A planar polygon $A_1A_2A_3 . . .A_{n-1}A_n$ ($n > 4$) is made of rigid rods that are connected by hinges. Is it possible to bend the polygon (at hinges only!) into a triangle?

For distinct positive integers $a, b<2012$, define $f(a, b)$ to be the number of integers $k$ with $1\le k<2012$ such that the remainder when $ak$ divided by $2012$ is greater than that of $bk$ divided by $2012$. Let $S$ be the minimum value of $f(a, b)$, where $a$ and $b$ range over all pairs of distinct positive integers less than $2012$. Determine $S$.

Let $x_1,x_2, ..., x_n$ be real numbers satisfying $\sum_{k=1}^{n-1} min(x_k; x_{k+1}) = min(x_1; x_n)$. Prove that $\sum_{k=2}^{n-1} x_k \ge 0$.

Find all triplets $(x, y, z)$ of real numbers that satisfy the equation $$2^{x^2-3y+z}+2^{y^2-3z+x}+2^{z^2-3x+y}=1,5.$$

João calculated the product of the non zero digits of each integer from $1$ to $10^{2009}$ and then he summed these $10^{2009}$ products. Which number did he obtain?

Canmoo is trying to do constructions, but doesn't have a ruler or compass. Instead, Canmoo has a device that, given four distinct points $A$, $B$, $C$, $P$ in the plane, will mark the isogonal conjugate of $P$ with respect to triangle $ABC$, if it exists. Show that if two points are marked on the plane, then Canmoo can construct their midpoint using this device, a pencil for marking additional points, and no other tools. (Recall that the [i]isogonal conjugate[/i] of $P$ with respect to triangle $ABC$ is the point $Q$ such that lines $AP$ and $AQ$ are reflections around the bisector of $\angle BAC$, lines $BP$ and $BQ$ are reflections around the bisector of $\angle CBA$, lines $CP$ and $CQ$ are reflections around the bisector of $\angle ACB$. Additional points marked by the pencil can be assumed to be in general position, meaning they don't lie on any line through two existing points or any circle through three existing points.) [i]Maxim Li[/i]

In the cyclic quadrilateral $ABCD$ with $AB = AD$, points $M$ and $N$ lie on the sides $CD$ and $BC$ respectively so that $MN = BN + DM$. Lines $AM$ and $AN$ meet the circumcircle of $ABCD$ again at points $P$ and $Q$ respectively. Prove that the orthocenter of the triangle $APQ$ lies on the segment $MN$.

The diagonals of a trapezoid $ ABCD $ whose bases are $ [AB] $ and $ [CD] $ intersect at $P.$ Prove that \[S_{PAB} + S_{PCD} > S_{PBC} + S_{PDA},\] Where $S_{XYZ} $ denotes the area of $\triangle XYZ $.

In the diagram, $OB_i$ is parallel and equal in length to $A_i A_{i + 1}$ for $i = 1$, 2, 3, and 4 ($A_5 = A_1$). Show that the area of $B_1 B_2 B_3 B_4$ is twice that of $A_1 A_2 A_3 A_4$. [asy] unitsize(1 cm); pair O; pair[] A, B; O = (0,0); A[1] = (0.5,-3); A[2] = (2,0); A[3] = (-0.2,0.5); A[4] = (-1,0); B[1] = A[2] - A[1]; B[2] = A[3] - A[2]; B[3] = A[4] - A[3]; B[4] = A[1] - A[4]; draw(A[1]--A[2]--A[3]--A[4]--cycle); draw(B[1]--B[2]--B[3]--B[4]--cycle); draw(O--B[1]); draw(O--B[2]); draw(O--B[3]); draw(O--B[4]); label("$A_1$", A[1], S); label("$A_2$", A[2], E); label("$A_3$", A[3], N); label("$A_4$", A[4], W); label("$B_1$", B[1], NE); label("$B_2$", B[2], W); label("$B_3$", B[3], SW); label("$B_4$", B[4], S); label("$O$", O, E); [/asy]

Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. Take five good haikus Scramble their lines randomly What are the chances That you end up with Five completely good haikus (With five, seven, five)? Your answer will be m over n where m,n Are numbers such that m,n positive Integers where gcd Of m,n is 1. Take this answer and Add the numerator and Denominator. [i]Proposed by Jeff Lin[/i]

Find all integer-valued polynomials $$f, g : \mathbb{N} \rightarrow \mathbb{N} \text{ such that} \; \forall \; x \in \mathbb{N}, \tau (f(x)) = g(x)$$ holds for all positive integer $x$, where $\tau (x)$ is the number of positive factors of $x$ [i]Proposed By - ckliao914[/i]

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[ f(x^2 - y^2) = x f(x) - y f(y) \] for all pairs of real numbers $x$ and $y$.

Let $n$ be a positive integer, which gives remainder $4$ of dividing by $8$. Numbers $1 = k_1 < k_2 < ... < k_m = n$ are all positive diivisors of $n$. Show that if $i \in \{ 1, 2, ..., m - 1 \}$ isn't divisible by $3$, then $k_{i + 1} \le 2k_{i}$.

On an escalator which is not moving, a person descends faster than he ascends. Is it faster for this person to descend and ascend once on an upward-moving escalator, or to descend and ascend once on a downward-moving escalator? (It is assumed that all the speeds mentioned here are constant, that the speed of the escalator is the same no matter if it is moving up or down and that the speed of the person relative to the escalator is always greater than the speed of the escalator.) (Folklore)

It is given a convex quadrilateral $ ABCD$ in which $ \angle B\plus{}\angle C < 180^0$. Lines $ AB$ and $ CD$ intersect in point E. Prove that $ CD*CE\equal{}AC^2\plus{}AB*AE \leftrightarrow \angle B\equal{} \angle D$

If the graph is a graph of POSITION vs. TIME, then the squirrel has the greatest speed at what time(s) or during what time interval(s)? (A) From A to B (B) From B to C only (C) From B to D (D) From C to D only (E) From D to E

Recall that $[x]$ denotes the largest integer not exceeding $x$ for real $x$. For integers $a, b$ in the interval $1 \leq a<b \leq 100$, find the number of ordered pairs $(a, b)$ satisfying the following equation. $$[a+\frac{b}{a}]=[b+\frac{a}{b}]$$

Consider the set of integers $ \left \{ 1, 2, \dots , 100 \right \} $. Let $ \left \{ x_1, x_2, \dots , x_{100} \right \}$ be some arbitrary arrangement of the integers $ \left \{ 1, 2, \dots , 100 \right \}$, where all of the $x_i$ are different. Find the smallest possible value of the sum $$S = \left | x_2 - x_1 \right | + \left | x_3 - x_2 \right | + \cdots+ \left |x_{100} - x_{99} \right | + \left |x_1 - x_{100} \right | .$$

Today is $12/14/24,$ which is of the form $ab/ac/bc$ for not necessarily distinct digits $a$, $b$, and $c$. Find the number of other dates in the $21$st century that can also be written in this form.

Let $a,b,c$ be positive real numbers that satisfy $a+b+c=abc$. Prove that $$\sqrt{(1+a^2)(1+b^2)}+\sqrt{(1+b^2)(1+c^2)}+\sqrt{(1+a^2)(1+c^2)}-\sqrt{(1+a^2)(1+b^2)(1+c^2)} \ge 4.$$

Two people A and B play a game in the $m \times n$ grid ($m,n \in \mathbb{N^*}$). Each person respectively (A plays first) draw a segment between two point of the grid such that this segment doesn't contain any point (except the 2 ends) and also the segment (except the 2 ends) doesn't intersect with any other segments. The last person who can't draw is the loser. Which one (of A and B) have the winning tactics?

Find all natural numbers $n, k$ such that $$ 2^n – 5^k = 7. $$

Let $x>0.$ Prove that $$2^{-x}+2^{-1/x} \leq 1.$$

[b]p1.[/b] Suppose $5$ bales of hay are weighted two at a time in all possible ways. The weights obtained are $110$, $112$, $113$, $114$, $115$, $116$, $117$, $118$, $120$, $121$. What is the difference between the heaviest and the lightest bale? [b]p2.[/b] Paul and Paula are playing a game with dice. Each have an $8$-sided die, and they roll at the same time. If the number is the same they continue rolling; otherwise the one who rolled a higher number wins. What is the probability that the game lasts at most $3$ rounds? [b]p3[/b]. Find the unique positive integer $n$ such that $\frac{n^3+5}{n^2-1}$ is an integer. [b]p4.[/b] How many numbers have $6$ digits, some four of which are $2, 0, 1, 4$ (not necessarily consecutive or in that order) and have the sum of their digits equal to $9$? [b]p5.[/b] The Duke School has $N$ students, where $N$ is at most $500$. Every year the school has three sports competitions: one in basketball, one in volleyball, and one in soccer. Students may participate in all three competitions. A basketball team has $5$ spots, a volleyball team has $6$ spots, and a soccer team has $11$ spots on the team. All students are encouraged to play, but $16$ people choose not to play basketball, $9$ choose not to play volleyball and $5$ choose not to play soccer. Miraculously, other than that all of the students who wanted to play could be divided evenly into teams of the appropriate size. How many players are there in the school? [b]p6.[/b] Let $\{a_n\}_{n\ge 1}$ be a sequence of real numbers such that $a_1 = 0$ and $a_{n+1} =\frac{a_n-\sqrt3}{\sqrt3 a_n+1}$ . Find $a_1 + a_2 +.. + a_{2014}$. [b]p7.[/b] A soldier is fighting a three-headed dragon. At any minute, the soldier swings her sword, at which point there are three outcomes: either the soldier misses and the dragon grows a new head, the soldier chops off one head that instantaneously regrows, or the soldier chops off two heads and none grow back. If the dragon has at least two heads, the soldier is equally likely to miss or chop off two heads. The dragon dies when it has no heads left, and it overpowers the soldier if it has at least five heads. What is the probability that the soldier wins [b]p8.[/b] A rook moves alternating horizontally and vertically on an infinite chessboard. The rook moves one square horizontally (in either direction) at the first move, two squares vertically at the second, three horizontally at the third and so on. Let $S$ be the set of integers $n$ with the property that there exists a series of moves such that after the $n$-th move the rock is back where it started. Find the number of elements in the set $S \cap \{1, 2, ..., 2014\}$. [b]p9.[/b] Find the largest integer $n$ such that the number of positive integer divisors of $n$ (including $1$ and $n$) is at least $\sqrt{n}$. [b]p10.[/b] Suppose that $x, y$ are irrational numbers such that $xy$, $x^2 + y$, $y^2 + x$ are rational numbers. Find $x + y$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].