Find all real $\alpha$ s.t. \[ [ \sqrt{n + \alpha} + \sqrt{n} ] = [ \sqrt{4n+1} ] \] holds for all natural numbers $n$

Prove that from $x + y = 1 \ (x, y \in \mathbb R)$ it follows that \[x^{m+1} \sum_{j=0}^n \binom{m+j}{j} y^j + y^{n+1} \sum_{i=0}^m \binom{n+i}{i} x^i = 1 \qquad (m, n = 0, 1, 2, \ldots ).\]

Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Compute $\displaystyle \sum_{i=1}^{\phi(2023)} \dfrac{\gcd(i,\phi(2023))}{\phi(2023)}$. [i]Proposed by Giacomo Rizzo[/i]

Consider function $ f: R\to R$ which satisfies the conditions for any mutually distinct real numbers $ a,b,c,d$ satisfying $ \frac {a \minus{} b}{b \minus{} c} \plus{} \frac {a \minus{} d}{d \minus{} c} \equal{} 0$, $ f(a),f(b),f(c),f(d)$ are mutully different and $ \frac {f(a) \minus{} f(b)}{f(b) \minus{} f(c)} \plus{} \frac {f(a) \minus{} f(d)}{f(d) \minus{} f(c)} \equal{} 0.$ Prove that function $ f$ is linear

Solve the equation $\cos^n{x}-\sin^n{x}=1$ where $n$ is a natural number.

For real numbers $x$, $y$ and $z$, solve the system of equations: $$x^3+y^3=3y+3z+4$$ $$y^3+z^3=3z+3x+4$$ $$x^3+z^3=3x+3y+4$$

In terms of real parameter $a$ solve inequality: $\log _{a} {x} + \mid a+\log _{a} {x} \mid \cdot \log _{\sqrt{x}} {a} \geq a\log _{x} {a}$ in set of real numbers

[b]p1.[/b] Find the sum of the seventeenth powers of the seventeen roots of the seventeeth degree polynomial equation $x^{17} - 17x + 17 = 0$. [b]p2.[/b] Four identical spherical cows, each of radius $17$ meters, are arranged in a tetrahedral pyramid (their centers are the vertices of a regular tetrahedron, and each one is tangent to the other three). The pyramid of cows is put on the ground, with three of them laying on it. What is the distance between the ground and the top of the topmost cow? [b]p3.[/b] If $a_n$ is the last digit of $\sum^{n}_{i=1} i$, what would the value of $\sum^{1000}_{i=1}a_i$ be? [b]p4.[/b] If there are $15$ teams to play in a tournament, $2$ teams per game, in how many ways can the tournament be organized if each team is to participate in exactly $5$ games against dierent opponents? [b]p5.[/b] For $n = 20$ and $k = 6$, calculate $$2^k {n \choose 0}{n \choose k}- 2^{k-1}{n \choose 1}{{n - 1} \choose {k - 1}} + 2^{k-2}{n \choose 2}{{n - 2} \choose {k - 2}} +...+ (-1)^k {n \choose k}{{n - k} \choose 0}$$ where ${n \choose k}$ is the number of ways to choose $k$ things from a set of $n$. [b]p6.[/b] Given a function $f(x) = ax^2 + b$, with a$, b$ real numbers such that $$f(f(f(x))) = -128x^8 + \frac{128}{3}x^6 - \frac{16}{22}x^2 +\frac{23}{102}$$ , find $b^a$. [b]p7.[/b] Simplify the following fraction $$\frac{(2^3-1)(3^3-1)...(100^3-1)}{(2^3+1)(3^3+1)...(100^3+1)}$$ [b]p8.[/b] Simplify the following expression $$\frac{\sqrt{3 + \sqrt5} + \sqrt{3 - \sqrt5}}{\sqrt{3 - \sqrt8}} -\frac{4}{ \sqrt{8 - 2\sqrt{15}}}$$ [b]p9.[/b] Suppose that $p(x)$ is a polynomial of degree $100$ such that $p(k) = k2^{k-1}$ , $k =1, 2, 3 ,... , 100$. What is the value of $p(101)$ ? [b]p10. [/b] Find all $17$ real solutions $(w, x, y, z)$ to the following system of equalities: $$ 2w + w^2x = x$$ $$ 2x + x^2y=y $$ $$ 2y + y^2z=z $$ $$ -2z+z^2w=w $$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p(x)$ and $q(x)$ be non-constant polynomial functions with integer coeffcients. It is known that the polynomial $p(x)q(x) - 2015$ has at least $33$ different integer roots. Prove that neither $p(x)$ nor $q(x)$ can be a polynomial of degree less than three.

[b]p1.[/b] What is the units digit of $1 + 9 + 9^2 +... + 9^{2015}$ ? [b]p2.[/b] In Fourtown, every person must have a car and therefore a license plate. Every license plate must be a $4$-digit number where each digit is a value between $0$ and $9$ inclusive. However $0000$ is not a valid license plate. What is the minimum population of Fourtown to guarantee that at least two people who have the same license plate? [b]p3.[/b] Two sides of an isosceles triangle $\vartriangle ABC$ have lengths $9$ and $4$. What is the area of $\vartriangle ABC$? [b]p4.[/b] Let $x$ be a real number such that $10^{\frac{1}{x}} = x$. Find $(x^3)^{2x}$. [b]p5.[/b] A Berkeley student and a Stanford student are going to visit each others campus and go back to their own campuses immediately after they arrive by riding bikes. Each of them rides at a constant speed. They first meet at a place $17.5$ miles away from Berkeley, and secondly $10$ miles away from Stanford. How far is Berkeley away from Stanford in miles? [b]p6.[/b] Let $ABCDEF$ be a regular hexagon. Find the number of subsets $S$ of $\{A,B,C,D,E, F\}$ such that every edge of the hexagon has at least one of its endpoints in $S$. [b]p7.[/b] A three digit number is a multiple of $35$ and the sum of its digits is $15$. Find this number. [b]p8.[/b] Thomas, Olga, Ken, and Edward are playing the card game SAND. Each draws a card from a $52$ card deck. What is the probability that each player gets a dierent rank and a different suit from the others? [b]p9.[/b] An isosceles triangle has two vertices at $(1, 4)$ and $(3, 6)$. Find the $x$-coordinate of the third vertex assuming it lies on the $x$-axis. [b]p10.[/b] Find the number of functions from the set $\{1, 2,..., 8\}$ to itself such that $f(f(x)) = x$ for all $1 \le x \le 8$. [b]p11.[/b] The circle has the property that, no matter how it's rotated, the distance between the highest and the lowest point is constant. However, surprisingly, the circle is not the only shape with that property. A Reuleaux Triangle, which also has this constant diameter property, is constructed as follows. First, start with an equilateral triangle. Then, between every pair of vertices of the triangle, draw a circular arc whose center is the $3$rd vertex of the triangle. Find the ratio between the areas of a Reuleaux Triangle and of a circle whose diameters are equal. [b]p12.[/b] Let $a$, $b$, $c$ be positive integers such that gcd $(a, b) = 2$, gcd $(b, c) = 3$, lcm $(a, c) = 42$, and lcm $(a, b) = 30$. Find $abc$. [b]p13.[/b] A point $P$ is inside the square $ABCD$. If $PA = 5$, $PB = 1$, $PD = 7$, then what is $PC$? [b]p14.[/b] Find all positive integers $n$ such that, for every positive integer $x$ relatively prime to $n$, we have that $n$ divides $x^2 - 1$. You may assume that if $n = 2^km$, where $m$ is odd, then $n$ has this property if and only if both $2^k$ and $m$ do. [b]p15.[/b] Given integers $a, b, c$ satisfying $$abc + a + c = 12$$ $$bc + ac = 8$$ $$b - ac = -2,$$ what is the value of $a$? [b]p16.[/b] Two sides of a triangle have lengths $20$ and $30$. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side? [b]p17.[/b] Find the number of non-negative integer solutions $(x, y, z)$ of the equation $$xyz + xy + yz + zx + x + y + z = 2014.$$ [b]p18.[/b] Assume that $A$, $B$, $C$, $D$, $E$, $F$ are equally spaced on a circle of radius $1$, as in the figure below. Find the area of the kite bounded by the lines $EA$, $AC$, $FC$, $BE$. [img]https://cdn.artofproblemsolving.com/attachments/7/7/57e6e1c4ef17f84a7a66a65e2aa2ab9c7fd05d.png[/img] [b]p19.[/b] A positive integer is called cyclic if it is not divisible by the square of any prime, and whenever $p < q$ are primes that divide it, $q$ does not leave a remainder of $1$ when divided by $p$. Compute the number of cyclic numbers less than or equal to $100$. [b]p20.[/b] On an $8\times 8$ chess board, a queen can move horizontally, vertically, and diagonally in any direction for as many squares as she wishes. Find the average (over all $64$ possible positions of the queen) of the number of squares the queen can reach from a particular square (do not count the square she stands on). PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

During the van ride from the Grand Canyon to the beach, Michael asks his dad about the costs of renewable energy resources. "How much more does it really cost for a family like ours to switch entirely to renewable energy?" Jerry explains, "Part of that depends on where the family lives. In the Western states, solar energy pays off more than it does where we live in the Southeast. But as technology gets better, costs of producing more photovoltaic power go down, so in just a few years more people will have reasonably inexpensive options for switching to clearner power sources. Even now most families could switch to biomass for between $\$200$ and $\$1000$ per year. The energy comes from sawdust, switchgrass, and even landfill gas. We pay that premium ourselves, but some families operate on a tighter budget, or don't understand the alternatives yet." "Ew, landfill gas!" Alexis complains mockingly. Wanting to save her own energy, Alexis decides to take a nap. She falls asleep and dreams of walking around a $2-\text{D}$ coordinate grid, looking for a wormhole that she believes will transport her to the beach (bypassing the time spent in the family van). In her dream, Alexis finds herself holding a device that she recognizes as a $\textit{tricorder}$ from one of the old $\textit{Star Trek}$ t.v. series. The tricorder has a button labeled "wormhole" and when Alexis presses the button, a computerized voice from the tricorder announces, "You are at the origin. Distance to the wormhole is $2400$ units. Your wormhole distance allotment is $\textit{two}$."' Unsure as to how to reach, Alexis begins walking forward. As she walks, the tricorder displays at all times her distance from her starting point at the origin. When Alexis is $2400$ units from the origin, she again presses the "wormhole" buttom. The same computerized voice as before begins, "Distance to the origin is $2400$ units. Distance to the wormhole is $3840$ units. Your wormhole distance allotment is $\textit{two}$." Alexis begins to feel disoriented. She wonders what is means that her $\textit{wormhole distance allotment is two}$, and why that number didn't change as she pushed the button. She puts her hat down to mark her position, then wanders aroud a bit. The tricorder shows her two readings as she walks. The first she recognizes as her distance to the origin. The second reading clearly indicates her distance from the point where her hat lies - where she last pressed the button that gave her distance to the wormhole. Alexis picks up her hat and begins walking around. Eventually Alexis finds herself at a spot $2400$ units from the origin and $3840$ units from where she last pressed the button. Feeling hopeful, Alexis presses the tricorder's wormhole button again. Nothing happens. She presses it again, and again nothing happens. "Oh," she thinks, "my wormhole allotment was $\textit{two}$, and I used it up already!" Despair fills poor Alexis who isn't sure what a wormhole looks like or how she's supposed to find it. Then she takes matters into her own hands. Alexis sits down and scribbles some notes and realizes where the wormhole must be. Alexis gets up and runs straight from her "third position" to the wormhole. As she gets closer, she sees the wormhole, which looks oddly like a huge scoop of icecream. Alexis runs into the wormhole, then wakes up. How many units did Alexis run from her third position to the wormhole?

Solve in natural numbers the system of equations $3x^2+6y^2+5z^2=1997$ and $3x+6y+5z=161$ .

Given an integer $a_1$($a_1 \neq -1$), find a real number sequence $\{ a_n \}$($a_i \neq 0, i=1,2,\cdots,5$) such that $x_1,x_2,\cdots,x_5$ and $y_1,y_2,\cdots,y_5$ satisfy $b_{i1}x_1+b_{i2}x_2+\cdots +b_{i5}x_5=2y_i$, $i=1,2,3,4,5$, then $x_1y_1+x_2y_2+\cdots+x_5y_5=0$, where $b_{ij}=\prod_{1 \leq k \leq i} (1+ja_k)$.

The infinite sequence $P_0(x),P_1(x),P_2(x),\ldots,P_n(x),\ldots$ is defined as \[P_0(x)=x,\quad P_n(x) = P_{n-1}(x-1)\cdot P_{n-1}(x+1),\quad n\ge 1.\] Find the largest $k$ such that $P_{2014}(x)$ is divisible by $x^k$.

Let $n$ be a natural number and $\alpha ,\beta ,\gamma$ be the angles of an acute triangle. Determine the least possible value of the sum: $T=tan^n \alpha+tan^n \beta+tan^n \gamma$.

You have a set of five weights, together with a balance that allows you to compare the weight of two things. The weights are known to be $10$, $20$,$30$,$40$ and $50$ grams, but are otherwise identical except for their labels. The $10$ and $50$ gram weights are clearly labelled, but the labels have been erased on the remaining weights. Using the balance exactly once, is it possible to determine what one of the three unlabelled weights is? If so, explain how, and if not, explain why not.

Find all real solutions to the equation $$(x+1)^{2001}+(x+1)^{2000}(x-2)+(x+1)^{1999}(x-2)^2+...+(x+1)^2(x-2)^{1999}+(x+1)^{2000}(x-2)+(x+1)^{2001}=0$$

Let $f(x) = \frac{4^x}{4^x+2}$ for $x > 0$. Evaluate $\sum_{k=1}^{1920}f\left(\frac{k}{1921}\right)$

Let $\{u_n \}$ be the sequence defined by its first two terms $u_0, u_1$ and the recursion formula \[u_{n+2 }= u_n - u_{n+1}.\] [b](a)[/b] Show that $u_n$ can be written in the form $u_n = \alpha a^n + \beta b^n$, where $a, b, \alpha, \beta$ are constants independent of $n$ that have to be determined. [b](b)[/b] If $S_n = u_0 + u_1 + \cdots + u_n$, prove that $S_n + u_{n-1}$ is a constant independent of $n.$ Determine this constant.

Proove that in every five positive numbers there is a pair, say $a,b$, for which $$\left| \frac{1}{a+25}- \frac{1}{b+25}\right| <\frac{1}{100}.$$

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

Let $\alpha, \beta, \gamma \in C$ be the roots of the polynomial $x^3 - 3x2 + 3x + 7$. For any complex number $z$, let $f(z)$ be defined as follows: $$f(z) = |z -\alpha | + |z - \beta|+ |z-\gamma | - 2 \underbrace{\max}_{w \in \{\alpha, \beta, \gamma}\} |z - w|.$$ Let $A$ be the area of the region bounded by the locus of all $z \in C$ at which $f(z)$ attains its global minimum. Find $\lfloor A \rfloor$.

Let $M,a,b,r$ be non-negative integers with $a,r\ge 2$, and suppose there exists a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ satisfying the following conditions: (1) For all $n\in \mathbb{Z}$, $f^{(r)}(n)=an+b$ where $f^{(r)}$ denotes the composition of $r$ copies of $f$ (2) For all $n\ge M$, $f(n)\ge 0$ (3) For all $n>m>M$, $n-m|f(n)-f(m)$ Show that $a$ is a perfect $r$-th power.

Let $a,b,c$ be positive real numbers. Prove that $\sqrt{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt{\frac{c}{a+b}+\frac{a}{b+c}}\ge 3$

The sum of non-negative numbers $ x $, $ y $ and $ z $ is $3$. Prove the inequality $$ {1 \over x ^ 2 + y + z} + {1 \over x + y ^ 2 + z} + {1 \over x + y + z ^ 2} \leq 1. $$