The two brothers, without waiting for the bus, decided to walk to the next stop. After passing $1/3$ of the way, they looked back and saw a bus approaching the stop. One of the brothers ran backwards, and the other ran forward at the same speed. It turned out that everyone ran to their stop exactly at the moment when the bus approached it. Find the speed of the brothers, if the bus speed is $30$ km / h, neglect the bus stop time.

Determine all values $\alpha\in\mathbb R$ with the following property: if positive numbers $(x,y,z)$ satisfy the inequality \[x^2+y^2+z^2\le\alpha(xy+yz+zx),\] then $x,y,z$ are sides of a triangle.

In every cell of the table $3 \times 3$ a monomial with a positive coefficient is written (cells (1,1); (2,3); (3,2) have the degree of two, cells (1,2);(2,1);(3,3) have a degree of one, cells (3,1);(2,2);(1,3) have a constant). Vuga added up monomials in every row and got three quadratic polynomials. It turned out that exactly $N$ of them have real roots. Leka added up monomials in every column and got three quadratic polynomials. It turned out that exactly $M$ of them have real roots. Find the maximum possible value of $N-M$.

Let \( f: \mathbb{R} \to \mathbb{R} \) be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of \( f \). It is known that the graph of \( f \) contains exactly \( N \) chords, one of which has length 2025. Find the minimum possible value of \( N \).

Let $k$ be a positive integer and $a_1, a_2,... , a_k$ be nonnegative real numbers. Initially, there is a sequence of $n \geq k$ zeros written on a blackboard. At each step, Nicole chooses $k$ consecutive numbers written on the blackboard and increases the first number by $a_1$, the second one by $a_2$, and so on, until she increases the $k$-th one by $a_k$. After a positive number of steps, Nicole managed to make all the numbers on the blackboard equal. Prove that all the nonzero numbers among $a_1, a_2, . . . , a_k$ are equal.

Evaluate \[\frac{1/2}{1+\sqrt2}+\frac{1/4}{1+\sqrt[4]2}+\frac{1/8}{1+\sqrt[8]2}+\frac{1/16}{1+\sqrt[16]2}+\cdots\] [i]Proposed by Ethan Tan[/i]

For each positive integer $ k$, find the smallest number $ n_{k}$ for which there exist real $ n_{k}\times n_{k}$ matrices $ A_{1}, A_{2}, \ldots, A_{k}$ such that all of the following conditions hold: (1) $ A_{1}^{2}= A_{2}^{2}= \ldots = A_{k}^{2}= 0$, (2) $ A_{i}A_{j}= A_{j}A_{i}$ for all $ 1 \le i, j \le k$, and (3) $ A_{1}A_{2}\ldots A_{k}\ne 0$.

Let $\alpha>1$ be a real number such that the sequence $a_n=\alpha\lfloor \alpha^n\rfloor- \lfloor \alpha^{n+1}\rfloor$, with $n\geq 1$, is periodic, that is, there is a positive integer $p$ such that $a_{n+p}=a_n$ for all $n$. Prove that $\alpha$ is an integer.

For a given real number $a$, define the sequence $(a_n)$ by $a_1 = a$ and $$a_{n+1} =\begin{cases} \dfrac12 \left(a_n -\dfrac{1}{a_n}\right) \,\,\, if \,\,\, a_n \ne 0, \\ 0 \,\,\, if \,\,\, a_n = 0 \end{cases}$$ Prove that the sequence $(a_n)$ contains infinitely many nonpositive terms.

Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[ xf(y) + f(xf(y)) - xf(f(y)) - f(xy) = 2x + f(y) - f(x+y)\] holds for all $x,y \in \mathbb{R}$.

For a real number $ x$ is $ [x]$ the next smaller integer to $ x$, that is the integer $ g$ with $ g\leqq<g+1$, and $ \{x\}=x-[x]$ is the “decimal part” of $ x$. Determine all triples $ (a,b,c)$ of real numbers, which fulfil the following system of equations: \[ \{a\}+[b]+\{c\}=2,9\]\[ \{b\}+[c]+\{a\}=5,3\]\[\{c\}+[a]+\{b\}=4,0\]

For a finite set $A$ of positive integers and its subset $B$, call $B$ a [i]half subset[/i] of $A$ when it satisfies the equation $\sum_{a\in A}a=2\sum_{b\in B}b$. For example, if $A=\{1,2,3\}$, then $\{1,2\}$ and $\{3\}$ are half subset of $A$. Determine all positive integers $n$ such that there exists a finite set $A$ which has exactly $n$ half subsets.

Find the real roots of the equation $$(5-x)^4+ (x-2)^ 4 = 17$$ and the real roots of a more general equation $$(a - x) ^4+ (x - b)^4 = c$$

Let $n$ ($n \ge 1$) be an integer. Consider the equation $2\cdot \lfloor{\frac{1}{2x}}\rfloor - n + 1 = (n + 1)(1 - nx)$, where $x$ is the unknown real variable. (a) Solve the equation for $n = 8$. (b) Prove that there exists an integer $n$ for which the equation has at least $2021$ solutions. (For any real number $y$ by $\lfloor{y} \rfloor$ we denote the largest integer $m$ such that $m \le y$.)

The numbers $ a_i $, $ b_i $, $ c_i $, $ d_i $ satisfy the conditions $ 0\leq c_i \leq a_i \leq b_i \leq d_i $ and $ a_i+b_i = c_i+d_i $ for $ i=1,2 ,\ldots,n$. Prove that $$ \prod_{i=1}^n a_i + \prod_{i=1}^n b_i \leq \prod_{i=1}^n c_i + \prod_{i=1}^n d_i$$

Let $S = \{\frac{\pi^n}{1992^m} | m,n \in \mathbb Z \}.$ Show that every real number $x \geq 0$ is an accumulation point of $S.$

Solve the equation ("$2$" encounters $1985$ times): $$\dfrac{x}{2+ \dfrac{x}{2+\dfrac{x}{2+... \dfrac{x}{2+\sqrt {1+x}}}}}=1$$

Find all $\mathbb Q$-linear maps $\Phi:\mathbb Q[x]\to\mathbb Q[x]$ such that for any irreducible polynomial $p\in\mathbb Q[x]$ the polynomial $\Phi(p)$ is also irreducible.

There are $ 7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ cards, which you place one by one in the boxes. The first card is placed in box #$1$, the second in box #$2$, and so forth up to the seventh card which is placed in box #$7$. You then start working back in the other direction, placing the eighth card in box #$6$, the ninth in box #$5$, up to the thirteenth card being placed in box #$1$. The fourteenth card is then placed in box #$2$, and this continues until every card is distributed. What box will the last card be placed in?

The lengths of the sides of the polygon are $a_1$, $a_2$,. $..$ ,$a_n$. The square trinomial $f(x)$ is such that $f(a_1) = f(a_2 +...+ a_n)$. Prove that if $A$ is the sum of the lengths of several sides of a polygon, $B$ is the sum of the lengths of its remaining sides, then $f(A) = f(B)$.

[b]p1.[/b] Roll two dice. What is the probability that the sum of the rolls is prime? [b]p2. [/b]Compute the sum of the first $20$ squares. [b]p3.[/b] How many integers between $0$ and $999$ are not divisible by $7, 11$, or $13$? [b]p4.[/b] Compute the number of ways to make $50$ cents using only pennies, nickels, dimes, and quarters. [b]p5.[/b] A rectangular prism has side lengths $1, 1$, and $2$. What is the product of the lengths of all of the diagonals? [b]p6.[/b] What is the last digit of $7^{6^{5^{4^{3^{2^1}}}}}$ ? [b]p7.[/b] Given square $ABCD$ with side length $3$, we construct two regular hexagons on sides $AB$ and $CD$ such that the hexagons contain the square. What is the area of the intersection of the two hexagons? [img]https://cdn.artofproblemsolving.com/attachments/f/c/b2b010cdd0a270bc10c6e3bb3f450ba20a03e7.png[/img] [b]p8.[/b] Brooke is driving a car at a steady speed. When she passes a stopped police officer, she begins decelerating at a rate of $10$ miles per hour per minute until she reaches the speed limit of $25$ miles per hour. However, when Brooke passed the police officer, he immediately began accelerating at a rate of $20$ miles per hour per minute until he reaches the rate of $40$ miles per hour. If the police officer catches up to Brooke after 3 minutes, how fast was Brooke driving initially? [b]p9.[/b] Find the ordered pair of positive integers $(x, y)$ such that $144x - 89y = 1$ and $x$ is minimal. [b]p10.[/b] How many zeroes does the product of the positive factors of $10000$ (including $1$ and $10000$) have? [b]p11.[/b] There is a square configuration of desks. It is known that one can rearrange these desks such that it has $7$ fewer rows but $10$ more columns, with $13$ desks remaining. How many desks are there in the square configuration? [b]p12.[/b] Given that there are $168$ primes with $3$ digits or less, how many numbers between $1$ and $1000$ inclusive have a prime number of factors? [b]p13.[/b] In the diagram below, we can place the integers from $1$ to $19$ exactly once such that the sum of the entries in each row, in any direction and of any size, is the same. This is called the magic sum. It is known that such a configuration exists. Compute the magic sum. [img]https://cdn.artofproblemsolving.com/attachments/3/4/7efaa5ba5ad250e24e5ad7ef03addbf76bcfb4.png[/img] [b]p14.[/b] Let $E$ be a random point inside rectangle $ABCD$ with side lengths $AB = 2$ and $BC = 1$. What is the probability that angles $ABE$ and $CDE$ are both obtuse? [b]p15.[/b] Draw all of the diagonals of a regular $13$-gon. Given that no three diagonals meet at points other than the vertices of the $13$-gon, how many intersection points lie strictly inside the $13$-gon? [b]p16.[/b] A box of pencils costs the same as $11$ erasers and $7$ pencils. A box of erasers costs the same as $6$ erasers and a pencil. A box of empty boxes and an eraser costs the same as a pencil. Given that boxes cost a penny and each of the boxes contain an equal number of objects, how much does it costs to buy a box of pencils and a box of erasers combined? [b]p17.[/b] In the following figure, all angles are right angles and all sides have length $1$. Determine the area of the region in the same plane that is at most a distance of $1/2$ away from the perimeter of the figure. [img]https://cdn.artofproblemsolving.com/attachments/6/2/f53ae3b802618703f04f41546e3990a7d0640e.png[/img] [b]p18.[/b] Given that $468751 = 5^8 + 5^7 + 1$ is a product of two primes, find both of them. [b]p19.[/b] Your wardrobe contains two red socks, two green socks, two blue socks, and two yellow socks. It is currently dark right now, but you decide to pair up the socks randomly. What is the probability that none of the pairs are of the same color? [b]p20.[/b] Consider a cylinder with height $20$ and radius $14$. Inside the cylinder, we construct two right cones also with height $20$ and radius $14$, such that the two cones share the two bases of the cylinder respectively. What is the volume ratio of the intersection of the two cones and the union of the two cones? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Denote by $\mathbb{S}$ the set of all proper subsets of $\mathbb{Z}_{>0}$. Find all functions $f : \mathbb{S} \mapsto \mathbb{Z}_{>0}$ that satisfy the following:\\ [color=#FFFFFF]___[/color]1. For all sets $A, B \in \mathbb{S}$ we have \[f(A \cap B) = \text{min}(f(A), f(B)).\] [color=#FFFFFF]___[/color]2. For all positive integers $n$ we have \[\sum \limits_{X \subseteq [1, n]} f(X) = 2^{n+1}-1.\] (Here, by a proper subset $X$ of $\mathbb{Z}_{>0}$ we mean $X \subset \mathbb{Z}_{>0}$ with $X \ne \mathbb{Z}_{>0}$. It is allowed for $X$ to have infinite size.) \\ [i]Proposed by MV Adhitya, Kanav Talwar, Siddharth Choppara, Archit Manas[/i]

Solve the system of equations $\begin{cases} x^3+\frac13 y=x^2+x -\frac43 \\ y^3+\frac14 z=y^2+y -\frac54 \\ z^3+\frac15 x=z^2+z -\frac65 \end{cases}$

prove for all $k> 1$ equation $(x+1)(x+2)...(x+k)=y^{2}$ has finite solutions.

Determine all pairs of non-constant polynomials $p(x)$ and $q(x)$, each with leading coefficient $1$, degree $n$, and $n$ roots which are non-negative integers, that satisfy $p(x)-q(x)=1$.