Determine, whether exists function $f$, which assigns each integer $k$, nonnegative integer $f(k)$ and meets the conditions: $f(0) > 0$, for each integer $k$ minimal number of the form $f(k - l) + f(l)$, where $l \in \mathbb{Z}$, equals $f(k)$.

An infinite sequence is given by $x_1=2, x_2=7, x_{n+1} = 4x_n - x_{n-1}$ for all $n \geq 2$. Does there exist a perfect square in this sequence? [hide="Remark"]During the test the initial value of $x_1$ was given as $1$, thus the problem was not graded[/hide]

The positive integers $a,b,c$ are less than $99$ and satisfy $a^2+b^2=c^2+99^2$. . Find the minimum and maximum value of $a+b+c$.

[u]Set 1 [/u] [b]1.1[/b] Compute the number of real numbers x such that the sequence $x$, $x^2$, $x^3$,$ x^4$, $x^5$, $...$ eventually repeats. (To be clear, we say a sequence “eventually repeats” if there is some block of consecutive digits that repeats past some point—for instance, the sequence $1$, $2$, $3$, $4$, $5$, $6$, $5$, $6$, $5$, $6$, $...$ is eventually repeating with repeating block $5$, $6$.) [b]1.2[/b] Let $T$ be the answer to the previous problem. Nicole has a broken calculator which, when told to multiply $a$ by $b$, starts by multiplying $a$ by $b$, but then multiplies that product by b again, and then adds $b$ to the result. Nicole inputs the computation “$k \times k$” into the calculator for some real number $k$ and gets an answer of $10T$. If she instead used a working calculator, what answer should she have gotten? [b]1.3[/b] Let $T$ be the answer to the previous problem. Find the positive difference between the largest and smallest perfect squares that can be written as $x^2 + y^2$ for integers $x, y$ satisfying $\sqrt{T} \le x \le T$ and $\sqrt{T} \le y \le T$. PS. You should use hide for answers.

[b]p1.[/b] How many two-digit primes have a units digit of $3$? [b]p2.[/b] How many ways can you arrange the letters $A$, $R$, and $T$ such that it makes a three letter combination? Each letter is used once. [b]p3.[/b] Hanna and Kevin are running a $100$ meter race. If Hanna takes $20$ seconds to finish the race and Kevin runs $15$ meters per second faster than Hanna, by how many seconds does Kevin finish before Hanna? [b]p4.[/b] It takes an ant $3$ minutes to travel a $120^o$ arc of a circle with radius $2$. How long (in minutes) would it take the ant to travel the entirety of a circle with radius $2022$? [b]p5.[/b] Let $\vartriangle ABC$ be a triangle with angle bisector $AD$. Given $AB = 4$, $AD = 2\sqrt2$, $AC = 4$, find the area of $\vartriangle ABC$. [b]p6.[/b] What is the coefficient of $x^5y^2$ in the expansion of $(x + 2y + 4)^8$? [b]p7.[/b] Find the least positive integer $x$ such that $\sqrt{20475x}$ is an integer. [b]p8.[/b] What is the value of $k^2$ if $\frac{x^5 + 3x^4 + 10x^2 + 8x + k}{x^3 + 2x + 4}$ has a remainder of $2$? [b]p9.[/b] Let $ABCD$ be a square with side length $4$. Let $M$, $N$, and $P$ be the midpoints of $\overline{AB}$, $\overline{BC}$ and $\overline{CD}$, respectively. The area of the intersection between $\vartriangle DMN$ and $\vartriangle ANP$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p10.[/b] Let $x$ be all the powers of two from $2^1$ to $2^{2023}$ concatenated, or attached, end to end ($x = 2481632...$). Let y be the product of all the powers of two from $2^1$ to $2^{2023}$ ($y = 2 \cdot 4 \cdot 8 \cdot 16 \cdot 32... $ ). Let 2a be the largest power of two that divides $x$ and $2^b$ be the largest power of two that divides $y$. Compute $\frac{b}{a}$ . [b]p11.[/b] Larry is making a s’more. He has to have one graham cracker on the top and one on the bottom, with eight layers in between. Each layer can made out of chocolate, more graham crackers, or marshmallows. If graham crackers cannot be placed next to each other, how many ways can he make this s’more? [b]p12.[/b] Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, $AC = 5$. Circle $O$ is centered at $B$ and has radius $\frac{8\sqrt{3}}{5}$ . The area inside the triangle but not inside the circle can be written as $\frac{a-b\sqrt{c}-d\pi}{e}$ , where $gcd(a, b, d, e) =1$ and $c$ is squarefree. Find $a + b + c + d + e$. [b]p13.[/b] Let $F(x)$ be a quadratic polynomial. Given that $F(x^2 - x) = F (2F(x) - 1)$ for all $x$, the sum of all possible values of $F(2022)$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p14.[/b] Find the sum of all positive integers $n$ such that $6\phi (n) = \phi (5n)+8$, where $\phi$ is Euler’s totient function. Note: Euler’s totient $(\phi)$ is a function where $\phi (n)$ is the number of positive integers less than and relatively prime to $n$. For example, $\phi (4) = 2$ since only $1$, $3$ are the numbers less than and relatively prime to $4$. [b]p15.[/b] Three numbers $x$, $y$, and $z$ are chosen at random from the interval $[0, 1]$. The probability that there exists an obtuse triangle with side lengths $x$, $y$, and $z$ can be written in the form $\frac{a\pi-b}{c}$ , where $a$, $b$, $c$ are positive integers with $gcd(a, b, c) = 1$. Find $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The degree-$6$ polynomial $f$ satisfies $f(7) - f(1) = 1, f(8) - f(2) = 16, f(9) - f(3) = 81, f(10) - f(4) = 256$ and $f(11) - f(5) = 625.$ Compute $f(15) - f(-3).$

Let $\dots, a_{-1}, a_0, a_1, a_2, \dots$ be a sequence of positive integers satisfying the folloring relations: $a_n = 0$ for $n < 0$, $a_0 = 1$, and for $n \ge 1$, \[a_n = a_{n - 1} + 2(n - 1)a_{n - 2} + 9(n - 1)(n - 2)a_{n - 3} + 8(n - 1)(n - 2)(n - 3)a_{n - 4}.\] Compute \[\sum_{n \ge 0} \frac{10^n a_n}{n!}.\]

Find the maximum value of $\frac{xyz}{(1 + 5x)(4x + 3y)(5y + 6z)(z + 18)}$ as $x, y$ and $z$ range over the set of all positive real numbers. Justify your answer.

Find all triples $(x,y, z)$ of real (but not necessarily positive) numbers satisfying $3(x^2 + y^2 + z^2) = 1$ , $x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3$.

$a$,$b$,$c$ are real. What is the highest value of $a+b+c$ if $a^2+4b^2+9c^2-2a-12b+6c+2=0$

Suppose $a,b$ and $c$ are real numbers that satisfy $a+b+c=5$ and $\tfrac{1}{a}+\tfrac{1}{b}+\tfrac{1}{c}=\tfrac15$. Find the greatest possible value of $a^3+b^3+c^3$.

Prove that exists a infinity of triplets $a, b, c\in\mathbb{R}$ satisfying simultaneously the relations $a+b+c=0$ and $a^4+b^4+c^4=50$. Moldova National Math Olympiad 2010, 12th grade

Given a sequence {$a_n$} such that $a_1=2$ and for all positive integer $n\geq 2$ $a_{n+1}=\frac{a_n^4+9}{16a_n}$. Prove that $\frac {4}{5}<a_n<\frac {5}{4}$

Let $a, b, c, d, e$ be real numbers satisfying the following conditions. \[a \le b \le c \le d \le e, \quad a+e=1, \quad b+c+d=3, \quad a^2+b^2+c^2+d^2+e^2=14\]Determine the maximum possible value of $ae$.

The set $A{}$ of positive integers satisfies the following conditions: [list=1] [*]If a positive integer $n{}$ belongs to $A{}$, then $2n$ also belongs to $A{}$; [*]For any positive integer $n{}$ there exists an element of $A{}$ divisible by $n{}$; [*]There exist finite subsets of $A{}$ with arbitrarily large sums of reciprocals of elements. [/list]Prove that for any positive rational number $r{}$ there exists a finite subset $B\subset A$ such that \[\sum_{x\in B}\frac{1}{x}=r.\]

Let $\mathbb{Z}$ be the set of integers. Find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that: $$2023f(f(x))+2022x^2=2022f(x)+2023[f(x)]^2+1$$ for each integer $x$.

Compute $$ 2022^{\left(2022^{\cdot ^ {\cdot ^{\cdot ^{\left(2022^{2022}\right)}}}}\right)} \pmod{111} $$ where there are $2022$ $2022$s. (Give the answer as an integer from $0$ to $110$). [i]Proposed by David Tang[/i]

If $(1+x+x^2)^{1000}=a_0+a_1x+a_2x^2+\cdots+a_{2000}x^{2000}$ ($a_0,a_1,\cdots,a_{2000}$ are coefficients), then the value of $a_0+a_3+a_6+\cdots+a_{1998}$ is $\text{(A)}3^{333}\qquad\text{(B)}3^{666}\qquad\text{(C)}3^{999}\qquad\text{(D)}3^{2001}$

Let the polynomial $P(x) = a_nx^n+a_{n-1}x^{n-1}+...+a_0$ has at least one real root and $a_0 \ne 0$. Prove that, consequently crossing out the monomials in the notation $P(x)$ in some order, we can obtain the number $a_0$ from it so that each intermediate polynomial also has at least one real root.

Let $p(x)$ be a monic integer polynomial of degree $n$ that has $n$ real roots, $\alpha_1,\alpha_2,\ldots, \alpha_n$. Let $q(x)$ be an arbitrary integer polynomial that is relatively prime to polynomial $p(x)$. Prove that \[\sum_{i=1}^n \left|q(\alpha_i)\right|\ge n.\] [i]Submitted by Dávid Matolcsi, Berkeley[/i]

Real numbers $x,y,z$ are not all equal and satisfy $x+\frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x}=k$. Find all possible values of $k$.

Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$. Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.

Prove that if coefficients of the quadratic equation $ ax^2\plus{}bx\plus{}c\equal{}0$ are odd integers, then the roots of the equation cannot be rational numbers.

The sequence $a_1, a_2,..., a_{2000}$ of real numbers satisfies the condition \[a_1^3+a_2^3+...+a_n^3=(a_1+a_2+...+a_n)^2\] for all $n$, $1\leq n \leq 2000$. Prove that every element of the sequence is an integer.

Let $a_1, a_2, \cdots, a_n$ be a sequence of real numbers with $a_1+a_2+\cdots+a_n=0$. Define the score $S(\sigma)$ of a permutation $\sigma=(b_1, \cdots b_n)$ of $(a_1, \cdots a_n)$ to be the minima of the sum $$(x_1-b_1)^2+\cdots+(x_n-b_n)^2$$ over all real numbers $x_1\le \cdots \le x_n$. Prove that $S(\sigma)$ attains the maxima over all permutations $\sigma$, if and only if for all $1\le k\le n$, $$b_1+b_2+\cdots+b_k\ge 0.$$ [i]Proposed by Anzo Teh Zhao Yang[/i]