Find the natural numbers $ a $ that have the property that there exists a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ such that $ f(f(n))=a+n, $ for any natural number $ n, $ and the function $ g:\mathbb{N}\longrightarrow\mathbb{N} $ defined as $ g(n)=f(n)-n $ is injective.

[b]p1[/b]. Is it possible to find $n$ positive numbers such that their sum is equal to $1$ and the sum of their squares is less than $\frac{1}{10}$? [b]p2.[/b] In the country of Sepulia, there are several towns with airports. Each town has a certain number of scheduled, round-trip connecting flights with other towns. Prove that there are two towns that have connecting flights with the same number of towns. [b]p3.[/b] A $4 \times 4$ magic square is a $4 \times 4$ table filled with numbers $1, 2, 3,..., 16$ - with each number appearing exactly once - in such a way that the sum of the numbers in each row, in each column, and in each diagonal is the same. Is it possible to complete $\begin{bmatrix} 2 & 3 & * & * \\ 4 & * & * & *\\ * & * & * & *\\ * & * & * & * \end{bmatrix}$ to a magic square? (That is, can you replace the stars with remaining numbers $1, 5, 6,..., 16$, to obtain a magic square?) [b]p4.[/b] Is it possible to label the edges of a cube with the numbers $1, 2, 3, ... , 12$ in such a way that the sum of the numbers labelling the three edges coming into a vertex is the same for all vertices? [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all real numbers $a,b,$ and $c$: (i) If $a+b+c\ge 0$ then $f(a^3)+f(b^3)+f(c^3)\ge 3f(abc).$ (ii) If $a+b+c\le 0$ then $f(a^3)+f(b^3)+f(c^3)\le 3f(abc).$ [i]Proposed by Ashwin Sah[/i]

Let \(c\) be a real number. For all \(x\) and \(y\) real numbers we have, \[f(x-f(y))=f(x-y)+c(f(x)-f(y))\] and \(f(x)\) is not constant. \(a)\) Find all possible values of \(c\). \(b)\) Can \(f\) be periodic?

For matrices $A=[a_{ij}]_{m \times m}$ and $B=[b_{ij}]_{m \times m}$ where $A,B \in \mathbb{Z} ^{m \times m}$ let $A \equiv B \pmod{n}$ only if $a_{ij} \equiv b_{ij} \pmod{n}$ for every $i,j \in \{ 1,2,...,m \}$, that's $A-B=nZ$ for some $Z \in \mathbb{Z}^{m \times m}$. (The symbol $A \in \mathbb{Z} ^{m \times m}$ means that every element in $A$ is an integer.) Prove that for $A \in \mathbb{Z} ^{m \times m}$ there is $B \in \mathbb{Z} ^{m \times m}$ , where $AB \equiv I \pmod{n }$ only if $(\det (A),n)=1$ and find the value of $B$ in the form of $A$ where $I$ represents the dimensional identity matrix $m \times m$. [i](PP-nine)[/i]

Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\] [i]Proposed by Japan[/i]

Let $a,b,c$ be positive real numbers satisfying $a+b+c=3$.Prove that $\sum_{sym} \frac{a^{2}}{(b+c)^{3}}\geq \frac{3}{8}$

Let $p$ be a polynomial with integer coefficients and let $a_1<a_2<\cdots <a_k$ be integers. Given that $p(a_i)\ne 0\forall\; i=1,2,\cdots, k$. [list] (a) Prove $\exists\; a\in \mathbb{Z}$ such that \[ p(a_i)\mid p(a)\;\;\forall i=1,2,\dots ,k \] (b) Does there exist $a\in \mathbb{Z}$ such that \[ \prod_{i=1}^{k}p(a_i)\mid p(a) \][/list]

Let $a, b, c$ be three non-zero integers. It is known that the sums $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are integers. Find these sums.

Show that there exists a positive real number $x\neq 2$ such that $\log_2x=\frac{x}{2}$. Hence obtain the set of real numbers $c$ such that \[\frac{\log_2x}{x}=c\] has only one real solution.

Let $a,b,c>0.$ Prove that$$\frac{a^3+b^2c+bc^2}{bc}+\frac{b^3+c^2a+ca^2}{ca}+\frac{c^3+a^2b+ab^2}{ab}\geq 3(a+b+c)$$ $$\frac{bc}{a^3+b^2c+bc^2}+\frac{ca}{b^3+c^2a+ca^2}+\frac{ab}{c^3+a^2b+ab^2}\leq \frac{1}{3}(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})$$

Let $T = \{1,2,...,10\}$. Find the number of bijective functions $f : T\to T$ that satises the following for all $x \in T$: $f(f(x)) = x$ $|f(x) - x| \ge 2$

Find the sum of all integers $n$ satisfying the following inequality: \[\frac{1}{4}<\sin\frac{\pi}{n}<\frac{1}{3}.\]

Find (with proof) all monic polynomials $f(x)$ with integer coefficients that satisfy the following two conditions. 1. $f (0) = 2004$. 2. If $x$ is irrational, then $f (x)$ is also irrational. (Notes: Apolynomial is monic if its highest degree term has coefficient $1$. Thus, $f (x) = x^4-5x^3-4x+7$ is an example of a monic polynomial with integer coefficients. A number $x$ is rational if it can be written as a fraction of two integers. A number $x$ is irrational if it is a real number which cannot be written as a fraction of two integers. For example, $2/5$ and $-9$ are rational, while $\sqrt2$ and $\pi$ are well known to be irrational.)

Suppose $z_0,z_1,\ldots,z_{n-1}$ are complex numbers such that $z_k=e^{2k\pi i/n}$ for $k=0,1,2,\ldots,n-1$. Prove that for any complex number $z$, $\sum_{k=0}^{n-1}|z-z_k|\ge n$.

Let $ A $ be the set of real solutions of the equation $ 3^x=x+2, $ and let be the set $ B $ of real solutions of the equation $ \log_3 (x+2) +\log_2 \left( 3^x-x \right) =3^x-1 . $ Prove the validity of the following subpoints: [b]a)[/b] $ A\subset B. $ [b]b)[/b] $ B\not\subset\mathbb{Q} \wedge B\not\subset \mathbb{R}\setminus\mathbb{Q} . $

For 3 real numbers $a,b,c$ let $s_n=a^{n}+b^{n}+c^{n}$. It is known that $s_1=2$, $s_2=6$ and $s_3=14$. Prove that for all natural numbers $n>1$, we have $|s^2_n-s_{n-1}s_{n+1}|=8$

Let $ (x_1,x_2,\cdots)$ be a sequence of positive numbers such that $ (8x_2 \minus{} 7x_1)x_1^7 \equal{} 8$ and \[ x_{k \plus{} 1}x_{k \minus{} 1} \minus{} x_k^2 \equal{} \frac {x_{k \minus{} 1}^8 \minus{} x_k^8}{x_k^7x_{k \minus{} 1}^7} \text{ for }k \equal{} 2,3,\ldots \] Determine real number $ a$ such that if $ x_1 > a$, then the sequence is monotonically decreasing, and if $ 0 < x_1 < a$, then the sequence is not monotonic.

Find all functions $f:R -\{0,2\} \to R$ that satisfy for all $x \ne 0,2$ $$f(x) \cdot \left(f\left(\sqrt[3]{\frac{2+x}{2-x}}\right) \right)^2=\frac{x^3}{4}$$

Let $f : (0,+\infty) \to \mathbb R$ be a function having the property that $f(x) = f\left(\frac{1}{x}\right)$ for all $x > 0.$ Prove that there exists a function $u : [1,+\infty) \to \mathbb R$ satisfying $u\left(\frac{x+\frac 1x }{2} \right) = f(x)$ for all $x > 0.$

Given that $a$ is a real solution to the polynomial equation $$nx^n-x^{n-1}-x^{n-2}-\cdots-x-1=0$$ where $n$ is a positive integer, show that $a=1$ or $-1<a<0$.

Let the sequence $(a_n)$ be constructed in the following way: $$ a_1=1,\mbox{ }a_2=1,\mbox{ }a_{n+2}=a_{n+1}+\frac{1}{a_n},\mbox{ }n=1,2,\ldots. $$ Prove that $a_{180}>19$. [i](Folklore)[/i]

[b]p1[/b]. Evaluate $1! + 2! + 3! + 4! + 5! $ (where $n!$ is the product of all integers from $1$ to $n$, inclusive). [b]p2.[/b] Harold opens a pack of Bertie Bott's Every Flavor Beans that contains $10$ blueberry, $10$ watermelon, $3$ spinach and $2$ earwax-flavored jelly beans. If he picks a jelly bean at random, then what is the probability that it is not spinach-flavored? [b]p3.[/b] Find the sum of the positive factors of $32$ (including $32$ itself). [b]p4.[/b] Carol stands at a flag pole that is $21$ feet tall. She begins to walk in the direction of the flag's shadow to say hi to her friends. When she has walked $10$ feet, her shadow passes the flag's shadow. Given that Carol is exactly $5$ feet tall, how long in feet is her shadow? [b]p5.[/b] A solid metal sphere of radius $7$ cm is melted and reshaped into four solid metal spheres with radii $1$, $5$, $6$, and $x$ cm. What is the value of $x$? [b]p6.[/b] Let $A = (2,-2)$ and $B = (-3, 3)$. If $(a,0)$ and $(0, b)$ are both equidistant from $A$ and $B$, then what is the value of $a + b$? [b]p7.[/b] For every flip, there is an $x^2$ percent chance of flipping heads, where $x$ is the number of flips that have already been made. What is the probability that my first three flips will all come up tails? [b]p8.[/b] Consider the sequence of letters $Z\,\,W\,\,Y\,\,X\,\,V$. There are two ways to modify the sequence: we can either swap two adjacent letters or reverse the entire sequence. What is the least number of these changes we need to make in order to put the letters in alphabetical order? [b]p9.[/b] A square and a rectangle overlap each other such that the area inside the square but outside the rectangle is equal to the area inside the rectangle but outside the square. If the area of the rectangle is $169$, then find the side length of the square. [b]p10.[/b] If $A = 50\sqrt3$, $B = 60\sqrt2$, and $C = 85$, then order $A$, $B$, and $C$ from least to greatest. [b]p11.[/b] How many ways are there to arrange the letters of the word $RACECAR$? (Identical letters are assumed to be indistinguishable.) [b]p12.[/b] A cube and a regular tetrahedron (which has four faces composed of equilateral triangles) have the same surface area. Let $r$ be the ratio of the edge length of the cube to the edge length of the tetrahedron. Find $r^2$. [b]p13.[/b] Given that $x^2 + x + \frac{1}{x} +\frac{1}{x^2} = 10$, find all possible values of $x +\frac{1}{x}$ . [b]p14.[/b] Astronaut Bob has a rope one unit long. He must attach one end to his spacesuit and one end to his stationary spacecraft, which assumes the shape of a box with dimensions $3\times 2\times 2$. If he can attach and re-attach the rope onto any point on the surface of his spacecraft, then what is the total volume of space outside of the spacecraft that Bob can reach? Assume that Bob's size is negligible. [b]p15.[/b] Triangle $ABC$ has $AB = 4$, $BC = 3$, and $AC = 5$. Point $B$ is reflected across $\overline{AC}$ to point $B'$. The lines that contain $AB'$ and $BC$ are then drawn to intersect at point $D$. Find $AD$. [b]p16.[/b] Consider a rectangle $ABCD$ with side lengths $5$ and $12$. If a circle tangent to all sides of $\vartriangle ABD$ and a circle tangent to all sides of $\vartriangle BCD$ are drawn, then how far apart are the centers of the circles? [b]p17.[/b] An increasing geometric sequence $a_0, a_1, a_2,...$ has a positive common ratio. Also, the value of $a_3 + a_2 - a_1 - a_0$ is equal to half the value of $a_4 - a_0$. What is the value of the common ratio? [b]p18.[/b] In triangle $ABC$, $AB = 9$, $BC = 11$, and $AC = 16$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{BC}$, respectively, such that $BE = BF = 4$. What is the area of triangle $CEF$? [b]p19.[/b] Xavier, Yuna, and Zach are running around a circular track. The three start at one point and run clockwise, each at a constant speed. After $8$ minutes, Zach passes Xavier for the first time. Xavier first passes Yuna for the first time in $12$ minutes. After how many seconds since the three began running did Zach first pass Yuna? [b]p20.[/b] How many unit fractions are there such that their decimal equivalent has a cycle of $6$ repeating integers? Exclude fractions that repeat in cycles of $1$, $2$, or $3$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] A number of Exonians took a math test. If all of their scores were positive integers and the mean of their scores was $8.6$, find the minimum possible number of students. [b]p2.[/b] Find the least composite positive integer that is not divisible by any of $3, 4$, and $5$. [b]p3.[/b] Five checkers are on the squares of an $8\times 8$ checkerboard such that no two checkers are in the same row or the same column. How many squares on the checkerboard share neither a row nor a column with any of the five checkers? [b]p4.[/b] Let the operation $x@y$ be $y - x$. Compute $((... ((1@2)@3)@ ...@ 2013)@2014)@2015$. [b]p5.[/b] In a town, each family has either one or two children. According to a recent survey, $40\%$ of the children in the town have a sibling. What fraction of the families in the town have two children? [b]p6.[/b] Equilateral triangles $ABE$, $BCF$, $CDG$ and $DAH$ are constructed outside the unit square $ABCD$. Eliza wants to stand inside octagon $AEBFCGDH$ so that she can see every point in the octagon without being blocked by an edge. What is the area of the region in which she can stand? [b]p7.[/b] Let $S$ be the string $0101010101010$. Determine the number of substrings containing an odd number of $1$'s. (A substring is defined by a pair of (not necessarily distinct) characters of the string and represents the characters between, inclusively, the two elements of the string.) [b]p8.[/b] Let the positive divisors of $n$ be $d_1, d_2, ...$ in increasing order. If $d_6 = 35$, determine the minimum possible value of $n$. [b]p9.[/b] The unit squares on the coordinate plane that have four lattice point vertices are colored black or white, as on a chessboard, shown on the diagram below. [img]https://cdn.artofproblemsolving.com/attachments/6/4/f400d52ae9e8131cacb90b2de942a48662ea8c.png[/img] For an ordered pair $(m, n)$, let $OXZY$ be the rectangle with vertices $O = (0, 0)$, $X = (m, 0)$, $Z = (m, n)$ and $Y = (0, n)$. How many ordered pairs $(m, n)$ of nonzero integers exist such that rectangle $OXZY$ contains exactly $32$ black squares? [b]p10.[/b] In triangle $ABC$, $AB = 2BC$. Given that $M$ is the midpoint of $AB$ and $\angle MCA = 60^o$, compute $\frac{CM}{AC}$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].