Let $P$ be the set of points in the Euclidean plane, and let $L$ be the set of lines in the same plane. Does there exist an one-to-one mapping (injective function) $f : L \to P$ such that for each $\ell \in L$ we have $f(\ell) \in \ell$?

Every $x, 0 \leq x \leq 1$, admits a unique representation $x = \sum_{j=0}^{\infty} a_j 2^{-j}$, where all the $a_j$ belong to $\{0, 1\}$ and infinitely many of them are $0$. If $b(0) = \frac{1+c}{2+c}, b(1) =\frac{1}{2+c},c > 0$, and \[f(x)=a_0 + \sum_{j=0}^{\infty}b(a_0) \cdots b(a_j) a_{j+1}\] show that $0 < f(x) -x < c$ for every $x, 0 < x < 1.$

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

Reals $a,b,c,x,y$ satisfy $a^2+b^2+c^2=x^2+y^2=1$. Find the maximum value of $$(ax+by)^2+(bx+cy)^2$$

Find the positive integers $x_1, x_2, \ldots, x_{29}$ at least one of which is greater that 1988 so that \[ x^2_1 + x^2_2 + \ldots x^2_{29} = 29 \cdot x_1 \cdot x_2 \ldots x_{29}. \]

Find all complex numbers $z$ such that $|z^3+2-2i|+z\overline z|z|=2\sqrt2.$

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

If $ P(x)$ is a polynomial with integer coefficients and $ a$, $ b$, $ c$, three distinct integers, then show that it is impossible to have $ P(a)\equal{}b$, $ P(b)\equal{}c$, $ P(c)\equal{}a$.

Let $f(x)$ be a polynomial with integer co-efficients. Assume that $3$ divides the value $f(n)$ for each integer $n$. Prove that when $f(x)$ is divided by $x^3-x$ , the remainder is of the form $3r(x)$ where $r(x)$ is a polynomial with integer coefficients.

Find the prime factorization of $\textstyle\sum_{1\le i < j \le 100}ij$.

Find all polynomials $p(x)$ with integer coefficients such that for each positive integer $n$, the number $2^n - 1$ is divisible by $p(n)$.

Let the AP of the form $4$, $9$, $\ldots$ be $\mathbf{A}$, and the AP of the form $16$, $25$, $\ldots$ be $\mathbf{B}$. Find the number of integers from $1$ to $2024$ inclusive, that appear in only one of the AP's. For clarification, the AP's $\mathbf{A}$ and $\mathbf{B}$ start from 4 and 16 respectively. [hide=Answer]584[/hide]

Given five real numbers $u_0, u_1, u_2, u_3, u_4$, prove that it is always possible to find five real numbers $v0, v_1, v_2, v_3, v_4$ that satisfy the following conditions: $(i)$ $u_i-v_i \in \mathbb N, \quad 0 \leq i \leq 4$ $(ii)$ $\sum_{0 \leq i<j \leq 4} (v_i - v_j)^2 < 4.$ [i]Proposed by Netherlands.[/i]

Let $ \alpha,\beta,\gamma$ be the angles of a triangle. Prove that $csc^2\frac{\alpha}{2}+csc^2\frac{\beta}{2}+csc^2\frac{\gamma}{2} \ge 12$ and find the conditions for equality.

The sum of the solutions to the equation $$x^{\log_2 x} =\frac{64}{x}$$ can be written as$ \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.

Prove that for all positive real numbers $a,b,c,d$, $$\frac{2}{(a+b)(c+d)+(b+c)(a+d)} \leq \frac{1}{(a+c)(b+d)+4ac}+\frac{1}{(a+c)(b+d)+4bd}$$ and determine when equality occurs.

Find all functions $f$, of the form $f(x) = x^3 +px^2 +qx+r$ with $p$, $q$ and $r$ integers, such that $f(s) = 506$ for some integer $s$ and $f(\sqrt3) = 0$.

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]D16.[/b] The cooking club at Blair creates $14$ croissants and $21$ danishes. Daniel chooses pastries randomly, stopping when he gets at least one croissant and at least two danishes. How many pastries must he choose to guarantee that he has one croissant and two danishes? [b]D17.[/b] Each digit in a $3$ digit integer is either $1, 2$, or $4$ with equal probability. What is the probability that the hundreds digit is greater than the sum of the tens digit and the ones digit? [b]D18 / Z11.[/b] How many two digit numbers are there such that the product of their digits is prime? [b]D19 / Z9.[/b] In the coordinate plane, a point is selected in the rectangle defined by $-6 \le x \le 4$ and $-2 \le y \le 8$. What is the largest possible distance between the point and the origin, $(0, 0)$? [b]D20 / Z10.[/b] The sum of two numbers is $6$ and the sum of their squares is $32$. Find the product of the two numbers. [u]Set 5[/u] [b]D21 / Z12.[/b] Triangle $ABC$ has area $4$ and $\overline{AB} = 4$. What is the maximum possible value of $\angle ACB$? [b]D22 / Z13.[/b] Let $ABCD$ be an iscoceles trapezoid with $AB = CD$ and M be the midpoint of $AD$. If $\vartriangle ABM$ and $\vartriangle MCD$ are equilateral, and $BC = 4$, find the area of trapezoid $ABCD$. [b]D23 / Z14.[/b] Let $x$ and $y$ be positive real numbers that satisfy $(x^2 + y^2)^2 = y^2$. Find the maximum possible value of $x$. [b]D24 / Z17.[/b] In parallelogram $ABCD$, $\angle A \cdot \angle C - \angle B \cdot \angle D = 720^o$ where all angles are in degrees. Find the value of $\angle C$. [b]D25.[/b] The number $12ab9876543$ is divisible by $101$, where $a, b$ represent digits between $0$ and $9$. What is $10a + b$? [u]Set 6[/u] [b]D26 / Z26.[/b] For every person who wrote a problem that appeared on the final MBMT tests, take the number of problems they wrote, and then take that number’s factorial, and finally multiply all these together to get $n$. Estimate the greatest integer $a$ such that $2^a$ evenly divides $n$. [b]D27 / Z27.[/b] Circles of radius $5$ are centered at each corner of a square with side length $6$. If a random point $P$ is chosen randomly inside the square, what is the probability that $P$ lies within all four circles? [b]D28 / Z28.[/b] Mr. Rose’s evil cousin, Mr. Caulem, has teaches a class of three hundred bees. Every week, he tries to disrupt Mr. Rose’s $4$th period by sending three of his bee students to fly around and make human students panic. Unfortunately, no pair of bees can fly together twice, as then Mr. Rose will become suspicious and trace them back to Mr. Caulem. What’s the largest number of weeks Mr. Caulem can disrupt Mr. Rose’s class? [b]D29 / Z29. [/b]Two blind brothers Beard and Bored are driving their tractors in the middle of a field facing north, and both are $10$ meters west from a roast turkey. Beard, can turn exactly $0.7^o$ and Bored can turn exactly $0.2^o$ degrees. Driving at a consistent $2$ meters per second, they drive straight until they notice the smell of the turkey getting farther away, and then turn right and repeat until they get to the turkey. Suppose Beard gets to the Turkey in about $818.5$ seconds. Estimate the amount of time it will take Bored. [b]D30 / Z30.[/b] Let a be the probability that $4$ randomly chosen positive integers have no common divisor except for $1$. Estimate $300a$. Note that the integers $1, 2, 3, 4$ have no common divisor except for $1$. Remark. This problem is asking you to find $300 \lim_{n\to \infty} a_n$, if $a_n$ is defined to be the probability that $4$ randomly chosen integers from $\{1, 2, ..., n\}$ have greatest common divisor $1$. PS. You should use hide for answers. D.1-15 / Z.1-8 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916240p26045561]here [/url]and Z.15-25 [url=https://artofproblemsolving.com/community/c3h2916258p26045774]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

Let $a,\,b,$ and $c$ be real numbers such that $a+b+c=\frac1{a}+\frac1{b}+\frac1{c}$ and $abc=5$. The value of $$\left(a-\frac1{b}\right)^3+\left(b-\frac1{c}\right)^3+\left(c-\frac1{a}\right)^3$$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Let $f:\mathbb{R}^{\geq 0}\to\mathbb{R}$ be a function such that $f(x)-3x$ and $f(x)-x^3$ are ascendant functions. Prove that $f(x)-x^2-x$ is an ascendant function, too. (We call the function $g(x)$ ascendant, when for every $x\leq{y}$ we have $g(x)\leq{g(y)}$.)

Let $ f_m: \mathbb R \to \mathbb R$, $ f_m(x)\equal{}(m^2\plus{}m\plus{}1)x^2\minus{}2(m^2\plus{}1)x\plus{}m^2\minus{}m\plus{}1,$ where $ m \in \mathbb R$. 1) Find the fixed common point of all this parabolas. 2) Find $ m$ such that the distance from that fixed point to $ Oy$ is minimal.

Let a sequence of positive reals $\{u_n\}^{\infty}_{n=1}$ be given. For every positive integer $n$, let $k_n$ be the least positive integer satisfying: \[\sum^{k_n}_{i=1} \frac{1}{i} \geq \sum^n_{i=1} u_i.\] Show that the sequence $\left\{\frac{k_{n+1}}{k_n}\right\}$ has finite limit if and only if $\{u_n\}$ does.

Prove that the polynom $ X^3-aX-a+1 $ has three integer roots, for an infinite number of integers $ a. $ [i]Liviu Parsan[/i]