For natural number $n$ and real numbers $\alpha$ and $x$ satisfy the inequalities $\alpha^{n+1}\le x\le1$ and $0<\alpha<1$. Prove that $$\prod_{k=1}^n\left|\frac{x-\alpha^k}{x+\alpha^k}\right|\le\prod_{k=1}^n\left|\frac{1-\alpha^k}{1+\alpha^k}\right|.$$ [i]Borislav Boyanov[/i]

Let $x$ be a real number. Find the greatest possible value of the following expression: $\frac{47^x}{\sqrt{43}}+\frac{43^x}{\sqrt{47}}-2021^x$.

Let $a\geq b\geq c\geq d$ be real numbers such that $(a-b)(b-c)(c-d)(d-a)=-3.$ [list=a] [*]If $a+b+c+d=6,$ prove that $d<0,36.$ [*]If $a^2+b^2+c^2+d^2=14,$ prove that $(a+c)(b+d)\leq 8.$ When does equality hold? [/list]

If reals $x, y, z $ satises $tan x + tan y + tan z = 2$ and $0 < x, y,z < \frac{\pi}{2}.$ Prove that $$sin^2 x + sin^2 y + sin^2 z < 1.$$

We call polynomial $S(x)\in\mathbb{R}[x]$ sadeh whenever it's divisible by $x$ but not divisible by $x^2$. For the polynomial $P(x)\in\mathbb{R}[x]$ we know that there exists a sadeh polynomial $Q(x)$ such that $P(Q(x))-Q(2x)$ is divisible by $x^2$. Prove that there exists sadeh polynomial $R(x)$ such that $P(R(x))-R(2x)$ is divisible by $x^{1401}$.

Hi guys, I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this: 1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though. 2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary. 3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions: A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh? B. Do NOT go back to the previous problem(s). This causes too much confusion. C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for. 4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving! Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D

Let $R^+$ be the set of positive real numbers. Determine all functions $f:R^+$ $\rightarrow$ $R^+$ such that for all positive real numbers $x$ and $y:$ \[f(x+f(xy))+y=f(x)f(y)+1\] [i]Ukraine[/i]

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Snowhite wrote on a piece of paper a whole number greater than $1$ and multiplied it by itself. She obtained a number, all digits of which are $1$: $n^2 = 111...111$ Does she know how to multiply? [b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a bishop on an arbitrary square. Then the second player can put another bishop on a free square that is not controlled by the first bishop. Then the first player can put a new bishop on a free square that is not controlled by the bishops on the board. Then the second player can do the same, etc. A player who cannot put a new bishop on the board loses the game. Who has a winning strategy? [b]p4.[/b] Four girls Marry, Jill, Ann and Susan participated in the concert. They sang songs. Every song was performed by three girls. Mary sang $8$ songs, more then anybody. Susan sang $5$ songs less then all other girls. How many songs were performed at the concert? [b]p5.[/b] Pinocchio has a $10\times 10$ table of numbers. He took the sums of the numbers in each row and each such sum was positive. Then he took the sum of the numbers in each columns and each such sum was negative. Can you trust Pinocchio's calculations? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 9[/u] [b]p25.[/b] Let $S$ be the region bounded by the lines $y = x/2$, $y = -x/2$, and $x = 6$. Pick a random point $P = (x, y)$ in $S$ and translate it $3$ units right to $P' = (x + 3, y)$. What is the probability that $P'$ is in $S$? [b]p26.[/b] A triangle with side lengths $17$, $25$, and $28$ has a circle centered at each of its three vertices such that the three circles are mutually externally tangent to each other. What is the combined area of the circles? [b]p27.[/b] Find all ordered pairs $(x, y)$ of integers such that $x^2 - 2x + y^2 - 6y = -9$. [u]Round 10[/u] [b]p28.[/b] In how many ways can the letters in the word $SCHAFKOPF$ be arranged if the two $F$’s cannot be next to each other and the $A$ and the $O$ must be next to each other? [b]p29.[/b] Let a sequence $a_0, a_1, a_2, ...$ be defined by $a_0 = 20$, $a_1 = 11$, $a_2 = 0$, and for all integers $n \ge 3$, $$a_n + a_{n-1 }= a_{n-2} + a_{n-3}.$$ Find the sum $a_0 + a_1 + a_2 + · · · + a_{2010} + a_{2011}$. [b]p30.[/b] Find the sum of all positive integers b such that the base $b$ number $190_b$ is a perfect square. [u]Round 11[/u] [b]p31.[/b] Find all real values of x such that $\sqrt[3]{4x -1} + \sqrt[3]{4x + 1 }= \sqrt[3]{8x}$. [b]p32.[/b] Right triangle $ABC$ has a right angle at B. The angle bisector of $\angle ABC$ is drawn and extended to a point E such that $\angle ECA = \angle ACB$. Let $F$ be the foot of the perpendicular from $E$ to ray $\overrightarrow{BC}$. Given that $AB = 4$, $BC = 2$, and $EF = 8$, find the area of triangle $ACE$. [b]p33.[/b] You are the soul in the southwest corner of a four by four grid of distinct souls in the Fields of Asphodel. You move one square east and at the same time all the other souls move one square north, south, east, or west so that each square is now reoccupied and no two souls switched places directly. How many end results are possible from this move? [u]Round 12[/u] [b]p34.[/b] A [i]Pythagorean [/i] triple is an ordered triple of positive integers $(a, b, c)$ with $a < b < c $and $a^2 + b^2 = c^2$ . A [i]primitive [/i] Pythagorean triple is a Pythagorean triple where all three numbers are relatively prime to each other. Find the number of primitive Pythagorean triples in which all three members are less than $100,000$. If $P$ is the true answer and $A$ is your team’s answer to this problem, your score will be $max \left\{15 -\frac{|A -P|}{500} , 0 \right\}$ , rounded to the nearest integer. [b]p35.[/b] According to the Enable2k North American word list, how many words in the English language contain the letters $L, M, T$ in order but not necessarily together? If $A$ is your team’s answer to this problem and $W$ is the true answer, the score you will receive is $max \left\{15 -100\left| \frac{A}{W}-1\right| , 0 \right\}$, rounded to the nearest integer. [b]p36.[/b] Write down $5$ positive integers less than or equal to $42$. For each of the numbers written, if no other teams put down that number, your team gets $3$ points. Otherwise, you get $0$ points. Any number written that does not satisfy the given requirement automatically gets $0$ points. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2952214p26434209]here[/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3133709p28395558]here[/url]. Rest Rounds soon. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.

Given is a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $|f(x+y)-f(x)-f(y)|\leq 1$. Prove the existence of an additive function $g:\mathbb{R}\rightarrow \mathbb{R}$ (that is $g(x+y)=g(x)+g(y)$) such that $|f(x)-g(x)|\leq 1$ for any $x \in \mathbb{R}$

Show that for any positive real numbers $a$, $b$, $c$ the following inequality takes place $$\frac{a}{\sqrt{7a^2+b^2+c^2}}+\frac{b}{\sqrt{a^2+7b^2+c^2}}+\frac{c}{\sqrt{a^2+b^2+7c^2}} \leq 1.$$

Let $f$ be a function defined on $(0, \infty )$ as follows: \[f(x)=x+\frac1x\] Let $h$ be a function defined for all $x \in (0,1)$ as \[h(x)=\frac{x^4}{(1-x)^6}\] Suppose that $g(x)=f(h(x))$ for all $x \in (0,1)$. (a) Show that $h$ is a strictly increasing function. (b) Show that there exists a real number $x_0 \in (0,1)$ such that $g$ is strictly decreasing in the interval $(0,x_0]$ and strictly increasing in the interval $[x_0,1)$.

Find all functions $f:\mathbb {Z}\to\mathbb Z$, satisfy that for any integer ${a}$, ${b}$, ${c}$, $$2f(a^2+b^2+c^2)-2f(ab+bc+ca)=f(a-b)^2+f(b-c)^2+f(c-a)^2$$

Given $a,b,c \in \mathbb{R}$ satisfying $a+b+c=a^2+b^2+c^2=1$, show that $\frac{-1}{4} \leq ab \leq \frac{4}{9}$.

Let $n\ge 3$ be a positive integer. Prove that a polynomial of the form \[x^n+a_{n-3}x^{n-3}+a_{n-4}x^{n-4}+\ldots +a_1x+a_0,\] where at least one of the real coefficients $a_0,a_1,\ldots ,a_{n-3}$ is nonzero, cannot have all real roots.

A binary operation $*$ on real numbers has the property that $(a * b) * c = a+b+c$ for all $a$, $b$, $c$. Prove that $a * b = a+b$.

[u]Round 5 [/u] [b]p13.[/b] How many ways can we form a group with an odd number of members (plural) from $99$ people? Express your answer in the form $a^b + c$, where $a, b$, and $c$ are integers and $a$ is prime. [b]p14.[/b] A cube is inscibed in a right circular cone such that the ratio of the height of the cone to the radius is $2:1$. Compute the fraction of the cone’s volume that the cube occupies. [b]p15.[/b] Let $F_0 = 1$, $F_1 = 1$ and $F_k = F_{k-1} + F_{k-2}$. Let $P(x) = \sum^{99}_{k=0} x^{F_k}$ . The remainder when $P(x)$ is divided by $x^3 - 1$ can be expressed as $ax^2 + bx + c$. Find $2a + b$. [u]Round 6 [/u] [b]p16.[/b] Ankit finds a quite peculiar deck of cards in that each card has n distinct symbols on it and any two cards chosen from the deck will have exactly one symbol in common. The cards are guaranteed to not have a certain symbol which is held in common with all the cards. Ankit decides to create a function f(n) which describes the maximum possible number of cards in a set given the previous constraints. What is the value of $f(10)$? [b]p17.[/b] If $|x| <\frac14$ and $$X = \sum^{\infty}_{N=0} \sum^{N}_{n=0} {N \choose n}x^{2n}(2x)^{N-n}.$$ then write $X$ in terms of $x$ without any summation or product symbols (and without an infinite number of ‘$+$’s, etc.). [b]p18.[/b] Dietrich is playing a game where he is given three numbers $a, b, c$ which range from $[0, 3]$ in a continuous uniform distribution. Dietrich wins the game if the maximum distance between any two numbers is no more than $1$. What is the probability Dietrich wins the game? [u]Round 7 [/u] [b]p19.[/b] Consider f defined by $$f(x) = x^6 + a_1x^5 + a_2x^4 + a_3x^3 + a_4x^2 + a_5x + a_6.$$ How many tuples of positive integers $(a_1, a_2, a_3, a_4, a_5, a_6)$ exist such that $f(-1) = 12$ and $f(1) = 30$? [b]p20.[/b] Let $a_n$ be the number of permutations of the numbers $S = \{1, 2, ... , n\}$ such that for all $k$ with $1 \le k \le n$, the sum of $k$ and the number in the $k$th position of the permutation is a power of $2$. Compute $a_1 + a_2 + a_4 + a_8 + ... + a_{1048576}$. [b]p21.[/b] A $4$-dimensional hypercube of edge length $1$ is constructed in $4$-space with its edges parallel to the coordinate axes and one vertex at the origin. Its coordinates are given by all possible permutations of $(0, 0, 0, 0)$,$(1, 0, 0, 0)$,$(1, 1, 0, 0)$,$(1, 1, 1, 0)$, and $(1, 1, 1, 1)$. The $3$-dimensional hyperplane given by $x+y+z+w = 2$ intersects the hypercube at $6$ of its vertices. Compute the 3-dimensional volume of the solid formed by the intersection. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2781335p24424563]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$C$ is the circle center $(0,1)$, radius $1$. $P$ is the parabola $y = ax^2$. They meet at $(0, 0)$. For what values of $a$ do they meet at another point or points?

[b]p1.[/b] An animal farm has geese and pigs with a total of $30$ heads and $84$ legs. Find the number of pigs and geese on this farm. [b]p2.[/b] What is the maximum number of $1 \times 1$ squares of a $7 \times 7$ board that can be colored black in such a way that the black squares don’t touch each other even at their corners? Show your answer on the figure below and explain why it is not possible to get more black squares satisfying the given conditions. [img]https://cdn.artofproblemsolving.com/attachments/d/5/2a0528428f4a5811565b94061486699df0577c.png[/img] [b]p3.[/b] Decide whether it is possible to divide a regular hexagon into three equal not necessarily regular hexagons? A regular hexagon is a hexagon with equal sides and equal angles. [img]https://cdn.artofproblemsolving.com/attachments/3/7/5d941b599a90e13a2e8ada635e1f1f3f234703.png[/img] [b]p4.[/b] A rectangle is subdivided into a number of smaller rectangles. One observes that perimeters of all smaller rectangles are whole numbers. Is it possible that the perimeter of the original rectangle is not a whole number? [b]p5.[/b] Place parentheses on the left hand side of the following equality to make it correct. $$ 4 \times 12 + 18 : 6 + 3 = 50$$ [b]p6.[/b] Is it possible to cut a $16\times 9$ rectangle into two equal parts which can be assembled into a square? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p(x)$ be a polynomial with integer coefficients such that $p(0) = 0$ and $0 \le p(1) \le 10^7$. Suppose that there exist positive integers $a,b$ such that $p(a) = 1999$ and $p(b) = 2001$. Determine all possible values of $p(1)$. (Note: $1999$ is a prime number.)

Let $p$ and $q$ be integers. Show that there exists an interval $I$ of length $1/q$ and a polynomial $P$ with integral coefficients such that \[ \left|P(x)-\frac pq \right| < \frac{1}{q^2}\]for all $x \in I.$

The sequence $a_0, a_1, a_2, ..$ is defined by $a_0 = a, a_{n+1} = a_n + L(a_n)$, where $L(m)$ is the last digit of $m$ (eg $L(14) = 4$). Suppose that the sequence is strictly increasing. Show that infinitely many terms must be divisible by $d = 3$. For what other d is this true?

A drunken horse moves on an infinite board whose squares are numbered in pairs $(a, b) \in \mathbb{Z}^2$. In each movement, the 8 possibilities $$(a, b) \rightarrow (a \pm 1, b \pm 2),$$ $$(a, b) \rightarrow (a \pm 2, b \pm 1)$$ are equally likely. Knowing that the knight starts at $(0, 0)$, calculate the probability that, after $2023$ moves, it is in a square $(a, b)$ with $a \equiv 4 \pmod 8$ and $b \equiv 5 \pmod 8$.

Let a; b; c be positive real numbers such that $\frac{1}{bc}+\frac{1}{ca}+\frac{1}{ab} \geq 1$. Prove that $\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab} \geq 1$.