Let $q$ be a positive rational number. Two ants are initially at the same point $X$ in the plane. In the $n$-th minute $(n = 1,2,...)$ each of them chooses whether to walk due north, east, south or west and then walks the distance of $q^n$ metres. After a whole number of minutes, they are at the same point in the plane (not necessarily $X$), but have not taken exactly the same route within that time. Determine all possible values of $q$. Proposed by Jeremy King, UK

Natural number $k$ has $n$ digits in its decimal notation. It was rounded up to tens, then the obtained number was rounded up to hundreds, and so on $(n-1)$ times. Prove that the obtained number $m$ satisfies inequality $m < \frac{18k}{13}$. (Examples of rounding: $191\to190\to 200, 135\to140\to 100$.)

Consider the set $\mathbb Q^2$ of points in $\mathbb R^2$, both of whose coordinates are rational. [b](a)[/b] Prove that the union of segments with vertices from $\mathbb Q^2$ is the entire set $\mathbb R^2$. [b](b)[/b] Is the convex hull of $\mathbb Q^2$ (i.e., the smallest convex set in $\mathbb R^2$ that contains $\mathbb Q^2$) equal to $\mathbb R^2$ ?

Find the amount of different values given by the following expression: $\frac{n^2-2}{n^2-n+2}$ where $ n \in \{1,2,3,..,100\}$

Find all real $x,y,z$ that \[\left\{\begin{array}{c}x+y+zx=\frac12\\ \\ y+z+xy=\frac12\\ \\ z+x+yz=\frac12\end{array}\right.\]

Find all polynomials $P(x)$ with integer coefficients, such that for all positive integers $m, n$, $$m+n \mid P^{(m)}(n)-P^{(n)}(m).$$ [i]Proposed by Navid Safaei, Iran[/i]

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f(x^3)+f(y^3)=(x+y)f(x^2)+f(y^2)- f(xy)\] for all $x\in\mathbb{R}$.

Simplify i) $1+\frac{2a+\dfrac{2}{a}}{a+\dfrac{1}{a}}$ ii) $\frac{3b+\dfrac{3}{b}+\dfrac{3}{b^2}}{b+\dfrac{1}{b}+\dfrac{1}{b^2}}$ iii) $\frac{\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{ab}\right)a^6b^2-a^6-a^5b}{a^4b}$

Let $x$ and $y$ are real numbers such that $x^2y^2+2yx^2+1=0.$ If $S=\frac{2}{x^2}+1+\frac{1}{x}+y(y+2+\frac{1}{x})$, find (a)max$S$ and (b) min$S$.

Show that there exists a finite set $A \subset \mathbb{R}^2$ such that for every $X \in A$ there are points $Y_1, Y_2, \ldots, Y_{1993}$ in $A$ such that the distance between $X$ and $Y_i$ is equal to 1, for every $i.$

Let $n$ be a natural number and $f(x)$ be a polynomial with real coefficients having $n$ different positive real roots. Is it possible the polynomial: $$x(x+1)(x+2)(x+4)f(x)+a$$ to be presented as the $k$-th power of a polynomial with real coefficients, for some natural $k\geq 2$ and real $a$?

[b]p1.[/b] How many nonzero digits are in the number $(5^{94} + 5^{92})(2^{94} + 2^{92})$? [b]p2.[/b] Suppose $A$ is a set of $2013$ distinct positive integers such that the arithmetic mean of any subset of $A$ is also an integer. Find an example of $A$. [b]p3.[/b] How many minutes until the smaller angle formed by the minute and hour hands on the face of a clock is congruent to the smaller angle between the hands at $5:15$ pm? Round your answer to the nearest minute. [b]p4.[/b] Suppose $a$ and $b$ are positive real numbers, $a + b = 1$, and $$1 +\frac{a^2 + 3b^2}{2ab}=\sqrt{4 +\frac{a}{b}+\frac{3b}{a}}.$$ Find $a$. [b]p5.[/b] Suppose $f(x) = \frac{e^x- 12e^{-x}}{ 2}$ . Find all $x$ such that $f(x) = 2$. [b]p6.[/b] Let $P_1$, $P_2$,$...$,$P_n$ be points equally spaced on a unit circle. For how many integer $n \in \{2, 3, ... , 2013\}$ is the product of all pairwise distances: $\prod_{1\le i<j\le n} P_iP_j$ a rational number? Note that $\prod$ means the product. For example, $\prod_{1\le i\le 3} i = 1\cdot 2 \cdot 3 = 6$. [b]p7.[/b] Determine the value $a$ such that the following sum converges if and only if $r \in (-\infty, a)$ : $$\sum^{\infty}_{n=1}(\sqrt{n^4 + n^r} - n^2).$$ Note that $\sum^{\infty}_{n=1}\frac{1}{n^s}$ converges if and only if $s > 1$. [b]p8.[/b] Find two pairs of positive integers $(a, b)$ with $a > b$ such that $a^2 + b^2 = 40501$. [b]p9.[/b] Consider a simplified memory-knowledge model. Suppose your total knowledge level the night before you went to a college was $100$ units. Each day, when you woke up in the morning you forgot $1\%$ of what you had learned. Then, by going to lectures, working on the homework, preparing for presentations, you had learned more and so your knowledge level went up by $10$ units at the end of the day. According to this model, how long do you need to stay in college until you reach the knowledge level of exactly $1000$? [b]p10.[/b] Suppose $P(x) = 2x^8 + x^6 - x^4 +1$, and that $P$ has roots $a_1$, $a_2$, $...$ , $a_8$ (a complex number $z$ is a root of the polynomial $P(x)$ if $P(z) = 0$). Find the value of $$(a^2_1-2)(a^2_2-2)(a^2_3-2)...(a^2_8-2).$$ [b]p11.[/b] Find all values of $x$ satisfying $(x^2 + 2x-5)^2 = -2x^2 - 3x + 15$. [b]p12.[/b] Suppose $x, y$ and $z$ are positive real numbers such that $$x^2 + y^2 + xy = 9,$$ $$y^2 + z^2 + yz = 16,$$ $$x^2 + z^2 + xz = 25.$$ Find $xy + yz + xz$ (the answer is unique). [b]p13.[/b] Suppose that $P(x)$ is a monic polynomial (i.e, the leading coefficient is $1$) with $20$ roots, each distinct and of the form $\frac{1}{3^k}$ for $k = 0,1,2,..., 19$. Find the coefficient of $x^{18}$ in $P(x)$. [b]p14.[/b] Find the sum of the reciprocals of all perfect squares whose prime factorization contains only powers of $3$, $5$, $7$ (i.e. $\frac{1}{1} + \frac{1}{9} + \frac{1}{25} + \frac{1}{419} + \frac{1}{811} + \frac{1}{215} + \frac{1}{441} + \frac{1}{625} + ...$). [b]p15.[/b] Find the number of integer quadruples $(a, b, c, d)$ which also satisfy the following system of equations: $$1+b + c^2 + d^3 =0,$$ $$a + b^2 + c^3 + d^4 =0,$$ $$a^2 + b^3 + c^4 + d^5 =0,$$ $$a^3+b^4+c^5+d^6 =0.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] What is the maximum number of points of intersection between a square and a triangle, assuming that no side of the triangle is parallel to any side of the square? [b]p2.[/b] Two angles of an isosceles triangle measure $80^o$ and $x^o$. What is the sum of all the possible values of $x$? [b]p3.[/b] Let $p$ and $q$ be prime numbers such that $p + q$ and p + $7q$ are both perfect squares. Find the value of $pq$. [b]p4.[/b] Anna, Betty, Carly, and Danielle are four pit bulls, each of which is either wearing or not wearing lipstick. The following three facts are true: (1) Anna is wearing lipstick if Betty is wearing lipstick. (2) Betty is wearing lipstick only if Carly is also wearing lipstick. (3) Carly is wearing lipstick if and only if Danielle is wearing lipstick The following five statements are each assigned a certain number of points: (a) Danielle is wearing lipstick if and only if Carly is wearing lipstick. (This statement is assigned $1$ point.) (b) If Anna is wearing lipstick, then Betty is wearing lipstick. (This statement is assigned $6$ points.) (c) If Betty is wearing lipstick, then both Anna and Danielle must be wearing lipstick. (This statement is assigned $10$ points.) (d) If Danielle is wearing lipstick, then Anna is wearing lipstick. (This statement is assigned $12$ points.) (e) If Betty is wearing lipstick, then Danielle is wearing lipstick. (This statement is assigned $14$ points.) What is the sum of the points assigned to the statements that must be true? (For example, if only statements (a) and (d) are true, then the answer would be $1 + 12 = 13$.) [b]p5.[/b] Let $f(x)$ and $g(x)$ be functions such that $f(x) = 4x + 3$ and $g(x) = \frac{x + 1}{4}$. Evaluate $g(f(g(f(42))))$. [b]p6.[/b] Let $A,B,C$, and $D$ be consecutive vertices of a regular polygon. If $\angle ACD = 120^o$, how many sides does the polygon have? [b]p7.[/b] Fred and George have a fair $8$-sided die with the numbers $0, 1, 2, 9, 2, 0, 1, 1$ written on the sides. If Fred and George each roll the die once, what is the probability that Fred rolls a larger number than George? [b]p8.[/b] Find the smallest positive integer $t$ such that $(23t)^3 - (20t)^3 - (3t)^3$ is a perfect square. [b]p9.[/b] In triangle $ABC$, $AC = 8$ and $AC < AB$. Point $D$ lies on side BC with $\angle BAD = \angle CAD$. Let $M$ be the midpoint of $BC$. The line passing through $M$ parallel to $AD$ intersects lines $AB$ and $AC$ at $F$ and $E$, respectively. If $EF =\sqrt2$ and $AF = 1$, what is the length of segment $BC$? (See the following diagram.) [img]https://cdn.artofproblemsolving.com/attachments/2/3/4b5dd0ae28b09f5289fb0e6c72c7cbf421d025.png[/img] [b]p10.[/b] There are $2011$ evenly spaced points marked on a circular table. Three segments are randomly drawn between pairs of these points such that no two segments share an endpoint on the circle. What is the probability that each of these segments intersects the other two? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine the range of $w(w + x)(w + y)(w + z)$, where $x, y, z$, and $w$ are real numbers such that \[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\]

The equation $x^3y+xy^3+xy=0$ represents $\textbf{(A)}~\text{a circle}$ $\textbf{(B)}~\text{a circle and a pair of straight lines}$ $\textbf{(C)}~\text{a rectangular hyperbola}$ $\textbf{(D)}~\text{a pair of straight lines}$

Let $f (x) = x^2 + ax + b$ be a quadratic function with real coefficients $a, b$. It is given that the equation $f (f (x)) = 0$ has $4$ distinct real roots and the sum of $2$ roots among these roots is equal to $-1$. Prove that $b \le -\frac14$

Does there exist a quadratic trinomial $f(x)$ such that $f(1/2017)=1/2018$, $f(1/2018)=1/2017$, and two of its coefficients are integers? (A. Khrabrov)

Find all functions $f, g: \mathbb{R} \rightarrow \mathbb{R} $ satisfying the following conditions: [list][*] $f$ is not a constant function and if $x \le y$ then $f(x)\le f(y)$ [*] For all real number $x$, $f(g(x))=g(f(x))=0$ [*] For all real numbers $x$ and $y$, $f(x)+f(y)+g(x)+g(y)=f(x+y)+g(x+y)$ [*] For all real numbers $x$ and $y$, $f(x)+f(y)+f(g(x)+g(y))=f(x+y)$ [/list]

Show that there are infintely many pairs $(a,b)$ of relatively prime integers (not necessarily positive) such that both the equations \begin{eqnarray*} x^2 +ax +b &=& 0 \\ x^2 + 2ax + b &=& 0 \\ \end{eqnarray*} have integer roots.

Prove the identity \[ 1 \plus{} \frac{1}{2} \minus{} \frac{2}{3} \plus{} \frac{1}{4} \plus{} \frac{1}{5} \minus{} \frac{2}{6} \plus{} \ldots \plus{} \frac{1}{478} \plus{} \frac{1}{479} \minus{} \frac{2}{480} \equal{} 2 \cdot \sum^{159}_{k\equal{}0} \frac{641}{(161\plus{}k) \cdot (480\minus{}k)}.\]

[color=blue][size=150]PERU TST IMO - 2006[/size] Saturday, may 20.[/color] [b]Question 02[/b] Find all pairs $(a,b)$ real positive numbers $a$ and $b$ such that: $[a[bn]]= n-1,$ for all $n$ positive integer. Note: [x] denotes the integer part of $x$. ---------- [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88510]Spanish version[/url] $\text{\LaTeX}{}$ed by carlosbr

Consider the graphs of $y=2\log{x}$ and $y=\log{2x}$. We may say that: $ \textbf{(A)}\ \text{They do not intersect}$ $ \qquad\textbf{(B)}\ \text{They intersect at 1 point only}$ $\qquad\textbf{(C)}\ \text{They intersect at 2 points only}$ $\qquad\textbf{(D)}\ \text{They intersect at a finite number of points but greater than 2 }$ ${\qquad\textbf{(E)}\ \text{They coincide} } $

Let $P_0(x)=x^3-4x$. Sequence of polynomials is defined as following:\\ $P_{n+1}=P_n(1+x)P_n(1-x)-1$.\\ Prove that $x^{2016}|P_{2016}(x)$.

Johnny writes down quadratic equation \[ax^2 + bx + c = 0.\] with positive integer coefficients $a, b, c$. Then Pete changes one, two, or none “$+$” signs to “$-$”. Johnny wins, if both roots of the (changed) equation are integers. Otherwise (if there are no real roots or at least one of them is not an integer), Pete wins. Can Johnny choose the coefficients in such a way that he will always win?

Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that $$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$ for all positive integers $n$. Show that $a_{2022}\leq 1$.