A and B plays the following game: they choose randomly $k$ integers from $\{1,2,\dots,100\}$; if their sum is even, A wins, else B wins. For what values of $k$ does A and B have the same chance of winning?

Find all the integer solutions of the equation: $$\sqrt{x} + \sqrt{y} = \sqrt{x+2023}$$

Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that for all reals $ x, y $,$$ f(x^2+f(x+y))=y+xf(x+1) $$

Let $P(x)$ be a polynomial with integer coefficients and roots $1997$ and $2010$. Suppose further that $|P(2005)|<10$. Determine what integer values $P(2005)$ can get.

Solve in $R$ the equation: $8x^3-4x^2-4x+1=0$

Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$ for all $n \in N$.

Solve the equation: \[3\sqrt{3x-1}=x^2+1\]

Show that $$\sqrt{n + \left[ \sqrt{n} +\frac12\right]}$$ is an irrational number, for every positive integer $n$.

Let $a,b,c&gt;0$ and $abc=1$. Prove that $$\frac{a}{b^2(c+a)(a+b)}+\frac{b}{c^2(a+b)(b+c)}+\frac{c}{a^2(c+a)(a+b)}\ge \frac{3}{4}.$$ [i](Zhuge Liang)[/i]

Having three sets $ A,B\subset C, $ solve the set equation $ (X\cup (C\setminus A))\cap ((C\setminus X)\cup A)=B. $

Find all the non-zero polynomials $P(x),Q(x)$ with real coefficients and the minimum degree,such that for all $x \in \mathbb{R}$: \[ P(x^2)+Q(x)=P(x)+x^5Q(x) \]

We have an infinite sequence of real numbers $x_0,x_1, x_2, ... $ such that $x_{n+1} = \sqrt{x_n -\frac14}$ holds for all natural $n$ and moreover $x_0 \in \frac12$. (a) Prove that for every natural $n$ holds: $x_n > \frac12$ (b) Prove that $\lim_{n \to \infty} x_n$ exists. Calculate this limit.

[b]p1.[/b] Compute $17^2 + 17 \cdot 7 + 7^2$. [b]p2.[/b] You have $\$1.17$ in the minimum number of quarters, dimes, nickels, and pennies required to make exact change for all amounts up to $\$1.17$. How many coins do you have? [b]p3.[/b] Suppose that there is a $40\%$ chance it will rain today, and a $20\%$ chance it will rain today and tomorrow. If the chance it will rain tomorrow is independent of whether or not it rained today, what is the probability that it will rain tomorrow? (Express your answer as a percentage.) [b]p4.[/b] A number is called boxy if the number of its factors is a perfect square. Find the largest boxy number less than $200$. [b]p5.[/b] Alice, Bob, Carl, and Dave are either lying or telling the truth. If the four of them make the following statements, who has the coin? [i]Alice: I have the coin. Bob: Carl has the coin. Carl: Exactly one of us is telling the truth. Dave: The person who has the coin is male.[/i] [b]p6.[/b] Vicky has a bag holding some blue and some red marbles. Originally $\frac23$ of the marbles are red. After Vicky adds $25$ blue marbles, $\frac34$ of the marbles are blue. How many marbles were originally in the bag? [b]p7.[/b] Given pentagon $ABCDE$ with $BC = CD = DE = 4$, $\angle BCD = 90^o$ and $\angle CDE = 135^o$, what is the length of $BE$? [b]p8.[/b] A Berkeley student decides to take a train to San Jose, stopping at Stanford along the way. The distance from Berkeley to Stanford is double the distance from Stanford to San Jose. From Berkeley to Stanford, the train's average speed is $15$ meters per second. From Stanford to San Jose, the train's average speed is $20$ meters per second. What is the train's average speed for the entire trip? [b]p9.[/b] Find the area of the convex quadrilateral with vertices at the points $(-1, 5)$, $(3, 8)$, $(3,-1)$, and $(-1,-2)$. [b]p10.[/b] In an arithmetic sequence $a_1$, $a_2$, $a_3$, $...$ , twice the sum of the first term and the third term is equal to the fourth term. Find $a_4/a_1$. [b]p11.[/b] Alice, Bob, Clara, David, Eve, Fred, Greg, Harriet, and Isaac are on a committee. They need to split into three subcommittees of three people each. If no subcommittee can be all male or all female, how many ways are there to do this? [b]p12.[/b] Usually, spaceships have $6$ wheels. However, there are more advanced spaceships that have $9$ wheels. Aliens invade Earth with normal spaceships, advanced spaceships, and, surprisingly, bicycles (which have $2$ wheels). There are $10$ vehicles and $49$ wheels in total. How many bicycles are there? [b]p13.[/b] If you roll three regular six-sided dice, what is the probability that the three numbers showing will form an arithmetic sequence? (The order of the dice does matter, but we count both $(1,3, 2)$ and $(1, 2, 3)$ as arithmetic sequences.) [b]p14.[/b] Given regular hexagon $ABCDEF$ with center $O$ and side length $6$, what is the area of pentagon $ABODE$? [b]p15.[/b] Sophia, Emma, and Olivia are eating dinner together. The only dishes they know how to make are apple pie, hamburgers, hotdogs, cheese pizza, and ice cream. If Sophia doesn't eat dessert, Emma is vegetarian, and Olivia is allergic to apples, how many dierent options are there for dinner if each person must have at least one dish that they can eat? [b]p16.[/b] Consider the graph of $f(x) = x^3 + x + 2014$. A line intersects this cubic at three points, two of which have $x$-coordinates $20$ and $14$. Find the $x$-coordinate of the third intersection point. [b]p17.[/b] A frustum can be formed from a right circular cone by cutting of the tip of the cone with a cut perpendicular to the height. What is the surface area of such a frustum with lower radius $8$, upper radius $4$, and height $3$? [b]p18.[/b] A quadrilateral $ABCD$ is dened by the points $A = (2,-1)$, $B = (3, 6)$, $C = (6, 10)$ and $D = (5,-2)$. Let $\ell$ be the line that intersects and is perpendicular to the shorter diagonal at its midpoint. What is the slope of $\ell$? [b]p19.[/b] Consider the sequence $1$, $1$, $2$, $2$, $3$, $3$, $3$, $5$, $5$, $5$, $5$, $5$, $...$ where the elements are Fibonacci numbers and the Fibonacci number $F_n$ appears $F_n$ times. Find the $2014$th element of this sequence. (The Fibonacci numbers are defined as $F_1 = F_2 = 1$ and for $n > 2$, $F_n = F_{n-1}+F_{n-2}$.) [b]p20.[/b] Call a positive integer top-heavy if at least half of its digits are in the set $\{7, 8, 9\}$. How many three digit top-heavy numbers exist? (No number can have a leading zero.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$x$ and $y$ are positive and relatively prime and $z$ is an integer. They satisfy $(5z-4x)(5z-4y)=25xy$. Show that at least one of $10z+x+y$ or quotient of this number divided by $3$ is a square number (i.e. prove that $10z+x+y$ or integer part of $\frac{10z+x+y}{3}$ is a square number).

[b]p1.[/b] The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats, and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were there in each bus? [b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $50$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a 6 cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $150$ coins? [b]p3.[/b] Pinocchio multiplied two $2$ digit numbers. But witch Masha erased some of the digits. The erased digits are the ones marked with a $*$. Could you help Pinocchio to restore all the erased digits? $\begin{tabular}{ccccc} & & & 9 & 5 \\ x & & & * & * \\ \hline & & & * & * \\ + & 1 & * & * & \\ \hline & * & * & * & * \\ \end{tabular}$ Find all solutions. [b]p4.[/b] There are $50$ senators and $435$ members of House of Representatives. On Friday all of them voted a very important issue. Each senator and each representative was required to vote either "yes" or "no". The announced results showed that the number of "yes" votes was greater than the number of "no" votes by $24$. Prove that there was an error in counting the votes. [b]p5.[/b] Was there a year in the last millennium (from $1000$ to $2000$) such that the sum of the digits of that year is equal to the product of the digits? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The polynomial $f(x) = x^3 + rx^2 + sx + t$ has $r, s$, and $t$ as its roots (with multiplicity), where $f(1)$ is rational and $t \ne 0$. Compute $|f(0)|$.

Let $a, b, c$ be different real numbers. prove that $$\left(\frac{2a-b}{a-b}\right)^2+ \left(\frac{2b- c}{b-c}\right)^2+ \left(\frac{2c-a}{c-a}\right)^2 \ge 5. $$

The sequence of integers $\{a_i\}_{i = 0}^{\infty}$ satisfies $a_0 = 3$, $a_1 = 4$, and \[a_{n+2} = a_{n+1} a_n + \left\lceil \sqrt{a_{n+1}^2 - 1} \sqrt{a_n^2 - 1}\right\rceil\] for $n \ge 0$. Evaluate the sum \[\sum_{n = 0}^{\infty} \left(\frac{a_{n+3}}{a_{n+2}} - \frac{a_{n+2}}{a_n} + \frac{a_{n+1}}{a_{n+3}} - \frac{a_n}{a_{n+1}}\right).\]

Consider a sequence of positive integers with total sum $2019$ such that no number and no sum of a set of consecutive num bers is equal to $40$. What is the greatest possible length of such a sequence? (Alexandr Shapovalov)

Let $m$ and $n$ be positive integers for which $n\leq m\leq 2n$. Find the number of all complex solutions $(z_1,z_2,...,z_m)$ that satisfy $$z_1^7+z_2^7+...+z_m^7=n$$ Such that $z_k^3-2z_k^2+2z_k-1=0$ for all $k=1,2,...,m$.

Given a number $\overline{abcd}$, where $a$, $b$, $c$, and $d$, represent the digits of $\overline{abcd}$, find the minimum value of \[\frac{\overline{abcd}}{a+b+c+d}\] where $a$, $b$, $c$, and $d$ are distinct [hide=Answer]$\overline{abcd}=1089$, minimum value of $\dfrac{\overline{abcd}}{a+b+c+d}=60.5$[/hide]

Define a sequence $a_n$ such that $a_n = a_{n-1} - a_{n-2}$. Let $a_1 = 6$ and $a_2 = 5$. Find $\Sigma_{n=1}^{1000}a_n$.

We are allowed to put several brackets in the expression $$\frac{29 : 28 : 27 : 26 :... : 17 : 16}{15 : 14 : 13 : 12 : ... : 3 : 2}$$ always in the same places below each other. (a) Find the smallest possible integer value we can obtain in that way. (b) Find all possible integer values that can be obtained. Remark: in this problem, $$\frac{(29 : 28) : 27 : ... : 16}{(15 : 14) : 13 : ... : 2},$$ is valid position of parenthesis, on the other hand $$\frac{(29 : 28) : 27 : ... : 16}{15 : (14 : 13) : ... : 2}$$ is forbidden.

Let $a$ and $b$ be real numbers for which the following is true: $acscx + b cot x \ge 1$, for all $0 <x < \pi$ Find the least value of $a^2 + b$.

For every positive integer $n$, the symbol $a_n/b_n$ is the simplest form of the fraction $1+1/2+...+1/n$. Prove that for every pair of positive integers $(M, N)$ we can always find a positive integer $m$ where $(a_n, N) = 1$ for all $n = m, m + 1, ...,m + M$.