Find all real values of the parameter $a$ for which the system of equations \[x^4 = yz - x^2 + a,\] \[y^4 = zx - y^2 + a,\] \[z^4 = xy - z^2 + a,\] has at most one real solution.

Zan starts with a rational number $\tfrac{a}{b}$ written on the board in lowest terms. Then, every second, Zan adds $1$ to both the numerator and denominator of the latest fraction and writes the result in lowest terms. Zan stops as soon as he writes a fraction of the form $\tfrac{n}{n+1}$, for some positive integer $n$. If $\tfrac{a}{b}$ started in that form, Zan does nothing. As an example, if Zan starts with $\tfrac{13}{19}$, then after one second he writes $\tfrac{14}{20} = \tfrac{7}{10}$, then after two seconds $\tfrac{8}{11}$, then $\tfrac{9}{12} = \tfrac{3}{4}$, at which point he stops. (a) Prove that Zan will stop in less than $b-a$ seconds. (b) Show that if $\tfrac{n}{n+1}$ is the final number, then \[\frac{n-1}{n} < \frac{a}{b} \le \frac{n}{n+1}.\] [i](Proposed by Michael Tang.)[/i]

Natural numbers $m$ and $n$ satisfy $$gcd(m,n)+lcm(m,n) = m+n. $$Prove that one of numbers $m,n$ divides the other.

Compute $\frac{55}{21}\times \frac{28} 5\times \frac 3 2$.

2008 boys and 2008 girls sit on 4016 chairs around a round table. Each boy brings a garland and each girl brings a chocolate. In an "activity", each person gives his/her goods to the nearest person on the left. After some activities, it turns out that all boys get chocolates and all girls get garlands. Find the number of possible arrangements.

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 \[\sum_{n \ge 0} \frac{10^n a_n}{n!}.\]

In $\triangle ABC$, $E\in AC$, $D\in AB$, $P=BE\cap CD$. Given that $S\triangle BPC=12$, while the areas of $\triangle BPD$, $\triangle CPE$ and quadrilateral $AEPD$ are all the same, which is $x$. Find the value of $x$.

Determine all real numbers $a,b,c$ and $d$ with the following property: The numbers $a$ and $b$ are distinct roots of $2x^2-3cx+8d$ and the numbers $c$ and $d$ are distinct roots of $2x^2-3ax+8b$.

Prove that the equation $x^2 + y^2 = 1979$ has no integer solutions.

[b]p1.[/b] Let $a_0, a_1,...,a_n$ be such that $a_n \ne 0$ and $$(1 + x + x^3)^{341}(1 + 2x + x^2 + 2x^3 + 2x^4 + x^6)^{342} =\sum^n_{i=0}a_ix^i,$$ Find the number of odd numbers in the sequence a0; a1; : : : an. [b]p2.[/b] Let $F_0 = 1$, $F_1 = 1$ and F$_k = F_{k-1} + F_{k-2}$. Let $P(x) =\sum^{99}_{k=0} x^{F_k}$ . The remainder when $P(x)$ is divided by $x^3 - 1$ can be expressed as $ax^2 + bx + c$. Find $2a + b$. [b]p3.[/b] Let $a_n$ be the number of permutations of the numbers $S = \{1, 2,...,n\}$ such that for all $k$ with $1 \le k \le n$, the sum of $k$ and the number in the $k$th position of the permutation is a power of $2$. Compute $a_{2^0} + a_{2^1} +... + a_{2^{20}}$ . [b]p4.[/b] Three identical balls are painted white and black, so that half of each sphere is a white hemisphere, and the other half is a black one. The three balls are placed on a plane surface, each with a random orientation, so that each ball has a point of contact with the other two. What is the probability that at at least one point of contact between two of the balls, both balls are the same color? [b]p5.[/b] Compute the greatest positive integer $n$ such that there exists an odd integer $a$, for which $\frac{a^{2^n}-1}{4^{4^4}}$ is not an integer. [b]p6.[/b] You are blind and cannot feel the difference between a coin that is heads up or tails up. There are $100$ coins in front of you and are told that exactly $10$ of them are heads up. On the back of this paper, explain how you can split the otherwise indistinguishable coins into two groups so that both groups have the same number of heads. [b]p7.[/b] On the back of this page, write the best math pun you can think of. You’ll get a point if we chuckle. [b]p8.[/b] Pick an integer between $1$ and $10$. If you pick $k$, and $n$ total teams pick $k$, then you’ll receive $\frac{k}{10n}$ points. [b]p9.[/b] There are four prisoners in a dungeon. Tomorrow, they will be separated into a group of three in one room, and the other in a room by himself. Each will be given a hat to wear that is either black or white – two will be given white and two black. None of them will be able to communicate with each other and none will see his or her own hat color. The group of three is lined up, so that the one in the back can see the other two, the second can see the first, but the first cannot see the others. If anyone is certain of their hat color, then they immediately shout that they know it to the rest of the group. If they can secretly prove it to the guard, they are saved. They only say something if they’re sure. Which person is sure to survive? [b]p10.[/b] Down the road, there are $10$ prisoners in a dungeon. Tomorrow they will be lined up in a single room and each given a black or white hat – this time they don’t know how many of each. The person in the back can see everyone’s hat besides his own, and similarly everyone else can only see the hats of the people in front of them. The person in the back will shout out a guess for his hat color and will be saved if and only if he is right. Then the person in front of him will have to guess, and this will continue until everyone has the opportunity to be saved. Each person can only say his or her guess of “white” or “black” when their turn comes, and no other signals may be made. If they have the night before receiving the hats to try to devise some sort of code, how many people at a minimum can be saved with the most optimal code? Describe the code on the back of this paper for full points. [b]p11.[/b] A few of the problems on this mixer contest were taken from last year’s event. One of them had fewer than $5$ correct answers, and most of the answers given were the same incorrect answer. Half a point will be given if you can guess the number of the problem on this test that corresponds to last year’s question, and another $.5$ points will be given if you can guess the very common incorrect answer. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be an acute triangle with $\angle ACB>2 \angle ABC$. Let $I$ be the incenter of $ABC$, $K$ is the reflection of $I$ in line $BC$. Let line $BA$ and $KC$ intersect at $D$. The line through $B$ parallel to $CI$ intersects the minor arc $BC$ on the circumcircle of $ABC$ at $E(E \neq B)$. The line through $A$ parallel to $BC$ intersects the line $BE$ at $F$. Prove that if $BF=CE$, then $FK=AD$.

Let $a_1,a_2,...$ be an arithmetic sequence with the common difference between terms is positive. Assume there are $k$ terms of this sequence creates an geometric sequence with common ratio $d$. Prove that $n\ge 2^{k-1}$.

How many different (non similar) triangles are there whose angles have an integer number of degrees?

Determine the prime numbers $p$ and $q$ that satisfy the equality: $p^3 + 107 = 2q (17q + 24)$ .

$[x-(y-x)] - [(x-y) - x] =$ $ \textbf{(A)}\ 2y \qquad \textbf{(B)}\ 2x \qquad \textbf{(C)}\ -2y \qquad \textbf{(D)}\ -2x \qquad \textbf{(E)}\ 0 $

Let $AA'BCC'B'$ be a convex cyclic hexagon such that $AC$ is tangent to the incircle of the triangle $A'B'C'$, and $A'C'$ is tangent to the incircle of the triangle $ABC$. Let the lines $AB$ and $A'B'$ meet at $X$ and let the lines $BC$ and $B'C'$ meet at $Y$. Prove that if $XBYB'$ is a convex quadrilateral, then it has an incircle.

Let $f(n) = 9n^5- 5n^3 - 4n$. Find the greatest common divisor of $f(17), f(18),... ,f(2009)$.

Find all integer triples $(x,y,z)$ such that \[ \begin{array}{rcl} x-yz &=& 11 \\ xz+y &=& 13. \end{array}\]

Which figure's shaded part satisfies the inequality $\log_x(\log_x y^2)>0$? [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNi83LzY2ZWE5OGJmZjlhNzI1NDM5ZjdiNjZmYTcyZTFkMzEzZjUzMzk5LnBuZw==&rn=NGRkLnBuZw==[/img]

The board features a triangle $ABC$, its center of the circle circumscribed is the point $O$, the midpoint of the side $BC$ is the point $F$, and also some point $K$ on side $AC$ (see fig.). Master knowing that $\angle BAC$ of this triangle is equal to the sharp angle $\alpha$ has separately drawn an angle equal to $\alpha$. After this teacher wiped the board, leaving only the points $O, F, K$ and the angle $\alpha$. Is it possible with a compass and a ruler to construct the triangle $ABC$ ? Justify the answer. (Grigory Filippovsky) [img]https://1.bp.blogspot.com/-RRPt8HbqW4I/XObthZFXyyI/AAAAAAAAKOo/zfHemPjUsI4XAfV_tcmKA6_al0i_gQ9iACK4BGAYYCw/s1600/Yasinsky%2B2019%2Bp6.png[/img]

Let $ x_0 \equal{} 1$ and for $ n\ge0,$ let $ x_{n \plus{} 1} \equal{} 3x_n \plus{} \left\lfloor x_n\sqrt {5}\right\rfloor.$ In particular, $ x_1 \equal{} 5,\ x_2 \equal{} 26,\ x_3 \equal{} 136,\ x_4 \equal{} 712.$ Find a closed-form expression for $ x_{2007}.$ ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)

Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that $$f(x + f(y)) = f(x) + f(y)$$ for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.

Let $ABC$ be an isosceles right-angled triangle, having the right angle at vertex $C$. Let us consider the line through $C$ which is parallel to $AB$ and let $D$ be a point on this line such that $AB = BD$ and $D$ is closer to $B$ than to $A$. Find the angle $\angle CBD$.

a) Is it possible that the sum of all the positive divisors of two different natural numbers are equal? b) Show that if the product of all the positive divisors of two natural numbers are equal, then the two numbers must be equal.

Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]