When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \minus{} 1$ the quotient is $ f(y)$ and the remainder is $ R_1$. When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \plus{} 1$ the quotient is $ g(y)$ and the remainder is $ R_2$. If $ R_1 \equal{} R_2$ then $ m$ is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \minus{} 1 \qquad \textbf{(E)}\ \text{an undetermined constant}$

Assume $n$ is a positive integer. Considers sequences $a_0, a_1, \ldots, a_n$ for which $a_i \in \{1, 2, \ldots , n\}$ for all $i$ and $a_n = a_0$. (a) Suppose $n$ is odd. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i \pmod{n}$ for all $i = 1, 2, \ldots, n$. (b) Suppose $n$ is an odd prime. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i, 2i \pmod{n}$ for all $i = 1, 2, \ldots, n$.

On the surface of the planet lives one inhabitant, that can move with the speed not greater than $u$. A spaceship approaches to the planet with its speed $v$. Prove that if $v/u > 10$ , the spaceship can find the inhabitant, even it is trying to hide.

$N$ in natural. There are natural numbers from $N^3$ to $N^3+N$ on the board. $a$ numbers was colored in red, $b$ numbers was colored in blue. Sum of red numbers in divisible by sum of blue numbers. Prove, that $b|a$

Let $a_1,a_2,\cdots,a_n$ be a sequence of real numbers lying between $1$ and $-1$, i.e. $-1<a_i<1$, for $1\leq i \leq n$ and such that (i) $a_1+a_2+\cdots+a_n=0$ (ii) $a_1^2+a_2^2+\cdots+a_n^2=40$ Determine the smallest possible value of $n$

Let $\mathbb{N}$ be the set of all positive integers. The function $f: \mathbb{N} \to \mathbb{N}$ satisfies $f(1)=3, f(mn)=f(m)f(n)-f(m+n)+2$ for all $m,n \in \mathbb{N}$. Prove that $f$ does not exist. Comment: The original problem asked for the value of $f(2006)$, which obviously does not exist when $f$ does not. This was probably a mistake by the Olympiad committee. Hence the modified problem.

Let $p$ and $q$ be real numbers. The quadratic equation $$x^2 + px + q = 0$$ has the real solutions $x_1$ and $x_2$. In addition, the following two conditions apply: (i) The numbers $x_1$ and $x_2$ differ from each other by exactly $ 1$. (ii) The numbers $p$ and $q$ differ from each other by exactly $ 1$. Show that then $p$, $q$, $x_1$ and $x_2$ are integers. (G. Kirchner, University of Innsbruck)

Two positive integers have the sum $2002$. Can $2002$ divide their product?

Prove that for each natural number $d$, There is a monic and unique polynomial of degree $d$ like $P$ such that $P(1)$≠$0$ and for each sequence like $a_{1}$,$a_{2}$, $...$ of real numbers that the recurrence relation below is true for them, there is a natural number $k$ such that $0=a_{k}=a_{k+1}= ...$ : $P(n)a_{1} + P(n-1)a_{2} + ... + P(1)a_{n}=0$ $n>1$

Leonard wrote three 3-digit numbers on the board whose sum is $1000$. All of the nine digits are different. Determine which digit does not appear on the board. [i]Proposed by Giorgi Arabidze, Georgia[/i]

For all real numbers $a,b,c>0$ such that $abc=1$, prove that $\frac{a}{1+b^3}+\frac{b}{1+c^3}+\frac{c}{1+a^3}\geq \frac{3}{2}$.

Find all polynomials $P\in \mathbb C[X]$ such that \[P(X^{2})=P(X)^{2}+2P(X)\]

[b]a)[/b] Find the number of infinite sequences of integers $ \left( a_n \right)_{n\ge 1} $ that have the property that $ a_na_{n+2}a_{n+3}=-1, $ for any natural number $ n. $ [b]b)[/b] Prove that there is no infinite sequence of integers $ \left( b_n \right)_{n\ge 1} $ that have the property that $ b_nb_{n+2}b_{n+3}=2005, $ for any natural number $ n. $

Suppose $ a$ and $ b$ are real numbers such that the roots of the cubic equation $ ax^3\minus{}x^2\plus{}bx\minus{}1$ are positive real numbers. Prove that: \[ (i)\ 0<3ab\le 1\text{ and }(i)\ b\ge \sqrt{3} \] [19 points out of 100 for the 6 problems]

Do positive reals $a, b, c, x$ such that $a^2+ b^2 = c^2$ and $(a + x)^2+ (b +x)^2 = (c + x)^2$ exist?

Suppose $f(x, y)$ is a function that takes in two integers and outputs a real number, such that it satisfies $$f(x, y) = \frac{f(x, y + 1) + f(x, y - 1)}{2}$$ $$f(x, y) = \frac{f(x + 1, y) + f(x - 1, y)}{2}$$ What is the minimum number of pairs $(x, y)$ we need to evaluate to be able to uniquely determine $f$?

[b]p1.[/b] In rectangle $ABMC$, $AB= 5$ and $BM= 8$. If point $X$ is the midpoint of side $AC$, what is the area of triangle $XCM$? [b]p2.[/b] Find the sum of all possible values of $a+b+c+d$ such that $(a, b, c, d)$ are quadruplets of (not necessarily distinct) prime numbers satisfying $a \cdot b \cdot c \cdot d = 4792$. [b]p3.[/b] How many integers from $1$ to $2022$ inclusive are divisible by $6$ or $24$, but not by both? [b]p4.[/b] Jerry begins his English homework at $07:39$ a.m. At $07:44$ a.m., he has finished $2.5\%$ of his homework. Subsequently, for every five minutes that pass, he completes three times as much homework as he did in the previous five minute interval. If Jerry finishes his homework at $AB : CD$ a.m., what is $A + B + C + D$? For example, if he finishes at $03:14$ a.m., $A + B + C + D = 0 + 3 + 1 + 4$. [b]p5.[/b] Advay the frog jumps $10$ times on Mondays, Wednesdays and Fridays. He jumps $7$ times on Tuesdays and Saturdays. He jumps $5$ times on Thursdays and Sundays. How many times in total did Advay jump in November if November $17$th falls on a Thursday? (There are $30$ days in November). [b]p6.[/b] In the following diagram, $\angle BAD\cong \angle DAC$, $\overline{CD} = 2\overline{BD}$, and $ \angle AEC$ and $\angle ACE$ are complementary. Given that $\overline{BA} = 210$ and $\overline{EC} = 525$, find $\overline{AE}$. [img]https://cdn.artofproblemsolving.com/attachments/5/3/8e11caf2d7dbb143a296573f265e696b4ab27e.png[/img] [b]p7.[/b] How many trailing zeros are there when $2021!$ is expressed in base $2021$? [b]p8.[/b] When two circular rings of diameter $12$ on the Olympic Games Logo intersect, they meet at two points, creating a $60^o$ arc on each circle. If four such intersections exist on the logo, and no region is in $3$ circles, the area of the regions of the logo that exist in exactly two circles is $a\pi - b\sqrt{c}$ where $a$, $b$, $c$ are positive integers and $\sqrt{c}$ is fully simplified find $a + b + c$. [b]p9.[/b] If $x^2 + ax - 3$ is a factor of $x^4 - x^3 + bx^2 - 5x - 3$, then what is $|a + b|$? [b]p10.[/b] Let $(x, y, z)$ be the point on the graph of $x^4 +2x^2y^2 +y^4 -2x^2 -2y^2 +z^2 +1 = 0$ such that $x+y +z$ is maximized. Find $a+b$ if $xy +xz +yz$ can be expressed as $\frac{a}{b}$ where $a$, $b$ are relatively prime positive integers. [b]p11.[/b] Andy starts driving from Pittsburgh to Columbus and back at a random time from $12$ pm to $3$ pm. Brendan starts driving from Pittsburgh to Columbus and back at a random time from $1$ pm to $4$ pm. Both Andy and Brendan take $3$ hours for the round trip, and they travel at constant speeds. The probability that they pass each other closer to Pittsburgh than Columbus is$ m/n$, for relatively prime positive integers $m$ and $n$. What is $m + n$? [b]p12.[/b] Consider trapezoid $ABCD$ with $AB$ parallel to $CD$ and $AB < CD$. Let $AD \cap BC = O$, $BO = 5$, and $BC = 11$. Drop perpendicular $AH$ and $BI$ onto $CD$. Given that $AH : AD = \frac23$ and $BI : BC = \frac56$ , calculate $a + b + c + d - e$ if $AB + CD$ can be expressed as $\frac{a\sqrt{b} + c\sqrt{d}}{e}$ where $a$, $b$, $c$, $d$, $e$ are integers with $gcd(a, c, e) = 1$ and $\sqrt{b}$, $\sqrt{d}$ are fully simplified. [b]p13.[/b] The polynomials $p(x)$ and $q(x)$ are of the same degree and have the same set of integer coefficients but the order of the coefficients is different. What is the smallest possible positive difference between $p(2021)$ and $q(2021)$? [b]p14.[/b] Let $ABCD$ be a square with side length $12$, and $P$ be a point inside $ABCD$. Let line $AP$ intersect $DC$ at $E$. Let line $DE$ intersect the circumcircle of $ADP$ at $F \ne D$. Given that line $EB$ is tangent to the circumcircle of $ABP$ at $B$, and $FD = 8$, find $m + n$ if $AP$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. [b]p15.[/b] A three digit number $m$ is chosen such that its hundreds digit is the sum of the tens and units digits. What is the smallest positive integer $n$ such that $n$ cannot divide $m$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

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

Consider a non-zero real number $a$ such that $\{a\} + \left\{\frac{1}{a}\right\}=1$, where $\{x\}$ denotes the fractional part of $x$. Prove that for any positive integer $n$, $\{a^n\} + \left\{\frac{1}{a^n}\right\}= 1$.

Let $0 < a_1 < a_2 <... < a_{16} < 122$ be $16$ integers. Prove that there exist integers $(p, q, r, s)$, with $1 \le p < r \le s < q \le 16$, such that $a_p + a_q = a_r + a_s$. An additional $2$ points will be awarded for this problem, if you can find a larger bound than $122$ (with proof).

Let $a, b, c > 0$ satisfy $a + b + c \ge \frac{1}{a} +\frac{1}{b} +\frac{1}{c}$. Prove that $a^3 + b^3 + c^3 \ge a + b + c$

Given a real polynomial $P(x)=a_{2024}x^{2024}+\cdots+a_1x+a_0$ with degree $2024$, such that for all positive reals $b_1, b_2,\cdots, b_{2025}$ with product $1$, then; $$P(b_1)+P(b_2)+\cdots +P(b_{2025})\ge 0$$ Suppose there exist positive reals $c_1, c_2, \cdots, c_{2025}$ with product $1$, such that; $$P(c_1)+P(c_2)+ \cdots +P(c_{2025})=0$$ Is it possible that the values $c_1, c_2, \cdots, c_{2025}$ are all distinct? [i]Proposed by Ivan Chan Kai Chin[/i]

Find all real numbers $x,y,z$ satisfying the following system of equations: \begin{align*} xy+z&=-30\\ yz+x &= 30\\ zx+y &=-18 \end{align*}

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

Denote by $R_{>0}$ the set of all positive real numbers. Find all functions $f: R_{>0} \to R_{>0}$ such that for all $x,y \in R_{>0}$ the following equation holds $$f(y)f(x+f(y))=f(1+xy)$$