Let $a,b$ be integers and $p$ be a prime number such that: (i) $p$ is the greatest common divisor of $a$ and $b$; (ii) $p^2$ divides $a$. Prove that the polynomial $x^{n+2}+ax^{n+1}+bx^{n}+a+b$ cannot be decomposed into the product of two polynomials with integer coefficients and degree greater than $1$.

Let $P(x)$ be a quadratic polynomial with real coefficients such that $P(3) = 7$ and \[P(x) = P(0) + P(1)x + P(2)x^2\] for all real $x$. What is $P(-1)$?

A positive integer \( r \) is given, find the largest real number \( C \) such that there exists a geometric sequence $\{ a_n \}_{n\ge 1}$ with common ratio \( r \) satisfying $$ \| a_n \| \ge C $$ for all positive integers \( n \). Here, $\| x \|$ denotes the distance from the real number \( x \) to the nearest integer.

Prove that the equation $(\sqrt5 +1)^{2x}+ (\sqrt5 -1)^{2x}=2^x(y^2+2)$ has an infinite number of solutions in natural numbers.

Show that the distance between a point on the hyperbola $xy=5$ and a point on the ellipse $x^{2}+6y^{2}=6$ is at least $\frac{9}{7}$.

find all polynomials with integer coefficients that $P(\mathbb{Z})= ${$p(a):a\in \mathbb{Z}$} has a Geometric progression.

In the governor elections, there were three candidates: $A$, $B$, and $C$. In the first round, $A$ received $44\%$ of the votes that were cast between $B$ and $C$. No candidate obtained the majority needed to win in the first round, and $C$ was the one who received the least votes of the three, so there was a runoff between $A$ and $B$. The voters for the runoff were the same as in the first round, except for $p\%$ of those who voted for $C$, who chose not to participate in the runoff; $p$ is an integer, $1 \leqslant p \leqslant 100$. It is also known that all those who voted for $B$ in the first round also voted for him again in the runoff, but it is unknown what those who voted for $A$ in the first round did. A journalist claims that, knowing all this, one can infer with certainty who will win the runoff. Determine for which values of $p$ the journalist is telling the truth. [b]Note:[/b] The winner of the runoff is the one who receives more than half of the total votes cast in the runoff.

Let $\{a_n\}$ be a sequence defined by $a_1=0$ and $$a_n=\frac{1}{n}+\frac{1}{\lceil \frac{n}{2} \rceil}\sum_{k=1}^{\lceil \frac{n}{2} \rceil}a_k$$ for any positive integer $n$. Find the maximal term of this sequence.

[u]Round 5[/u] [b]p13.[/b] A $2016 \times 2016$ chess board is cut into $k \ge 1$ rectangle(s) with positive integer sidelengths. Let $p$ be the sum of the perimeters of all $k$ rectangles. Additionally, let $m$ and $M$ be the minimum and maximum possible value of $\frac{p}{k}$, respectively. Determine the ordered pair $(m,M)$. [b]p14.[/b] For nonnegative integers $n$, let $f (n)$ be the product of the digits of $n$. Compute $\sum^{1000}_{i=1}f (i )$. [b]p15.[/b] How many ordered pairs of positive integers $(m,n)$ have the property that $mn$ divides $2016$? [u]Round 6[/u] [b]p16.[/b] Let $a,b,c$ be distinct integers such that $a +b +c = 0$. Find the minimum possible positive value of $|a^3 +b^3 +c^3|$. [b]p17.[/b] Find the greatest positive integer $k$ such that $11^k -2^k$ is a perfect square. [b]p18.[/b] Find all ordered triples $(a,b,c)$ with $a \le b \le c$ of nonnegative integers such that $2a +2b +2c = ab +bc +ca$. [u]Round 7[/u] [b]p19.[/b] Let $f :N \to N$ be a function such that $f ( f (n))+ f (n +1) = n +2$ for all positive integers $n$. Find $f (20)+ f (16)$. [b]p20.[/b] Let $\vartriangle ABC$ be a triangle with area $10$ and $BC = 10$. Find the minimum possible value of $AB \cdot AC$. [b]p21.[/b] Let $\vartriangle ABC$ be a triangle with sidelengths $AB = 19$, $BC = 24$, $C A = 23$. Let $D$ be a point on minor arc $BC$ of the circumcircle of $\vartriangle ABC$ such that $DB =DC$. A circle with center $D$ that passes through $B$ and $C$ interests $AC$ again at a point $E \ne C$. Find the length of $AE$. [u]Round 8[/u] [b]p22.[/b] Let $m =\frac12 \sqrt{2+\sqrt{2+... \sqrt2}}$, where there are $2014$ square roots. Let $f_1(x) =2x^2 -1$ and let $f_n(x) = f_1( f_{n-1}(x))$. Find $f_{2015}(m)$. [b]p23.[/b] How many ordered triples of integers $(a,b,c)$ are there such that $0 < c \le b \le a \le 2016$, and $a +b-c = 2016$? [b]p24.[/b] In cyclic quadrilateral $ABCD$, $\angle B AD = 120^o$,$\angle ABC = 150^o$,$CD = 8$ and the area of $ABCD$ is $6\sqrt3$. Find the perimeter of $ABCD$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158461p28714996]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3162282p28763571]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The sum of all even positive integers less than $100$ those are not divisible by $3$ is (A): $938$, (B): $940$, (C): $1634$, (D): $1638$, (E): None of the above.

$u$ is a real parameter such that $0<u<1$. For $0\le x \le u$, $f(x)=0$. For $u\le x \le n$, $f(x)=1-\left(\sqrt{ux}+\sqrt{(1-u)(1-x)}\right)^2$. The sequence $\{u_n\}$ is define recursively as follows: $u_1=f(1)$ and $u_n=f(u_{n-1})$ $\forall n\in \mathbb{N}, n\neq 1$. Show that there exists a positive integer $k$ for which $u_k=0$.

Determine all positive integers $ k,n $ for which $ 2^k+10n^2+n^4 $ is a perfect square. [i]Japan EGMO 2016 Shortlist[/i]

Show that there is some rational number in the interval $(0,1)$ that can be expressed as a sum of $2021$ reciprocals of positive integers, but cannot be expressed as a sum of $2020$ reciprocals of positive integers.

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [b]Z15.[/b] Let $AOB$ be a quarter circle with center $O$ and radius $4$. Let $\omega_1$ and $\omega_2$ be semicircles inside $AOB$ with diameters $OA$ and $OB$, respectively. Find the area of the region within $AOB$ but outside of $\omega_1$ and $\omega_2$. [u]Set 4[/u] [b]Z16.[/b] Integers $a, b, c$ form a geometric sequence with an integer common ratio. If $c = a + 56$, find $b$. [b]Z17 / D24.[/b] In parallelogram $ABCD$, $\angle A \cdot \angle C - \angle B \cdot \angle D = 720^o$ where all angles are in degrees. Find the value of $\angle C$. [b]Z18.[/b] Steven likes arranging his rocks. A mountain formation is where the sequence of rocks to the left of the tallest rock increase in height while the sequence of rocks to the right of the tallest rock decrease in height. If his rocks are $1, 2, . . . , 10$ inches in height, how many mountain formations are possible? For example: the sequences $(1-3-5-6-10-9-8-7-4-2)$ and $(1-2-3-4-5-6-7-8-9-10)$ are considered mountain formations. [b]Z19.[/b] Find the smallest $5$-digit multiple of $11$ whose sum of digits is $15$. [b]Z20.[/b] Two circles, $\omega_1$ and $\omega_2$, have radii of $2$ and $8$, respectively, and are externally tangent at point $P$. Line $\ell$ is tangent to the two circles, intersecting $\omega_1$ at $A$ and $\omega_2$ at $B$. Line $m$ passes through $P$ and is tangent to both circles. If line $m$ intersects line $\ell$ at point $Q$, calculate the length of $P Q$. [u]Set 5[/u] [b]Z21.[/b] Sen picks a random $1$ million digit integer. Each digit of the integer is placed into a list. The probability that the last digit of the integer is strictly greater than twice the median of the digit list is closest to $\frac{1}{a}$, for some integer $a$. What is $a$? [b]Z22.[/b] Let $6$ points be evenly spaced on a circle with center $O$, and let $S$ be a set of $7$ points: the $6$ points on the circle and $O$. How many equilateral polygons (not self-intersecting and not necessarily convex) can be formed using some subset of $S$ as vertices? [b]Z23.[/b] For a positive integer $n$, define $r_n$ recursively as follows: $r_n = r^2_{n-1} + r^2_{n-2} + ... + r^2_0$,where $r_0 = 1$. Find the greatest integer less than $$\frac{r_2}{r^2_1}+\frac{r_3}{r^2_2}+ ...+\frac{r_{2023}}{r^2_{2022}}.$$ [b]Z24.[/b] Arnav starts at $21$ on the number line. Every minute, if he was at $n$, he randomly teleports to $2n^2$, $n^2$, or $\frac{n^2}{4}$ with equal chance. What is the probability that Arnav only ever steps on integers? [b]Z25.[/b] Let $ABCD$ be a rectangle inscribed in circle $\omega$ with $AB = 10$. If $P$ is the intersection of the tangents to $\omega$ at $C$ and $D$, what is the minimum distance from $P$ to $AB$? PS. You should use hide for answers. D.1-15 / Z.1-8 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916240p26045561]here [/url]and D.16-30/Z.9-14, 17, 26-30 [url=https://artofproblemsolving.com/community/c3h2916250p26045695]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that for any natural number n, n^3 + (n + 1)^3 + (n + 2)^3 is divisible by 9.

Let $a_1 < a_2< ...< a_n$ are real numbers, $$f(x) = \sum_{i=1}^n|x-a_i|,$$ for $n$ even. Find the minimum of this function.

Let $\mathbb R^\ast$ denote the set of nonzero reals. Find all functions $f: \mathbb R^\ast \to \mathbb R^\ast$ satisfying \[ f(x^2+y)+1=f(x^2+1)+\frac{f(xy)}{f(x)} \] for all $x,y \in \mathbb R^\ast$ with $x^2+y\neq 0$. [i]Proposed by Ryan Alweiss[/i]

(a) Let $a, b, c, d$ be integers such that $ad\ne bc$. Show that is always possible to write the fraction $\frac{1}{(ax+b)(cx+d)}$in the form $\frac{r}{ax+b}+\frac{s}{cx+d}$ (b) Find the sum $$\frac{1}{1 \cdot 4}+\frac{1}{4 \cdot 7}+\frac{1}{7 \cdot 10}+...+\frac{1}{1995 \cdot 1996}$$

A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have $ 3$ rows of small congruent equilateral triangles, with $ 5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $ 2003$ small equilateral triangles? [asy]unitsize(15mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair Ap=(0,0), Bp=(1,0), Cp=(2,0), Dp=(3,0), Gp=dir(60); pair Fp=shift(Gp)*Bp, Ep=shift(Gp)*Cp; pair Hp=shift(Gp)*Gp, Ip=shift(Gp)*Fp; pair Jp=shift(Gp)*Hp; pair[] points={Ap,Bp,Cp,Dp,Ep,Fp,Gp,Hp,Ip,Jp}; draw(Ap--Dp--Jp--cycle); draw(Gp--Bp--Ip--Hp--Cp--Ep--cycle); for(pair p : points) { fill(circle(p, 0.07),white); } pair[] Cn=new pair[5]; Cn[0]=centroid(Ap,Bp,Gp); Cn[1]=centroid(Gp,Bp,Fp); Cn[2]=centroid(Bp,Fp,Cp); Cn[3]=centroid(Cp,Fp,Ep); Cn[4]=centroid(Cp,Ep,Dp); label("$1$",Cn[0]); label("$2$",Cn[1]); label("$3$",Cn[2]); label("$4$",Cn[3]); label("$5$",Cn[4]); for (pair p : Cn) { draw(circle(p,0.1)); }[/asy] $ \textbf{(A)}\ 1,\!004,\!004 \qquad \textbf{(B)}\ 1,\!005,\!006 \qquad \textbf{(C)}\ 1,\!507,\!509 \qquad \textbf{(D)}\ 3,\!015,\!018 \qquad \textbf{(E)}\ 6,\!021,\!018$

Suppose that for a function $f: \mathbb{R}\to \mathbb{R}$ and real numbers $a<b$ one has $f(x)=0$ for all $x\in (a,b).$ Prove that $f(x)=0$ for all $x\in \mathbb{R}$ if \[\sum^{p-1}_{k=0}f\left(y+\frac{k}{p}\right)=0\] for every prime number $p$ and every real number $y.$

Let $a$ and $b$ be complex non-zero numbers and $z_1,z_2$ the roots of the polynomials $X^2+aX+b$. Show that $|z_1+z_2|=|z_1|+|z_2|$ if and only if there exists a real number $\lambda\ge 4$ such that $a^2=\lambda b$.

Let $n\in\mathbb{N}\setminus\left\{0\right\}$ be a positive integer. Find all the functions $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying that : $$f(x+y^{2n})=f(f(x))+y^{2n-1}f(y),(\forall)x,y\in\mathbb{R},$$ and $f(x)=0$ has an unique solution.

Indians find those sequences of non-negative real numbers $x_0, x_1,...$ [i]mystical [/i]t hat satisfy $x_0 < 2021$, $x_{i+1} = \lfloor x_i \rfloor \{x_i\}$ for every $i \ge 0$, furthermore the sequence contains an integer different from $0$. How many sequences are mystical according to the Indians?

p1. Let $U_n$ be a sequence of numbers that satisfy: $U_1=1$, $U_n=1+U_1U_2U_3...U_{n-1}$ for $n=2,3,...,2020$ Prove that $\frac{1}{U_1}+\frac{1}{U_2}+...+\frac{1}{U_{2019}}<2$ p2. If $a= \left \lceil \sqrt{2020+\sqrt{2020+...+\sqrt{2020}}} \right\rceil$ , $b= \left \lfloor \sqrt{1442+\sqrt{1442+...+\sqrt{1442}}} \right \rfloor$, and $c=a-b$, then determine the value of $c$. p3. Fajar will buy a pair of koi fish in the aquarium. If he randomly picks $2$ fish, then the probability that the $2$ fish are of the same sex is $1/2$. Prove that the number of koi fish in the aquarium is a perfect square. p4. A pharmacist wants to put $155$ ml of liquid into $3$ bottles. There are 3 bottle choices, namely a. Bottle A $\bullet$ Capacity: $5$ ml $\bullet$ The price of one bottle is $10,000$ Rp $\bullet$ If you buy the next bottle, you will get a $20\%$ discount, up to the $4$th purchase or if you buy $4$ bottles, get $ 1$ free bottle A b. Bottle B $\bullet$ Capacity: $8$ ml $\bullet$ The price of one bottle is $15.000$ Rp $\bullet$ If you buy $2$ : $20\%$ discount $\bullet$ If you buy $3$ : Free $ 1$ bottle of B c. Bottle C $\bullet$ Capacity : $14$ ml $\bullet$ Buy $ 1$ : $25.000$ Rp $\bullet$ Buy $2$ : Free $ 1$ bottle of A $\bullet$ Buy $3$ : Free $ 1$ bottle of B If in one purchase, you can only buy a maximum of $4$ bottles, then look for the possibility of pharmacists putting them in bottles so that the cost is minimal (bottles do not have to be filled to capacity). p5. Two circles, let's say $L_1$ and $L_2$ have the same center, namely at point $O$. Radius of $L_1$ is $10$ cm and radius of $L_2$ is $5$ cm. The points $A, B, C, D, E, F$ lie on $L_1$ so the arcs $AB,BC,CD,DE,EF,FA$ are equal. The points $P, Q, R$ lie on $L_2$ so that the arcs $PQ,QR,RS$ are equal and $PA=PF=QB=QC=RD=RD$ . Determine the area of ​​the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/b/5/0729eca97488ddfc82ab10eda02c708fecd7ae.png[/img]

Let $0 \leq b \leq c \leq d \leq a$ and $a>14$ are integers. Prove, that there is such natural $n$ that can not be represented as $$n=x(ax+b)+y(ay+c)+z(az+d)$$ where $x,y,z$ are some integers. [i]K. Kohas[/i]