Found problems: 85335
The volumes of the water containing in each of three big enough containers are integers. You are allowed only to relocate some times from one container to another the same volume of the water, that the destination already contains. Prove that you are able to discharge one of the containers.
Solve the system $\begin{cases} x+y=a \\
x^5 +y^5 = b^5
\end{cases}$
Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ continuous functions such that $\lim_{x\rightarrow \infty} f(x) =\infty$ and $\forall x,y\in \mathbb{R}, |x-y|>\varphi, \exists n<\varphi^{2023}, n\in \mathbb{N}$ such that
$$f^n(x)+f^n(y)=x+y$$
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that$$f(xf(y)-y) + f(xy-x) + f(x+y) = 2xy$$for all $x, y \in \mathbb{R}$.
Find all the real numbers $k$ that have the following property: For any non-zero real numbers $a$ and $b$, it is true that at least one of the following numbers: $$a, b,\frac{5}{a^2}+\frac{6}{b^3}$$is less than or equal to $k$.
Given a positive integer whose base-$10$ representation is $\overline{d_k\ldots d_0}$ for some integer $k \geq 0$, where $d_k \neq 0$, a move consists of selecting some integers $0 \leq i \leq j \leq k$, such that the digits $d_j,\ldots,d_i$ are not all $0$, erasing them from $n$, and replacing them with a divisor of $\overline{d_j\ldots d_i}$ (this divisor need not have the same number of digits as $\overline{d_j\ldots d_i}$).
Prove that for all sufficiently large even integers $n$, we may apply some sequence of moves to $n$ to transform it into $2024$.
[i]Allen Wang[/i]
A positive integer $t$ is called a Jane's integer if $t = x^3+y^2$ for some positive integers $x$ and $y$. Prove
that for every integer $n \ge 2$ there exist infinitely many positive integers $m$ such that the set of $n^2$ consecutive
integers $\{m+1,m+2,\dotsc,m+n^2\}$ contains exactly $n + 1$ Jane's integers.
Consider the set $S = \{ q + \frac{1}{q}, \text{ where } q \text{ ranges over all positive rational numbers} \}$.
(a) Let $N$ be a positive integer. Show that $N$ is the sum of two elements of $S$ if and only if $N$ is the product of two elements of $S$.
(b) Show that there exist infinitely many positive integers $N$ that cannot be written as the sum of two elements of $S$.
(c)Show that there exist infinitely many positive integers $N$ that can be written as the sum of two elements of $S$.
Le be a real number $ |a|<1, $ a natural number $ n\ge 2, $ and a $ 2\times 2 $ real matrix $ A $ that verifies
$$ \det \left( A^{2n} -aA^{2n-1} -aA+I \right)=0. $$ Show that $ \det A=1. $
Given a positive integrer number $n$ ($n\ne 0$), let $f(n)$ be the average of all the positive divisors of $n$. For example, $f(3)=\frac{1+3}{2}=2$, and $f(12)=\frac{1+2+3+4+6+12}{6}=\frac{14}{3}$.
[b]a[/b] Prove that $\frac{n+1}{2} \ge f(n)\ge \sqrt{n}$.
[b]b[/b] Find all $n$ such that $f(n)=\frac{91}{9}$.
Let $K, L, M, N$ be the midpoints of sides $AB, BC, CD, DA$ of a cyclic quadrilateral $ABCD$. Prove that the orthocentres of triangles $ANK, BKL, CLM, DMN$ are the vertices of a parallelogram.
For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$
[i]Proposed by Shahjalal Shohag, Bangladesh[/i]
Let $x \neq 0$ be a real number satisfying $ax^2+bx+c=0$ with $a,b,c \in \mathbb{Z}$ obeying $|a|+|b|+|c| > 1$. Then prove \[ |x| \geq \frac{1}{|a|+|b|+|c|-1}. \]
An array $ n\times n$ is given, consisting of $ n^2$ unit squares. A [i]pawn[/i] is placed arbitrarily on a unit square. The pawn can move from a square of the $ k$-th column to any square of the $ k$-th row. Show that there exists a sequence of $ n^2$ moves of the pawn so that all the unit squares of the array are visited once, the pawn returning to its original position.
[b]Dinu Serbanescu[/b]
Let $a,b,c$ be positive real numbers such that $(a+2b)(b+2c)=9$. Prove that\[\sqrt{\frac{a^2+b^2}{2}}+2\sqrt[3]{\frac{b^3+c^3}{2}}\geq 3.\]
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$,
\[f(a+d)+f(b-c)=f(a-d)+f(b+c).\]
[i]Calvin Deng.[/i]
Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $ (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0$. Prove that, for $ j \equal{} 1,2,3$, $\min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1$ holds.
Let $f(x)$ and $g(x)$ be given by
$f(x) = \frac{1}{x} + \frac{1}{x-2} + \frac{1}{x-4} + \cdots + \frac{1}{x-2018}$
$g(x) = \frac{1}{x-1} + \frac{1}{x-3} + \frac{1}{x-5} + \cdots + \frac{1}{x-2017}$.
Prove that $|f(x)-g(x)| >2$ for any non-integer real number $x$ satisfying $0 < x < 2018$.
In a contest, there are $m$ candidates and $n$ judges, where $n\geq 3$ is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most $k$ candidates. Prove that \[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]
Let $X$ be an $n$-element set and let $A_1,\ldots ,A_m$ be subsets of $X$ such that
i) $|A_i|=3$ for each $i=1,\ldots ,m$.
ii) $|A_i\cap A_j|\le 1$ for any two distinct indices $i,j$.
Show that there exists a subset of $X$ with at least $\lfloor\sqrt{2n}\rfloor$ elements which does not contain any of the $A_i$’s.
In a $2\times n$ array we have positive reals s.t. the sum of the numbers in each of the $n$ columns is $1$. Show that we can select a number in each column s.t. the sum of the selected numbers in each row is at most $\frac{n+1}4$.
Let $A=1,B=2,\ldots,Z=26$. Compute $BONANZA$, where the result is the product of the numbers represented by each letter.
[i]2018 CCA Math Bonanza Lightning Round #1.1[/i]
[list]
[*] A test was conducted in class. It is known that at least $\frac{2}{3}$ of the problems were hard. Each such problems were not solved by at least $\frac{2}{3}$ of the students. It is also known that at least $\frac{2}{3}$ of the students passed the test. Each such student solved at least $\frac{2}{3}$ of the suggested problems. Is this possible?
[*] Previous problem with $\frac{2}{3}$ replaced by $\frac{3}{4}$.
[*] Previous problem with $\frac{2}{3}$ replaced by $\frac{7}{10}$.[/list]
Isabella is making sushi. She slices a piece of salmon into the shape of a solid triangular prism. The prism is $2$ cm thick, and its triangular faces have side lengths $7$ cm, $ 24$cm, and $25$ cm. Find the volume of this piece of salmon in cm$^3$.
[i]Proposed by Isabella Li[/i]
[b]p1.[/b] Consider the equation $$x_1x_2 + x_2x_3 + x_3x_4 + · · · + x_{n-1}x_n + x_nx_1 = 0$$ where $x_i \in \{1,-1\}$ for $i = 1, 2, . . . , n$.
(a) Show that if the equation has a solution, then $n$ is even.
(b) Suppose $n$ is divisible by $4$. Show that the equation has a solution.
(c) Show that if the equation has a solution, then $n$ is divisible by $4$.
[b]p2.[/b] (a) Find a polynomial $f(x)$ with integer coefficients and two distinct integers $a$ and $b$ such that $f(a) = b$ and $f(b) = a$.
(b) Let $f(x)$ be a polynomial with integer coefficients and $a$, $b$, and $c$ be three integers. Suppose $f(a) = b$, $f(b) = c$, and $f(c) = a$. Show that $a = b = c$.
[b]p3.[/b] (a) Consider the triangle with vertices $M$ $(0, 2n + 1)$, $S$ $(1, 0)$, and $U \left(0, \frac{1}{2n^2}\right)$, where $n$ is a positive integer. If $\theta = \angle MSU$, prove that $\tan \theta = 2n - 1$.
(b) Find positive integers $a$ and $b$ that satisfy the following equation. $$arctan \frac18 = arctan \,\,a - arctan \,\, b$$
(c) Determine the exact value of the following infinite sum.
$$arctan \frac12 + arctan \frac18 + arctan \frac{1}{18} + arctan \frac{1}{32}+ ... + arctan \frac{1}{2n^2}+ ...$$
[b]p4.[/b] (a) Prove: $(55 + 12\sqrt{21})^{1/3} +(55 - 12\sqrt{21})^{1/3}= 5$.
(b) Completely factor $x^8 + x^6 + x^4 + x^2 + 1$ into polynomials with integer coefficients, and explain why your factorization is complete.
[b]p5.[/b] In this problem, we simulate a hula hoop as it gyrates about your waist. We model this situation by representing the hoop with a rotating a circle of radius $2$ initially centered at $(-1, 0)$, and representing your waist with a fixed circle of radius $1$ centered at the origin. Suppose we mark the point on the hoop that initially touches the fixed circle with a black dot (see the left figure).
As the circle of radius $2$ rotates, this dot will trace out a curve in the plane (see the right figure). Let $\theta$ be the angle between the positive x-axis and the ray that starts at the origin and goes through the point where the fixed circle and circle of radius $2$ touch. Determine formulas for the coordinates of the position of the dot, as functions $x(\theta)$ and $y(\theta)$. The left figure shows the situation when $\theta = 0$ and the right figure shows the situation when $\theta = 2pi/3$.
[img]https://cdn.artofproblemsolving.com/attachments/8/6/d15136872118b8e14c8f382bc21b41a8c90c66.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].