Show that $abc \le (ab + bc + ca)(a^2 + b^2 + c^2)^2$ for all positive real numbers $a, b$ and $c$ such that $a + b + c = 1$.

If $x$ is a positive rational number show that $x$ can be uniquely expressed in the form $x = \sum^n_{k=1} \frac{a_k}{k!}$ where $a_1, a_2, \ldots$ are integers, $0 \leq a_n \leq n - 1$, for $n > 1,$ and the series terminates. Show that $x$ can be expressed as the sum of reciprocals of different integers, each of which is greater than $10^6.$

Find all triples $(a, b, c)$ of positive integers such that: $$a + b + c = 24$$ $$a^2 + b^2 + c^2 = 210$$ $$abc = 440$$

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide] [u] Set 1[/u] [b]R1.1 / P1.1[/b] Find $291 + 503 - 91 + 492 - 103 - 392$. [b]R1.2[/b] Let the operation $a$ & $b$ be defined to be $\frac{a-b}{a+b}$. What is $3$ & $-2$? [b]R1.3[/b]. Joe can trade $5$ apples for $3$ oranges, and trade $6$ oranges for $5$ bananas. If he has $20$ apples, what is the largest number of bananas he can trade for? [b]R1.4[/b] A cone has a base with radius $3$ and a height of $5$. What is its volume? Express your answer in terms of $\pi$. [b]R1.5[/b] Guang brought dumplings to school for lunch, but by the time his lunch period comes around, he only has two dumplings left! He tries to remember what happened to the dumplings. He first traded $\frac34$ of his dumplings for Arman’s samosas, then he gave $3$ dumplings to Anish, and lastly he gave David $\frac12$ of the dumplings he had left. How many dumplings did Guang bring to school? [u]Set 2[/u] [b]R2.6 / P1.3[/b] In the recording studio, Kanye has $10$ different beats, $9$ different manuscripts, and 8 different samples. If he must choose $1$ beat, $1$ manuscript, and $1$ sample for his new song, how many selections can he make? [b]R2.7[/b] How many lines of symmetry does a regular dodecagon (a polygon with $12$ sides) have? [b]R2.8[/b] Let there be numbers $a, b, c$ such that $ab = 3$ and $abc = 9$. What is the value of $c$? [b]R2.9[/b] How many odd composite numbers are there between $1$ and $20$? [b]R2.10[/b] Consider the line given by the equation $3x - 5y = 2$. David is looking at another line of the form ax - 15y = 5, where a is a real number. What is the value of a such that the two lines do not intersect at any point? [u]Set 3[/u] [b]R3.11[/b] Let $ABCD$ be a rectangle such that $AB = 4$ and $BC = 3$. What is the length of BD? [b]R3.12[/b] Daniel is walking at a constant rate on a $100$-meter long moving walkway. The walkway moves at $3$ m/s. If it takes Daniel $20$ seconds to traverse the walkway, find his walking speed (excluding the speed of the walkway) in m/s. [b]R3.13 / P1.3[/b] Pratik has a $6$ sided die with the numbers $1, 2, 3, 4, 6$, and $12$ on the faces. He rolls the die twice and records the two numbers that turn up on top. What is the probability that the product of the two numbers is less than or equal to $12$? [b]R3.14 / P1.5[/b] Find the two-digit number such that the sum of its digits is twice the product of its digits. [b]R3.15[/b] If $a^2 + 2a = 120$, what is the value of $2a^2 + 4a + 1$? PS. You should use hide for answers. R16-30 /P6-10/ P26-30 have been posted [url=https://artofproblemsolving.com/community/c3h2786837p24497019]here[/url], and P11-25 [url=https://artofproblemsolving.com/community/c3h2786880p24497350]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ n$ be an even positive integer. Prove that there exists a positive inter $ k$ such that \[ k \equal{} f(x) \cdot (x\plus{}1)^n \plus{} g(x) \cdot (x^n \plus{} 1)\] for some polynomials $ f(x), g(x)$ having integer coefficients. If $ k_0$ denotes the least such $ k,$ determine $ k_0$ as a function of $ n,$ i.e. show that $ k_0 \equal{} 2^q$ where $ q$ is the odd integer determined by $ n \equal{} q \cdot 2^r, r \in \mathbb{N}.$ Note: This is variant A6' of the three variants given for this problem.

Let $x,y,z$ be non-negative real numbers such that $xyz=1$. Prove that $$(x^3+2y)(y^3+2z)(z^3+2x) \ge 27.$$

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 $x,y \in \mathbb{R}$.

Find all functions $f: R\to R$, such that $f(xy + x + y) + f(xy-x-y)=2f (x) + 2f (y)$ for all $x, y \in R$.

Determine all pairs $ (a,b)$ of positive integers, each integer having two decimal digits, such that $ 100a\plus{}b$ and $ 201a\plus{}b$ are both perfect squares.

Let $a$ and $b$ be real numbers. It is known that the graph of the parabola $y =ax^2 +b$ cuts the graph of the curve $y = x+1/x$ in exactly three points. Prove that $3ab < 1$.

Consider the sequence $a_1, a_2, a_3, ...$ defined by $a_1 = 9$ and $a_{n + 1} = \frac{(n + 5)a_n + 22}{n + 3}$ for $n \ge 1$. Find all natural numbers $n$ for which $a_n$ is a perfect square of an integer.

Let $n \ge 2$ be an integer. Let $a$ be the greatest positive integer such that $2^a | 5^n - 3^n$. Let $b$ be the greatest positive integer such that $2^b \le n$. Prove that $a \le b + 3$.

Let $a,b,c$ be positive real numbers such that $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}=3$. Prove that $$\frac{a+b}{a^2+ab+b^2}+ \frac{b+c}{b^2+bc+c^2}+ \frac{c+a}{c^2+ca+a^2}\le 2$$ When is the equality valid?

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

Solve for $x$: $x = 2 + \frac{4(2^6)}{11-3}$

There are $100$ numbers from $(0,1)$ on the board. On every move we replace two numbers $a,b$ with roots of $x^2-ax+b=0$(if it has two roots). Prove that process is not endless.

Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(y)+yf(z)+zf(x))=yf(x)+zf(y)+xf(z)$

[b]p1.[/b] There are eight different dominoes in the box (fig.), but the boundaries between them are not visible. Draw the boundaries. [img]https://cdn.artofproblemsolving.com/attachments/6/f/6352b18c25478d68a23820e32a7f237c9f2ba9.png[/img] [b]p2.[/b] The teacher drew a quadrilateral $ABCD$ on the board. Vanya and Vitya marked points $X$ and $Y$ inside it, from which all sides of the quadrilateral are visible at equal angles. What is the distance between points $X$ and $Y$? (From point $X$, side $AB$ is visible at angle $AXB$.) [b]pЗ.[/b] Several identical black squares, perhaps partially overlapping, were placed on a white plane. The result was a black polygonal figure, possibly with holes or from several pieces. Could it be that this figure does not have a single right angle? [b]p4.[/b] The bus ticket number consists of six digits (the first digits may be zeros). A ticket is called [i]lucky [/i] if the sum of the first three digits is equal to the sum of the last three. Prove that the sum of the numbers of all lucky tickets is divisible by $13$. [b]p5.[/b] The Meandrovka River, which has many bends, crosses a straight highway under thirteen bridges. Prove that there are two neighboring bridges along both the highway and the river. (Bridges are called river neighbors if there are no other bridges between them on the river section; bridges are called highway neighbors if there are no other bridges between them on the highway section.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

2) Let $a, b, c$ be real numbers such that $$a+\frac{b}{c}=b+\frac{c}{a}=c+\frac{a}{b}=1$$a) Prove that $ab+bc+ca=0$ and $a+b+c=3$. b) Prove that $|a|+|b|+|c|< 5$

Find all $n>1$ and integers $a_1,a_2,\dots,a_n$ satisfying the following three conditions: (i) $2<a_1\le a_2\le \cdots\le a_n$ (ii) $a_1,a_2,\dots,a_n$ are divisors of $15^{25}+1$. (iii) $2-\frac{2}{15^{25}+1}=\left(1-\frac{2}{a_1}\right)+\left(1-\frac{2}{a_2}\right)+\cdots+\left(1-\frac{2}{a_n}\right)$

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

Consider a polynomial in $n$ variables with real coefficients. We know that if every variable is $\pm1$, the value of the polynomial is positive, or negative if the number of $-1$'s is even, or odd, respectively. Prove that the degree of this polynomial is at least $n$.

Can one find 4004 positive integers such that the sum of any 2003 of them is not divisible by 2003?

If $x^3-3\sqrt3 x^2 +9x - 3\sqrt3 -64=0$ find the value of $x^6-8x^5+13x^4-5x^3+49x^2-137x+2015$ .

[b]p1.[/b] Find the sum of all $3$-digit positive integers $\overline{abc}$ that satisfy $$\overline{abc} = {n \choose a}+{n \choose b}+ {n \choose c}$$ for some $n \le 10$. [b]p2.[/b] Feng and Trung play a game. Feng chooses an integer $p$ from $1$ to $90$, and Trung tries to guess it. In each round, Trung asks Feng two yes-or-no questions about $p$. Feng must answer one question truthfully and one question untruthfully. After $15$ rounds, Trung concludes there are n possible values for $p$. What is the least possible value of $n$, assuming Feng chooses the best strategy to prevent Trung from guessing correctly? [b]p3.[/b] A hypercube $H_n$ is an $n$-dimensional analogue of a cube. Its vertices are all the points $(x_1, .., x_n)$ that satisfy $x_i = 0$ or $1$ for all $1 \le i \le n$ and its edges are all segments that connect two adjacent vertices. (Two vertices are adjacent if their coordinates differ at exactly one $x_i$ . For example, $(0,0,0,0)$ and $(0,0,0,1)$ are adjacent on $H_4$.) Let $\phi (H_n)$ be the number of cubes formed by the edges and vertices of $H_n$. Find $\phi (H_4) + \phi (H_5)$. [b]p4.[/b] Denote the legs of a right triangle as $a$ and $b$, the radius of the circumscribed circle as $R$ and the radius of the inscribed circle as $r$. Find $\frac{a+b}{R+r}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].