + y + z = 37$? [b]p9.[/b] Let $N$ be a number with $2021$ digits that has a remainder of $1$ when divided by $9$. $S(N)$ is the sum of the digits of $N$. What is the value of $S(S(S(S(N))))$? [b]p10.[/b] Ayana rolls a standard die $10$ times. If the probability that the sum of the $10$ die is divisible by $6$ is $\frac{m}{n}$ for relatively prime positive integers $m$, $n$, what is $m + n$? [b]p11.[/b] In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. The inscribed circle touches the side $BC$ at point $D$. The line $AI$ intersects side $BC$ at point $K$ given that $I$ is the incenter of triangle $ABC$. What is the area of the triangle $KID$? [b]p12.[/b] Given the cubic equation $2x^3+8x^2-42x-188$, with roots $a, b, c$, evaluate $|a^2b+a^2c+ab^2+b^2c+c^2a+bc^2|$. [b]p13.[/b] In tetrahedron $ABCD$, $AB=6$, $BC=8$, $CA=10$, and $DA$, $DB$, $DC=20$. If the volume of $ABCD$ is $a\sqrt{b}$ where $a$, $b$ are positive integers and in simplified radical form, what is $a + b$? [b]p14.[/b] A $2021$-digit number starts with the four digits $2021$ and the rest of the digits are randomly chosen from the set $0$,$1$,$2$,$3$,$4$,$5$,$6$. If the probability that the number is divisible by $14$ is $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. what is $m + n$? [b]p15.[/b] Let $ABCD$ be a cyclic quadrilateral with circumcenter $O_1$ and circumradius $20$, Let the intersection of $AC$ and $BD$ be $E$. Let the circumcenter of $\vartriangle EDC$ be $O_2$. Given that the circumradius of 4EDC is $13$; $O_1O_2 = 11$, $BE = 11 \sqrt2$, find $O_1E^2$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].