Let $a, b$ and $c$ be pairwise different natural numbers. Prove $\frac{a^3 + b^3 + c^3}{3} \ge abc + a + b + c$. When does equality holds? (Karl Czakler)

Let $r\geqslant 0$ be a real number and define $f(x)=1/(1+x^2)^r$. Prove that \[|f^{(k)}(x)|\leqslant\frac{2r\cdot(2r+1)\cdots(2r+k-1)}{(1+x^2)^{r+k/2}},\]for every natural number $k{}$. Here, $f^{(k)}(x)$ denotes the $k^{\text{th}}$ derivative of $f$.

Given a sequence $a_0, a_1, a_2, ... , a_n$, let its [i]arithmetic approximant[/i] be the arithmetic sequence $b_0, b_1, ... , b_n$ that minimizes the quantity $\sum_{i=0}^{n}(b_i -a_i)^2$, and denote this quantity the sequence’s anti-arithmeticity. Denote the number of integer sequences whose arithmetic approximant is the sequence $4$, $8$, $12$, $16$ and whose anti-arithmeticity is at most $20$.

Let be three real numbers $ a,b,c\in [0,1] $ satisfying the condition $ ab+bc+ca=1. $ Prove that $$ a^2+b^2+c^2\le 2, $$ and determine the cases in which equality is attained.

Given that three roots of $f(x)=x^4+ax^2+bx+c$ are $2$, $-3$, and $5$, what is the value of $a+b+c$?

Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$

Find all functions $f:\mathbb R\to\mathbb R$ such that $$f\left( x^2+xf(y)\right)=xf(x+y)$$ for all reals $x,y$.

Let's call a set of numbers [i]lucky[/i] if it cannot be divided into two nonempty groups so that the product of the sum of the numbers in one group and the sum of the numbers in the other is positive. The teacher wrote several integers on the blackboard. Prove that the children can add another integer to the existing ones so that the resulting set is lucky. [i]A. Kuznetsov[/i]

Given real numbers $x,y,z,a$ satisfying: $$ x+y+z = a$$ $$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z} = \frac{1}{a} $$ Prove that at least one of the numbers $x,y,z$ is equal to $a$.

show that thee is no function f definedonthe positive real numbes such that : $f(y) > (y-x)f(x)^2$

Let $f(x) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$. Find the remainder when $f(x^7)$ is divided by $f(x)$.

[b]p1[/b]. Evaluate $1! + 2! + 3! + 4! + 5! $ (where $n!$ is the product of all integers from $1$ to $n$, inclusive). [b]p2.[/b] Harold opens a pack of Bertie Bott's Every Flavor Beans that contains $10$ blueberry, $10$ watermelon, $3$ spinach and $2$ earwax-flavored jelly beans. If he picks a jelly bean at random, then what is the probability that it is not spinach-flavored? [b]p3.[/b] Find the sum of the positive factors of $32$ (including $32$ itself). [b]p4.[/b] Carol stands at a flag pole that is $21$ feet tall. She begins to walk in the direction of the flag's shadow to say hi to her friends. When she has walked $10$ feet, her shadow passes the flag's shadow. Given that Carol is exactly $5$ feet tall, how long in feet is her shadow? [b]p5.[/b] A solid metal sphere of radius $7$ cm is melted and reshaped into four solid metal spheres with radii $1$, $5$, $6$, and $x$ cm. What is the value of $x$? [b]p6.[/b] Let $A = (2,-2)$ and $B = (-3, 3)$. If $(a,0)$ and $(0, b)$ are both equidistant from $A$ and $B$, then what is the value of $a + b$? [b]p7.[/b] For every flip, there is an $x^2$ percent chance of flipping heads, where $x$ is the number of flips that have already been made. What is the probability that my first three flips will all come up tails? [b]p8.[/b] Consider the sequence of letters $Z\,\,W\,\,Y\,\,X\,\,V$. There are two ways to modify the sequence: we can either swap two adjacent letters or reverse the entire sequence. What is the least number of these changes we need to make in order to put the letters in alphabetical order? [b]p9.[/b] A square and a rectangle overlap each other such that the area inside the square but outside the rectangle is equal to the area inside the rectangle but outside the square. If the area of the rectangle is $169$, then find the side length of the square. [b]p10.[/b] If $A = 50\sqrt3$, $B = 60\sqrt2$, and $C = 85$, then order $A$, $B$, and $C$ from least to greatest. [b]p11.[/b] How many ways are there to arrange the letters of the word $RACECAR$? (Identical letters are assumed to be indistinguishable.) [b]p12.[/b] A cube and a regular tetrahedron (which has four faces composed of equilateral triangles) have the same surface area. Let $r$ be the ratio of the edge length of the cube to the edge length of the tetrahedron. Find $r^2$. [b]p13.[/b] Given that $x^2 + x + \frac{1}{x} +\frac{1}{x^2} = 10$, find all possible values of $x +\frac{1}{x}$ . [b]p14.[/b] Astronaut Bob has a rope one unit long. He must attach one end to his spacesuit and one end to his stationary spacecraft, which assumes the shape of a box with dimensions $3\times 2\times 2$. If he can attach and re-attach the rope onto any point on the surface of his spacecraft, then what is the total volume of space outside of the spacecraft that Bob can reach? Assume that Bob's size is negligible. [b]p15.[/b] Triangle $ABC$ has $AB = 4$, $BC = 3$, and $AC = 5$. Point $B$ is reflected across $\overline{AC}$ to point $B'$. The lines that contain $AB'$ and $BC$ are then drawn to intersect at point $D$. Find $AD$. [b]p16.[/b] Consider a rectangle $ABCD$ with side lengths $5$ and $12$. If a circle tangent to all sides of $\vartriangle ABD$ and a circle tangent to all sides of $\vartriangle BCD$ are drawn, then how far apart are the centers of the circles? [b]p17.[/b] An increasing geometric sequence $a_0, a_1, a_2,...$ has a positive common ratio. Also, the value of $a_3 + a_2 - a_1 - a_0$ is equal to half the value of $a_4 - a_0$. What is the value of the common ratio? [b]p18.[/b] In triangle $ABC$, $AB = 9$, $BC = 11$, and $AC = 16$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{BC}$, respectively, such that $BE = BF = 4$. What is the area of triangle $CEF$? [b]p19.[/b] Xavier, Yuna, and Zach are running around a circular track. The three start at one point and run clockwise, each at a constant speed. After $8$ minutes, Zach passes Xavier for the first time. Xavier first passes Yuna for the first time in $12$ minutes. After how many seconds since the three began running did Zach first pass Yuna? [b]p20.[/b] How many unit fractions are there such that their decimal equivalent has a cycle of $6$ repeating integers? Exclude fractions that repeat in cycles of $1$, $2$, or $3$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f : \mathbb{Z} \rightarrow \mathbb{Z}^+$ be a function, and define $h : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}^+$ by $h(x, y) = \gcd (f(x), f(y))$. If $h(x, y)$ is a two-variable polynomial in $x$ and $y$, prove that it must be constant.

1) Two nonnegative real numbers $x, y$ have constant sum $a$. Find the minimum value of $x^m + y^m$, where m is a given positive integer. 2) Let $m, n$ be positive integers and $k$ a positive real number. Consider nonnegative real numbers $x_1, x_2, . . . , x_n$ having constant sum $k$. Prove that the minimum value of the quantity $x^m_1+ ... + x^m_n$ occurs when $x_1 = x_2 = ... = x_n$.

Suppose that $4^{n}+2^{n}+1$ is prime for some positive integer $n$. Show that $n$ must be a power of $3$.

$(CZS 6)$ Let $d$ and $p$ be two real numbers. Find the first term of an arithmetic progression $a_1, a_2, a_3, \cdots$ with difference $d$ such that $a_1a_2a_3a_4 = p.$ Find the number of solutions in terms of $d$ and $p.$

Indicate (justifying your answer) if there exists a function $f: R \to R$ such that for all $x \in R$ fulfills that i) $\{f(x))\} \sin^2 x + \{x\} cos (f(x)) cosx =f (x)$ ii) $f (f(x)) = f(x)$ where $\{m\}$ denotes the fractional part of $m$. That is, $\{2.657\} = 0.657$, and $\{-1.75\} = 0.25$.

[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].

The [i]runs[/i] of a decimal number are its increasing or decreasing blocks of digits. Thus $024379$ has three [i]runs[/i] : $024, 43$, and $379$. Determine the average number of runs for a decimal number in the set $\{d_1d_2 \cdots d_n | d_k \neq d_{k+1}, k = 1, 2,\cdots, n - 1\}$, where $n \geq 2.$

A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities $$P(x, y, z)=P(x, y, xy-z)=P(x, zx-y, z)=P(yz-x, y, z).$$ Prove that there exists a polynomial $F(t)$ in one variable such that $$P(x,y,z)=F(x^2+y^2+z^2-xyz).$$

Mykhailo chose three distinct real numbers \( a, b, c \) and wrote the following numbers on the board: \[ a + b, \quad b + c, \quad c + a, \quad ab, \quad bc, \quad ca. \]What is the minimum possible number of distinct numbers that can be written on the board? [i]Proposed by Anton Trygub[/i]

Find all second degree polynomials $P(x)$ such that for all $a \in\mathbb{R} , a \geq 1$, then $P(a^2+a) \geq a.P(a+1)$

A collection of $n > 2$ numbers is called [i]crowded [/i] if each of them is less than their sum divided by $n - 1$ . Let $\{a, b, c, ,...\}$ be a crowded collection of $n$ numbers whose sum equals $S$. Prove that: (a) each of the numbers is positive, (b) we always have $a + b > c$, (c) we always have $a + b \ge \frac{S}{n-1}$ . (Regina Schleifer)

It is midnight on April $29$th, and Abigail is listening to a song by her favorite artist while staring at her clock, which has an hour, minute, and second hand. These hands move continuously. Between two consecutive midnights, compute the number of times the hour, minute, and second hands form two equal angles and no two hands overlap.

Let $ k$ be a positive integer. Prove that there exist polynomials $ P_0(n),P_1(n),\dots,P_{k\minus{}1}(n)$ (which may depend on $ k$) such that for any integer $ n,$ \[ \left\lfloor\frac{n}{k}\right\rfloor^k\equal{}P_0(n)\plus{}P_1(n)\left\lfloor\frac{n}{k}\right\rfloor\plus{} \cdots\plus{}P_{k\minus{}1}(n)\left\lfloor\frac{n}{k}\right\rfloor^{k\minus{}1}.\] ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)