Let $\mathbb N$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that: (i) The greatest common divisor of the sequence $f(1), f(2), \dots$ is $1$. (ii) For all sufficiently large integers $n$, we have $f(n) \neq 1$ and \[ f(a)^n \mid f(a+b)^{a^{n-1}} - f(b)^{a^{n-1}} \] for all positive integers $a$ and $b$. [i]Proposed by Yang Liu[/i]

For a scalene triangle $ABC$ with an incenter $I$, let its incircle meets the sides $BC, CA, AB$ at $D, E, F$, respectively. Denote by $P$ the intersection of the lines $AI$ and $DF$, and $Q$ the intersection of the lines $BI$ and $EF$. Prove that $\overline{PQ}=\overline{CD}$.

Determine all pairs $(p,q)$ of primes for which both $p^2+q^3$ and $q^2+p^3$ are perfect squares.

As shown in figure , in the pentagon $ABCDE$, $BC = DE$, $CD \parallel BE$, $AB>AE$. If $\angle BAC = \angle DAE$ and $\frac{AB}{BD}=\frac{AE}{ED}$. Prove that $AC$ bisects the line segment $BE$. [img]https://cdn.artofproblemsolving.com/attachments/3/2/5ce44f1e091786b865ae4319bda56c3ddbb8d7.png[/img]

Find all tuples $ (x_1,x_2,x_3,x_4,x_5,x_6)$ of non-negative integers, such that the following system of equations holds: $ x_1x_2(1\minus{}x_3)\equal{}x_4x_5 \\ x_2x_3(1\minus{}x_4)\equal{}x_5x_6 \\ x_3x_4(1\minus{}x_5)\equal{}x_6x_1 \\ x_4x_5(1\minus{}x_6)\equal{}x_1x_2 \\ x_5x_6(1\minus{}x_1)\equal{}x_2x_3 \\ x_6x_1(1\minus{}x_2)\equal{}x_3x_4$

Let $n$ be an integer greater than $1$ such that $n$ could be represented as a sum of the cubes of two rational numbers, prove that $n$ is also the sum of the cubes of two non-negative rational numbers. Proposed by [i]Navid Safaei[/i]

This is rather simple, but I liked it :). Show that a disk cannot be partitioned into two congruent subsets.

There are given 2050 points in a unit cube. Show that there are 5 points lying in an (open) ball with the radius 1/9.

Let $f$ be a real function such that for all $x\ne 0$, $x\ne 1$, $$f (x) + f \left(- \frac{1}{x - 1} \right) =\frac{9}{4x^2} + f\left(1 - \frac{1}{x} \right) .$$ Compute $f \left( \frac{1}{2}\right).$ .

Let be given an acute triangle $ABC$. Show that there exist unique points $A_1 \in BC$, $B_1 \in CA$, $C_1 \in AB$ such that each of these three points is the midpoint of the segment whose endpoints are the orthogonal projections of the other two points on the corresponding side. Prove that the triangle $A_1B_1C_1$ is similar to the triangle whose side lengths are the medians of $\triangle ABC$.

Bertha has $ 6$ daughters and no sons. Some of her daughters have $ 6$ daughters, and the rest have none. Bertha has a total of $ 30$ daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no children? $ \textbf{(A)}\ 22 \qquad \textbf{(B)}\ 23 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 25\qquad \textbf{(E)}\ 26$

Let $G_0, G_1,\ldots$ be infinite open subsets of a Hausdorff space. Prove that there exist some infinite pairwise disjoint open sets $V_0,V_1,\ldots$ and some indices $n_0<n_1<\cdots$ such that $V_i\subseteq G_{n_i}$ for every $i\geqslant 0.$

I have a four-digit palindrome $\underline{a} \ \underline{b} \ \underline{b} \ \underline{a}$ that is divisible by $b$ and is also divisible by the two-digit number $\underline{b} \ \underline{b}.$ Find the number of palindromes satisfying both of these properties.

For a natural number $ n\ge 2, $ calculate the integer part of $ \sqrt[n]{1+n}-\sqrt {2/n} . $ [i]Dan Nedeianu[/i]

How many multiples of $12$ divide $12!$ and have exactly $12$ divisors? [i]Proposed by Adam Bertelli[/i]

Robert tiles a $420 \times 420$ square grid completely with $1 \times 2$ blocks, then notices that the two diagonals of the grid pass through a total of $n$ blocks. Find the sum of all possible values of $n$.

Let us say that a pair of distinct positive integers is nice if their arithmetic mean and their geometric mean are both integer. Is it true that for each nice pair there is another nice pair with the same arithmetic mean? (The pairs $(a, b)$ and $(b, a)$ are considered to be the same pair.) [i]Boris Frenkin[/i]

Let the altitude of $\vartriangle ABC$ from $A$ intersect the circumcircle of $\vartriangle ABC$ at $D$. Let $E$ be a point on line $AD$ such that $E \ne A$ and $AD = DE$. If $AB = 13$, $BC = 14$, and $AC = 15$, what is the area of quadrilateral $BDCE$?

For a real parameter $a$, solve the equation $x^4-2ax^2+x+a^2-a=0$. Find all $a$ for which all solutions are real.

In the city of $\textrm{Las Cobayas}$, the houses are arranged in a rectangular grid of $3$ rows and $n\geq 2$ columns, as illustrated in the figure. $\textrm{Mich}$ plans to move there and wants to tour the city to visit some of the houses in a way that he visits at least one house from each column and does not visit the same house more than once. During his tour, $\textrm{Mich}$ can move between adjacent houses, that is, after visiting a house, he can continue his journey by visiting one of the neighboring houses to the north, south, east, or west, which are at most four. The figure illustrates one $\textrm{Mich´s}$ position (circle), and the houses to which he can move (triangles). Let $f(n)$ be the number of ways $\textrm{Mich}$ can complete his tour starting from a house in the first column and ending at a house in the last column. Prove that $f(n)$ is odd. [asy]size(200); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((2,0)--(3,0)--(3,1)--(2,1)--cycle); draw((4,0)--(5,0)--(5,1)--(4,1)--cycle); draw((0,2)--(1,2)--(1,3)--(0,3)--cycle); draw((2,2)--(3,2)--(3,3)--(2,3)--cycle); draw((4,2)--(5,2)--(5,3)--(4,3)--cycle); draw((0,4)--(1,4)--(1,5)--(0,5)--cycle); draw((2,4)--(3,4)--(3,5)--(2,5)--cycle); draw((4,4)--(5,4)--(5,5)--(4,5)--cycle); fill(circle((0.5,2.5), 0.4), black); fill((0.1262,4.15)--(0.8738,4.15)--(0.5,4.7974)--cycle, black); fill((0.1262,0.15)--(0.8738,0.15)--(0.5,0.7974)--cycle, black); fill((2.1262,2.15)--(2.8738,2.15)--(2.5,2.7974)--cycle, black); fill(circle((6,0.5), 0.07), black); fill(circle((6.3,0.5), 0.07), black); fill(circle((6.6,0.5), 0.07), black); fill(circle((6,2.5), 0.07), black); fill(circle((6.3,2.5), 0.07), black); fill(circle((6.6,2.5), 0.07), black); fill(circle((6,4.5), 0.07), black); fill(circle((6.3,4.5), 0.07), black); fill(circle((6.6,4.5), 0.07), black); draw((8,0)--(9,0)--(9,1)--(8,1)--cycle); draw((10,0)--(11,0)--(11,1)--(10,1)--cycle); draw((8,2)--(9,2)--(9,3)--(8,3)--cycle); draw((10,2)--(11,2)--(11,3)--(10,3)--cycle); draw((8,4)--(9,4)--(9,5)--(8,5)--cycle); draw((10,4)--(11,4)--(11,5)--(10,5)--cycle); draw((0,-0.2)--(0,-0.5)--(5.5,-0.5)--(5.5,-0.8)--(5.5,-0.5)--(11,-0.5)--(11,-0.5)--(11,-0.2)); label("$n$", (5.22,-1.15), dir(0), fontsize(10)); label("$\textrm{West}$", (-2,2.5), dir(0), fontsize(10)); label("$\textrm{East}$", (11.1,2.5), dir(0), fontsize(10)); label("$\textrm{North}$", (4.5,5.7), dir(0), fontsize(10)); label("$\textrm{South}$", (4.5,-2), dir(0), fontsize(10)); draw((0.5,2.5)--(2,2.5)--(1.8,2.7)--(2,2.5)--(1.8,2.3)); draw((0.5,2.5)--(0.5,4)--(0.3,3.7)--(0.5,4)--(0.7,3.7)); draw((0.5,2.5)--(0.5,1)--(0.3,1.3)--(0.5,1)--(0.7,1.3)); [/asy]

Find all pairs $(p,q)$ of real numbers such that $p+q=1998$ and the solutions of the equation $x^2+px+q=0$ are integers.

What is the least integer a greater than $14$ so that the triangle with side lengths $a - 1$, $a$, and $a + 1$ has integer area?

a) In the decimal expression of a positive number, $a$, all decimals beginning with the third after the decimal point, are deleted (i.e., we take an approximation of $a$ with accuracy to $0.01$ with deficiency). The number obtained is divided by $a$ and the quotient is similarly approximated with the same accuracy by a number $b$. What numbers $b$ can be thus obtained? Write all their possible values. b) same as (a) but with accuracy to $0.001$ c) same as (a) but with accuracy to $0.0001$

For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$. (1) Find the minimum value of $f(x)$. (2) Evaluate $\int_0^1 f(x)\ dx$. [i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(f(x)+y)+xf(y)=f(xy+y)+f(x)$$ for reals $x, y$.