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].

Find all natural numbers $N$ consisting of exactly $1112$ digits (in decimal notation) such that: (a) The sum of the digits of $N$ is divisible by $2000$; (b) The sum of the digits of $N+1$ is divisible by $2000$; (c) $1$ is a digit of $N$.

Color all points on a plane in red or blue. Prove that there exists two similar triangles, their similarity ratio is $1995$, and apexes of both triangles are in the same color.

For every natural number $n$ we define the derived number $n'$ as follows: $\bullet$ $0' = 1' = 0$ $\bullet$ if $n$ is prime, then $n' = 1$ $\bullet$ if $n = a \cdot b$, then $n' = a' b + a b'$ . For example: $15' = 3' 5 + 3 5' = 1\cdot 5 + 3\cdot 1 = 8$. Determine all natural numbers $n$ for which $n = n'$.

Let $ABC$ be the right-angled isosceles triangle whose equal sides have length 1. $P$ is a point on the hypotenuse, and the feet of the perpendiculars from $P$ to the other sides are $Q$ and $R$. Consider the areas of the triangles $APQ$ and $PBR$, and the area of the rectangle $QCRP$. Prove that regardless of how $P$ is chosen, the largest of these three areas is at least $2/9$.

Let $n$ be a fixed positive integer. Let $$A=\begin{bmatrix} a_{11} & a_{12} & \cdots &a_{1n} \\ a_{21} & a_{22} & \cdots &a_{2n} \\ \vdots & \vdots & \cdots & \vdots \\ a_{n1} & a_{n2} & \cdots &a_{nn} \end{bmatrix}\quad \text{and} \quad B=\begin{bmatrix} b_{11} & b_{12} & \cdots &b_{1n} \\ b_{21} & b_{22} & \cdots &b_{2n} \\ \vdots & \vdots & \cdots & \vdots \\ b_{n1} & b_{n2} & \cdots &b_{nn} \end{bmatrix}\quad$$ be two $n\times n$ tables such that $\{a_{ij}|1\le i,j\le n\}=\{b_{ij}|1\le i,j\le n\}=\{k\in N^*|1\le k\le n^2\}$. One can perform the following operation on table $A$: Choose $2$ numbers in the same row or in the same column of $A$, interchange these $2$ numbers, and leave the remaining $n^2-2$ numbers unchanged. This operation is called a [b]transposition[/b] of $A$. Find, with proof, the smallest positive integer $m$ such that for any tables $A$ and $B$, one can perform at most $m$ transpositions such that the resulting table of $A$ is $B$.

Find all functions $f:\mathbb N\to\mathbb N$ satisfying $$\operatorname{lcm}(f(x),y)\gcd(f(x),f(y))=f(x)f(f(y))$$ for all $x,y\in\mathbb N$.

In square $ABCD$ with side length $2$, let $M$ be the midpoint of $AB$. Let $N$ be a point on $AD$ such that $AN = 2ND$. Let point $P$ be the intersection of segment $MN$ and diagonal $AC$. Find the area of triangle $BPM$. [i]Proposed by Jacob Xu[/i]

We have $31$ pieces where $1$ is written on two of them, $2$ is written on eight of them, $3$ is written on twelve of them, $4$ is written on four of them, and $5$ is written on five of them. We place $30$ of them into a $5\times 6$ chessboard such that the sum of numbers on any row is equal to a fixed number and the sum of numbers on any column is equal to a fixed number. What is the number written on the piece which is not placed? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

Find all groups of positive integers $ (a,x,y,n,m)$ that satisfy $ a(x^n \minus{} x^m) \equal{} (ax^m \minus{} 4) y^2$ and $ m \equiv n \pmod{2}$ and $ ax$ is odd.

The solution set of $6x^2+5x<4$ is the set of all values of $x$ such that $\textbf{(A) }\textstyle -2<x<1\qquad\textbf{(B) }-\frac{4}{3}<x<\frac{1}{2}\qquad\textbf{(C) }-\frac{1}{2}<x<\frac{4}{3}\qquad$ $\textbf{(D) }x<\textstyle\frac{1}{2}\text{ or }x>-\frac{4}{3}\qquad\textbf{(E) }x<-\frac{4}{3}\text{ or }x>\frac{1}{2}$

Given 2n+3 points in the plane, no three on a line and no four on a circle, prove that it is always possible to find a circle C that goes through three of the given points and splits the other 2n in half, that is, has n on the inside and n on the outside.