Let $n$ be a positive integer. Two players, Alice and Bob, are playing the following game: - Alice chooses $n$ real numbers; not necessarily distinct. - Alice writes all pairwise sums on a sheet of paper and gives it to Bob. (There are $\frac{n(n-1)}{2}$ such sums; not necessarily distinct.) - Bob wins if he finds correctly the initial $n$ numbers chosen by Alice with only one guess. Can Bob be sure to win for the following cases? a. $n=5$ b. $n=6$ c. $n=8$ Justify your answer(s). [For example, when $n=4$, Alice may choose the numbers 1, 5, 7, 9, which have the same pairwise sums as the numbers 2, 4, 6, 10, and hence Bob cannot be sure to win.]

Santiago, Daniel and Fátima practice for the Math Olympics. Santiago thinks of a regular polygon and Daniel of another, without telling Fatima what the polygons are. They just tell you that one of the polygons has $3$ more sides than the other and that an angle of one of the polygons measures $10$ degrees more than one angle of the other. From this, and knowing that each interior angle of a regular polygon of $n$ sides measures $\frac{180(n-2)}{n}$ degrees, Fatima identifies what the polygons are. How many sides do the polygons that James and Daniel chose, have?

[u]Set 4[/u] [b]4.1[/b] Quadrilateral $ABCD$ (with $A, B, C$ not collinear and $A, D, C$ not collinear) has $AB = 4$, $BC = 7$, $CD = 10$, and $DA = 5$. Compute the number of possible integer lengths $AC$. [img]https://cdn.artofproblemsolving.com/attachments/1/6/4f43873a64bc00a0e6173002ccd80e8f1529a9.png[/img] [b]4.2[/b] Let $T$ be the answer from the previous part. $2T$ congruent isosceles triangles with base length $b$ and leg length $\ell$ are arranged to form a parallelogram as shown below (not necessarily the correct number of triangles). If the total length of all drawn line segments (not double counting overlapping sides) is exactly three times the perimeter of the parallelogram, find $\frac{\ell}{b}$. [img]https://cdn.artofproblemsolving.com/attachments/5/c/744f503ed822bc43acafe2633e6108022f2c88.png[/img] [b]4.3[/b] Let $T$ be the answer from the previous part. Rectangle $R$ has length $T$ times its width. $R$ is inscribed in a square $S$ such that the diagonals of $ S$ are parallel to the sides of $R$. What proportion of the area of $S$ is contained within $R$? [img]https://cdn.artofproblemsolving.com/attachments/a/1/0928dd1ffbeb4d7dee9b697fdb7696cc70c03d.png[/img] PS. You should use hide for answers.

In Theresa's first $8$ basketball games, she scored $7, 4, 3, 6, 8, 3, 1$ and $5$ points. In her ninth game, she scored fewer than $10$ points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than $10$ points and her points-per-game average for the $10$ games was also an integer. What is the product of the number of points she scored in the ninth and tenth games? $\textbf{(A)}\ 35\qquad \textbf{(B)}\ 40\qquad \textbf{(C)}\ 48\qquad \textbf{(D)}\ 56\qquad \textbf{(E)}\ 72$

Let $S = \{1,2,3,\ldots,n\}$. Consider a function $f\colon S\to S$. A subset $D$ of $S$ is said to be invariant if for all $x\in D$ we have $f(x)\in D$. The empty set and $S$ are also considered as invariant subsets. By $\deg (f)$ we define the number of invariant subsets $D$ of $S$ for the function $f$. [b]i)[/b] Show that there exists a function $f\colon S\to S$ such that $\deg (f)=2$. [b]ii)[/b] Show that for every $1\leq k\leq n$ there exists a function $f\colon S\to S$ such that $\deg (f)=2^{k}$.

For every positive integer $n$, let $\operatorname{mod_5}(n)$ be the remainder obtained when $n$ is divided by $5$. Define a function $f : \{0, 1, 2, 3, \dots\} \times \{0, 1, 2, 3, 4\} \to \{0, 1, 2, 3, 4\}$ recursively as follows: \[f(i, j) = \begin{cases} \operatorname{mod_5}(j+1) & \text{if }i=0\text{ and }0\leq j\leq 4 \\ f(i-1, 1) & \text{if }i\geq 1\text{ and }j=0 \text{, and}\\ f(i-1, f(i, j-1)) & \text{if }i\geq 1\text{ and }1\leq j\leq 4 \end{cases}\] What is $f(2015, 2)$? $\textbf{(A) }0 \qquad\textbf{(B) }1 \qquad\textbf{(C) }2 \qquad\textbf{(D) }3 \qquad\textbf{(E) }4$

Find the largest real number $k$, such that for any positive real numbers $a,b$, $$(a+b)(ab+1)(b+1)\geq kab^2$$

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 $\{n,p\}\in\mathbb{N}\cup \{0\}$ such that $2p\le n$. Prove that $\frac{(n-p)!}{p!}\le \left(\frac{n+1}{2}\right)^{n-2p}$. Determine all conditions under which equality holds.

[i]In this round, problems will depend on the answers to other problems. A bolded letter is used to denote a quantity whose value is determined by another problem's answer.[/i] [u]Part I[/u] [b]p1.[/b] Let F be the answer to problem number $6$. You want to tile a nondegenerate square with side length $F$ with $1\times 2$ rectangles and $1 \times 1$ squares. The rectangles can be oriented in either direction. How many ways can you do this? [b]p2.[/b] Let [b]A[/b] be the answer to problem number $1$. Triangle $ABC$ has a right angle at $B$ and the length of $AC$ is [b]A[/b]. Let $D$ be the midpoint of $AB$, and let $P$ be a point inside triangle $ABC$ such that $PA = PC = \frac{7\sqrt5}{4}$ and $PD = \frac74$ . The length of $AB^2$ is expressible as $m/n$ for $m, n$ relatively prime positive integers. Find $m$. [b]p3.[/b] Let [b]B[/b] be the answer to problem number $2$. Let $S$ be the set of positive integers less than or equal to [b]B[/b]. What is the maximum size of a subset of $S$ whose elements are pairwise relatively prime? [b]p4.[/b] Let [b]C[/b] be the answer to problem number $3$. You have $9$ shirts and $9$ pairs of pants. Each is either red or blue, you have more red shirts than blue shirts, and you have same number of red shirts as blue pants. Given that you have [b]C[/b] ways of wearing a shirt and pants whose colors match, find out how many red shirts you own. [b]p5.[/b] Let [b]D[/b] be the answer to problem number $4$. You have two odd positive integers $a, b$. It turns out that $lcm(a, b) + a = gcd(a, b) + b =$ [b]D[/b]. Find $ab$. [b]p6.[/b] Let [b]E[/b] be the answer to problem number $5$. A function $f$ defined on integers satisfies $f(y)+f(12-y) = 10$ and $f(y) + f(8 - y) = 4$ for all integers $y$. Given that $f($ [b]E[/b] $) = 0$, compute $f(4)$. [u]Part II[/u] [b]p7.[/b] Let [b]L[/b] be the answer to problem number $12$. You want to tile a nondegenerate square with side length [b]L[/b] with $1\times 2$ rectangles and $7\times 7$ squares. The rectangles can be oriented in either direction. How many ways can you do this? [b]p8.[/b] Let [b]G[/b] be the answer to problem number $7$. Triangle $ABC$ has a right angle at $B$ and the length of $AC$ is [b]G[/b]. Let $D$ be the midpoint of $AB$, and let $P$ be a point inside triangle $ABC$ such that $PA = PC = \frac12$ and $PD = \frac{1}{2010}$ . The length of $AB^2$ is expressible as $m/n$ for $m, n$ relatively prime positive integers. Find $m$. [b]p9.[/b] Let [b]H[/b] be the answer to problem number $8$. Let $S$ be the set of positive integers less than or equal to [b]H[/b]. What is the maximum size of a subset of $S$ whose elements are pairwise relatively prime? [b]p10.[/b] Let [b]I[/b] be the answer to problem number $9$. You have $391$ shirts and $391$ pairs of pants. Each is either red or blue, you have more red shirts than blue shirts, and you have same number of red shirts as red pants. Given that you have [b]I[/b] ways of wearing a shirt and pants whose colors match, find out how many red shirts you own. [b]p11.[/b] Let [b]J[/b] be the answer to problem number $10$. You have two odd positive integers $a, b$. It turns out that $lcm(a, b) + 2a = 2 gcd(a, b) + b = $ [b]J[/b]. Find $ab$. [b]p12.[/b] Let [b]K[/b] be the answer to problem number $11$. A function $f$ defined on integers satisfies $f(y)+f(7-y) = 8$ and $f(y) + f(5 - y) = 4$ for all integers $y$. Given that $f($ [b]K[/b] $) = 453$, compute $f(2)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Solve equation in real numbers $\sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{…+\sqrt{4^nx+3}}}}}-\sqrt{x}=1$

Given two concentric circles and a point $A$. Through point $A$, draw a secant such that its segment contained by the larger circle is divided by the smaller circle into three equal parts.

Let $G$ be a group and let $a$ be a constant member of it. Prove that \[G_a = \{x | \exists n \in \mathbb Z , x=a^n\}\] Is a subgroup of $G.$

A $k$-set is a set with exactly $k$ elements. For a $6$-set $A$ and any collection $\mathcal{F}$ of $4$-sets, we say that $A$ is [i]$\mathcal{F}$-good[/i] if there are exactly three elements $B_1, B_2, B_3$ in $\mathcal{F}$ that are subsets of $A$, and they furthermore satisfy $$(A \backslash B_1) \cup (A \backslash B_2) \cup (A \backslash B_3) = A.$$ Find all $n \geq 6$ so that there exists a collection $\mathcal{F}$ of $4$-subsets of $\{1, 2, \ldots , n\}$ such that every $6$-set $A \subseteq \{1, 2, \ldots , n\}$ is $\mathcal{F}$-good. [i] Proposed by usjl[/i]

Let $L,M$ and $N$ be points on sides $AC,AB$ and $BC$ of triangle $ABC$, respectively, such that $BL$ is the bisector of angle $ABC$ and segments $AN,BL$ and $CM$ have a common point. Prove that if $\angle ALB=\angle MNB$ then $\angle LNM=90^{\circ}$.

Given is a triangle $ABC$ and a point $K$ within the triangle. The point $K$ is mirrored in the sides of the triangle: $P , Q$ and $R$ are the mirrorings of $K$ in $AB , BC$ and $CA$, respectively . $M$ is the center of the circle passing through the vertices of triangle $PQR$. $M$ is mirrored again in the sides of triangle $ABC$: $P', Q'$ and $R'$ are the mirror of $M$ in $AB$ respectively, $BC$ and $CA$. a. Prove that $K$ is the center of the circle passing through the vertices of triangle $P'Q'R'$ . b. Where should you choose $K$ within triangle $ABC$ so that $M$ and $K$ coincide? Prove your answer.

Let $P(x)=x+1$ and $Q(x)=x^2+1.$ Consider all sequences $\langle(x_k,y_k)\rangle_{k\in\mathbb{N}}$ such that $(x_1,y_1)=(1,3)$ and $(x_{k+1},y_{k+1})$ is either $(P(x_k), Q(y_k))$ or $(Q(x_k),P(y_k))$ for each $k. $ We say that a positive integer $n$ is nice if $x_n=y_n$ holds in at least one of these sequences. Find all nice numbers.

Let $\Gamma$ be a circle centered at $O$ with radius $R$. Let $X$ and $Y$ be points on $\Gamma$ such that $XY<R$. Let $I$ be a point such that $IX = IY$ and $XY = OI$. Describe how to construct with ruler and compass a triangle which has circumcircle $\Gamma$, incenter $I$ and Euler line $OX$. Prove that this triangle is unique.

The circle inscribed in a triangle $ABC$ touches the sides $BC,CA,AB$ in $D,E, F$, respectively, and $X, Y,Z$ are the midpoints of $EF, FD,DE$, respectively. Prove that the centers of the inscribed circle and of the circles around $XYZ$ and $ABC$ are collinear.

The sum of 5 consecutive even integers is $ 4$ less than the sum of the first $ 8$ consecutive odd counting numbers. What is the smallest of the even integers? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14$

Determine the real value of $t$ that minimizes the expression \[ \sqrt{t^2 + (t^2 - 1)^2} + \sqrt{(t-14)^2 + (t^2 - 46)^2}. \]

There are $n$ distinct points in the plane. Given a circle in the plane containing at least one of the points in its interior. At each step one moves the center of the circle to the barycenter of all the points in the interior of the circle. Prove that this moving process terminates in the finite number of steps. what does barycenter of n distinct points mean?

The time-varying temperature of a certain body is given by a polynomial in the time of degree at most three. Show that the average temperature of the body between $9$ am and $3$ pm can always be found by taking the average of the temperatures at two fixed times, which are independent of the polynomial. Also, show that these two times are $10\colon \! 16$ am and $1\colon \!44$ pm to the nearest minute.

Circle $ X$ has a radius of $ \pi$. Circle $ Y$ has a circumference of $ 8\pi$. Circle $ Z$ has an area of $ 9\pi$. List the circles in order from smallest to largest radius. $ \textbf{(A)}\ X, Y, Z \qquad \textbf{(B)}\ Z, X, Y \qquad \textbf{(C)}\ Y, X, Z \qquad \textbf{(D)}\ Z, Y, X \qquad \textbf{(E)}\ X, Z, Y$

Henry starts a trip when the hands of the clock are together between $ 8$ a.m. and $ 9$ a.m. He arrives at his destination between $ 2$ p.m. and $ 3$ p.m. when the hands of the clock are exactly $ 180^\circ$ apart. The trip takes: $ \textbf{(A)}\ \text{6 hr.} \qquad \textbf{(B)}\ \text{6 hr. 43\minus{}7/11 min.} \qquad \textbf{(C)}\ \text{5 hr. 16\minus{}4/11 min.} \qquad \textbf{(D)}\ \text{6 hr. 30 min.} \qquad \textbf{(E)}\ \text{none of these}$