Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.

Three metals, $A, B $and $C$, with solutions of their respective cations are tested in a voltaic cell with the following results: $A$ and $B$: $A$ is the cathode $B$ and $C$: $C$ is the cathode $A$ and $C$: $A$ is the anode What is the order of the reduction potentials from highest to lowest for the cations of these metals? $ \textbf{(A)}\ A>B>C \qquad\textbf{(B)}\ B>C>A\qquad$ ${\textbf{(C)}\ C>A>B\qquad\textbf{(D}}\ B>A>C\qquad$

(1) Find the smallest positive integer $a$ such that $221|3^a -2^a$, (2) Let $A=\{n\in N^*: 211|1+2^n+3^n+4^n\}$. Are there infinitely many numbers $n$ such that both $n$ and $n+1$ belong to set $A$?

On a blackboard, there are $17$ integers not divisible by $17$. Alice and Bob play a game. Alice starts and they alternately play the following moves: $\bullet$ Alice chooses a number $a$ on the blackboard and replaces it with $a^2$ $\bullet$ Bob chooses a number $b$ on the blackboard and replaces it with $b^3$. Alice wins if the sum of the numbers on the blackboard is a multiple of $17$ after a finite number of steps. Prove that Alice has a winning strategy. (Daniel Holmes)

Circles $A,B,C,D$ are given on the plane such that circles $A$ and $B$ are externally tangent at $P, B$ and $C$ at $Q, C$ and $D$ at $R$, and $D$ and $A$ at $S$. Circles $A$ and $C$ do not meet, and so do not $B$ and $D$. (a) Prove that the points $P,Q,R,S$ lie on a circle. (b) Suppose that $A$ and $C$ have radius $2, B$ and $D$ have radius $3$, and the distance between the centers of $A$ and $C$ is $6$. Compute the area of the quadrilateral $PQRS$.

Let $A=\{1,2,\cdots ,100\}$. Let $S$ be a subset of power set of $A$ such that any two elements of $S$ has nonzero intersection (Note that elements of $S$ are actually some subsets of $A$). Then the maximum possible cardinality of $S$ is [list=1] [*] $2^{99}$ [*] $2^{99}+1$ [*] $2^{99}+2^{98}$ [*] None of these [/list]

Prove that there is no $1989$-digit natural number at least three of whose digits are equal to $5$ and such that the product of its digits equals their sum.

A differentiable function $f$ with $f(0) = f(1) = 0$ is defined on the interval $[0,1]$. Prove that there exists a point $y \in [0,1]$ such that $| f' (y)| = 4 \int _0^1 | f(x)|dx$.

Find the number of sequences of $2005$ terms with the following properties: (i) No three consecutive terms of the sequence are equal, (ii) Every term equals either $1$ or $-1$, (iii) The sum of all terms of the sequence is at least $666$.

Function $F(x)=|\cos^2x+2\sin x\cos x-\sin^2x+Ax+B|$, where $A,B$ are two real numbers, $x\in[0,\frac{3}{2}\pi]$. $M$ is the maximun value of $F(x)$. Find the minumum value of $M$.

There are $15$ people, including Petruk, Gareng, and Bagong, which will be partitioned into $6$ groups, randomly, that consists of $3, 3, 3, 2, 2$, and $2$ people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group.

Does there exist a set $ M$ of points in space such that every plane intersects $ M$ at a finite but nonzero number of points?

A finite set $S{}$ of $n{}$ points is given in the plane. No three points lie on the same line. The number of non-self-intersecting closed $n{}$-link polylines with vertices at these points will be denoted by $f(S).$ Prove that [list=a] [*]$f(S)>0$ for all sets $S{};$ [*]$f(S)=1$ if and only if all the points of $S{}$ lie on the convex hull of $S{};$ [*]if $f(S)>1$ then $f(S)\geqslant n-1$, with equality if and only if one point of $S$ lies inside the convex hull; [*]if exactly two points of $S{}$ lie inside the convex hull, then\[f(S)\geqslant\frac{(n-2)(n-3)}{2}.\] [/list]Let $n\geqslant 3.$ Denote by $F(n)$ the largest possible value of the function $f(S)$ over all admissible sets $S{}$ of $n{}$ points. Prove that \[F(n)\geqslant3\cdot 2^{(n-8)/3}.\][i]Proposed by E. Bakaev and D. Magzhanov[/i]

Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$, \[f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y).\]

Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs.

Let be three distinct squares $ ABCD, BCEF, EFGH. $ Show that $ \angle EDF +\angle HDG =45^{\circ } . $

Three unit vectors $a,b,c$ are given on the plane. Prove that one can choose the signs in the expression $x=\pm a\pm b\pm c$ so as to obtain a vector $x$ with $|x|\le\sqrt2$.

If $ a$, $ b$, $ c$, and $ d$ are the solutions of the equation $ x^4 \minus{} bx \minus{} 3 \equal{} 0$, then an equation whose solutions are \[ \frac {a \plus{} b \plus{} c}{d^2}, \frac {a \plus{} b \plus{} d}{c^2}, \frac {a \plus{} c \plus{} d}{b^2}, \frac {b \plus{} c \plus{} d}{a^2} \]is $ \textbf{(A)}\ 3x^4 \plus{} bx \plus{} 1 \equal{} 0\qquad \textbf{(B)}\ 3x^4 \minus{} bx \plus{} 1 \equal{} 0\qquad \textbf{(C)}\ 3x^4 \plus{} bx^3 \minus{} 1 \equal{} 0$ $ \textbf{(D)}\ 3x^4 \minus{} bx^3 \minus{} 1 \equal{} 0\qquad \textbf{(E)}\ \text{none of these}$

Let $a_{1}, a_{2}, \cdots, a_{k}$ be relatively prime positive integers. Determine the largest integer which cannot be expressed in the form \[x_{1}a_{2}a_{3}\cdots a_{k}+x_{2}a_{1}a_{3}\cdots a_{k}+\cdots+x_{k}a_{1}a_{2}\cdots a_{k-1}\] for some nonnegative integers $x_{1}, x_{2}, \cdots, x_{k}$.

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 $3ab < 1$.

New license plates consist of two letters, three digits, and two letters (from the English alphabet of$ 26$ letters). What is the largest possible number of such license plates if it is required that every two of them differ at no less than two positions?

Let $a, b, c$ be real numbers, and define $S_n = a^n + b^n + c^n$ for positive integers $n$. Suppose that $S_1, S_2, S_3$ are integers satisfying $6 | 5S_1 - 3S_2 - 2S_3$. Show that $S_n$ is an integer for all positive integers $n$.

[u]Round 6[/u] [b]p16.[/b] Let $a_1, a_2, ... , a_{2011}$ be a sequence of numbers such that $a_1 = 2011$ and $a_1+a_2+...+a_n = n^2 \cdot a_n$ for $n = 1, 2, ... 2011$. (That is, $a_1 = 1^2\cdot a_1$, $a_1 + a_2 = 2^2 \cdot a_2$, $...$) Compute $a_{2011}$. [b]p17.[/b] Three rectangles, with dimensions $3 \times 5$, $4 \times 2$, and $6 \times 4$, are each divided into unit squares which are alternately colored black and white like a checkerboard. Each rectangle is cut along one of its diagonals into two triangles. For each triangle, let m be the total black area and n the total white area. Find the maximum value of $|m - n|$ for the $6$ triangles. [b]p18.[/b] In triangle $ABC$, $\angle BAC = 90^o$, and the length of segment $AB$ is $2011$. Let $M$ be the midpoint of $BC$ and $D$ the midpoint of $AM$. Let $E$ be the point on segment $AB$ such that $EM \parallel CD$. What is the length of segment $BE$? [u]Round 7[/u] [b]p19.[/b] How many integers from $1$ to $100$, inclusive, can be expressed as the difference of two perfect squares? (For example, $3 = 2^2 - 1^2$). [b]p20.[/b] In triangle $ABC$, $\angle ABC = 45$ and $\angle ACB = 60^o$. Let $P$ and $Q$ be points on segment $BC$, $F$ a point on segment $AB$, and $E$ a point on segment $AC$ such that $F Q \parallel AC$ and $EP \parallel AB$. Let $D$ be the foot of the altitude from $A$ to $BC$. The lines $AD$, $F Q$, and $P E$ form a triangle. Find the positive difference, in degrees, between the largest and smallest angles of this triangle. [b]p21.[/b] For real number $x$, $\lceil x \rceil$ is equal to the smallest integer larger than or equal to $x$. For example, $\lceil 3 \rceil = 3$ and $\lceil 2.5 \rceil = 3$. Let $f(n)$ be a function such that $f(n) = \left\lceil \frac{n}{2}\right\rceil + f\left( \left\lceil \frac{n}{2}\right\rceil\right)$ for every integer $n$ greater than $1$. If $f(1) = 1$, find the maximum value of $f(k) - k$, where $k$ is a positive integer less than or equal to $2011$. [u]Round 8[/u] The answer to each of the three questions in this round depends on the answer to one of the other questions. There is only one set of correct answers to these problems; however, each question will be scored independently, regardless of whether the answers to the other questions are correct. [b]p22.[/b] Let $W$ be the answer to problem 24 in this guts round. Let $f(a) = \frac{1}{1 -\frac{1}{1- \frac{1}{a}}}$. Determine$|f(2) + ... + f(W)|$. [b]p23.[/b] Let $X$ be the answer to problem $22$ in this guts round. How many odd perfect squares are less than $8X$? [b]p24.[/b] Let $Y$ be the answer to problem $23$ in this guts round. What is the maximum number of points of intersections of two regular $(Y - 5)$-sided polygons, if no side of the first polygon is parallel to any side of the second polygon? [u]Round 9[/u] [b]p25.[/b] Cross country skiers $s_1, s_2, s_3, ..., s_7$ start a race one by one in that order. While each skier skis at a constant pace, the skiers do not all ski at the same rate. In the course of the race, each skier either overtakes another skier or is overtaken by another skier exactly two times. Find all the possible orders in which they can finish. Write each possible finish as an ordered septuplet $(a, b, c, d, e, f, g)$ where $a, b, c, d, e, f, g$ are the numbers $1-7$ in some order. (So a finishes first, b finishes second, etc.) [b]p26.[/b] Archie the Alchemist is making a list of all the elements in the world, and the proportion of earth, air, fire, and water needed to produce each. He writes the proportions in the form E:A:F:W. If each of the letters represents a whole number from $0$ to $4$, inclusive, how many different elements can Archie list? Note that if Archie lists wood as $2:0:1:2$, then $4:0:2:4$ would also produce wood. In addition, $0:0:0:0$ does not produce an element. [b]p27.[/b] Let $ABCD$ be a rectangle with $AB = 10$ and $BC = 12$. Let $M$ be the midpoint of $CD$, and $P$ be the point on $BM$ such that $DP = DA$. Find the area of quadrilateral $ABPD$. [u]Round 10[/u] [b]p28.[/b] David the farmer has an infinitely large grass-covered field which contains a straight wall. He ties his cow to the wall with a rope of integer length. The point where David ties his rope to the wall divides the wall into two parts of length $a$ and $b$, where $a > b$ and both are integers. The rope is shorter than the wall but is longer than $a$. Suppose that the cow can reach grass covering an area of $\frac{165\pi}{2}$. Find the ratio $\frac{a}{b}$ . You may assume that the wall has $0$ width. [b]p29.[/b] Let $S$ be the number of ordered quintuples $(a, b, x, y, n)$ of positive integers such that $$\frac{a}{x}+\frac{b}{y}=\frac{1}{n}$$ $$abn = 2011^{2011}$$ Compute the remainder when $S$ is divided by $2012$. [b]p30.[/b] Let $n$ be a positive integer. An $n \times n$ square grid is formed by $n^2$ unit squares. Each unit square is then colored either red or blue such that each row or column has exactly $10$ blue squares. A move consists of choosing a row or a column, and recolor each unit square in the chosen row or column – if it is red, we recolor it blue, and if it is blue, we recolor it red. Suppose that it is possible to obtain fewer than $10n$ blue squares after a sequence of finite number of moves. Find the maximum possible value of $n$. PS. You should use hide for answers. First rounds have been posted [url=https://artofproblemsolving.com/community/c4h2786905p24497746]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For real $a, b$ define the sequence $x_{0}, x_{1}, x_{2}, ...$ by $x_{0}= a, x_{n+1}= x_{n}+b \sin x_{n}$. If $b = 1$, show that the sequence converges to a finite limit for all $a$. If $b > 2$, show that the sequence diverges for some $a$.

Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$. [i]Proposed by Nikolay Beluhov, Bulgaria[/i]