A sequence $(a_k)_{k=1}^{\infty}$ has the property that there is a natural number $n$ such that $a_1 + a_2 +...+ a_n = 0$ and $a_{n+k} = a_k$ for all $k$. Prove that there exists a natural number $N$ such that $$\sum_{i=N}^{N+k} a_i \ge 0 \,\, \,\, for \,\,\,\, k = 0,1,2...$$

A traffic light installed at a main junction of a road, in which you circulate in both directions, it remains red for $30$ s and green for another $30$ s, alternately. You want to install another traffic light on the same road, for a secondary crossing, located $400$ m away from the first, which works with the same period of $1$ min duration. It is wanted that the cars that circulate at $60$ km/h on the road in any of the two senses and that they do not have to stop if there was only the traffic light of the main intersection. They also don't have to stop after installing the secondary crossover. How many seconds can red be on at the secondary traffic light? Note: It is suggested to reason on a Cartesian representation of the march of the cars, taking an axis of distances and another of times.

Coefficients $p$ and $q$ of the equation $x^2+px+q = 0$ are changed and the new ones differ from the old ones by $0.001$ or less. Can the greater root of the new equation differ from that of the old one by $1000$ or more?

[b]p1.[/b] Let $2^k$ be the largest power of $2$ dividing $30! = 30 \cdot 29 \cdot 28 ... 2 \cdot 1$. Find $k$. [b]p2.[/b] Let $d(n)$ be the total number of digits needed to write all the numbers from $1$ to $n$ in base $10$, for example, $d(5) = 5$ and $d(20) = 31$. Find $d(2012)$. [b]p3.[/b] Jim and TongTong play a game. Jim flips $10$ coins and TongTong flips $11$ coins, whoever gets the most heads wins. If they get the same number of heads, there is a tie. What is the probability that TongTong wins? [b]p4.[/b] There are a certain number of potatoes in a pile. When separated into mounds of three, two remain. When divided into mounds of four, three remain. When divided into mounds of five, one remain. It is clear there are at least $150$ potatoes in the pile. What is the least number of potatoes there can be in the pile? [b]p5.[/b] Call an ordered triple of sets $(A, B, C)$ nice if $|A \cap B| = |B \cap C| = |C \cap A| = 2$ and $|A \cap B \cap C| = 0$. How many ordered triples of subsets of $\{1, 2, · · · , 9\}$ are nice? [b]p6.[/b] Brett has an $ n \times n \times n$ cube (where $n$ is an integer) which he dips into blue paint. He then cuts the cube into a bunch of $ 1 \times 1 \times 1$ cubes, and notices that the number of un-painted cubes (which is positive) evenly divides the number of painted cubes. What is the largest possible side length of Brett’s original cube? Note that $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$. [b]p7.[/b] Choose two real numbers $x$ and $y$ uniformly at random from the interval $[0, 1]$. What is the probability that $x$ is closer to $1/4$ than $y$ is to $1/2$? [b]p8. [/b] In triangle $ABC$, we have $\angle BAC = 20^o$ and $AB = AC$. $D$ is a point on segment $AB$ such that $AD = BC$. What is $\angle ADC$, in degree. [b]p9.[/b] Let $a, b, c, d$ be real numbers such that $ab + c + d = 2012$, $bc + d + a = 2010$, $cd + a + b = 2013$, $da + b + c = 2009$. Find $d$. [b]p10. [/b]Let $\theta \in [0, 2\pi)$ such that $\cos \theta = 2/3$. Find $\sum_{n=0}^{\infty}\frac{1}{2^n}\cos(n \theta)$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f(x) = x^3 + 3x^2 - 1$ have roots $a,b,c$. (a) Find the value of $a^3 + b^3 + c^3$ (b) Find all possible values of $a^2b + b^2c + c^2a$

Find a irreducible fraction with denominator not greater than 2005, that is closest to $\frac{9}{25}$ but is not $\frac{9}{25}$

Points $(\sqrt{\pi}, a)$ and $(\sqrt{\pi}, b)$ are distinct points on the graph of $y^2+x^4=2x^2y+1$. What is $|a-b|$? $ \textbf{(A) }1\qquad\textbf{(B) }\dfrac{\pi}{2}\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{1+\pi}\qquad\textbf{(E) }1+\sqrt{\pi} $

It is given that $a_{11}, a_{22}$ are real numbers, that $x_1, x_2, a_{12}, b_1, b_2$ are complex numbers, and that $a_{11}a_{22}=a_{12}\overline{a_{12}}$ (Where $\overline{a_{12}}$ is he conjugate of $a_{12}$). We consider the following system in $x_1, x_2$: \[\overline{x_1}(a_{11}x_1 + a_{12}x_2) = b_1,\]\[\overline{x_2}(a_{12}x_1 + a_{22}x_2) = b_2.\] [b](a) [/b]Give one condition to make the system consistent. [b](b) [/b]Give one condition to make $\arg x_1 - \arg x_2 = 98^{\circ}.$

Find all integers $b$ and $c$ for which the equation $x^2-bx+c=0$ has two real roots $x_{1}$ and $x_{2}$ satisfying $x_{1}^2+x_{2}^2=5$.

For any natural number $m \ge 1$ and any real number $x \ge 0$ we define expression $$E (x, m) = \frac{(1^4 + x) (3^4 + x) (5^4 + x) ... [(2m -1)^ 4 + x]}{(2^4 + x) (4^4 + x) (6^4 + x) ... [(2m )^ 4 + x]}.$$ It is known that $E\left(\frac{1}{4},m\right)=\frac{1}{1013}.$ . Determine the value of $m$

Let $a$, $b$, $c$ be positive real numbers for which \[ \frac{5}{a} = b+c, \quad \frac{10}{b} = c+a, \quad \text{and} \quad \frac{13}{c} = a+b. \] If $a+b+c = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Evan Chen[/i]

Find all functions $ f $ defined on real numbers and taking values in the set of real numbers such that $ f(x+y)+f(y+z)+f(z+x) \geq f(x+2y+3z) $ for all real numbers $ x,y,z $. [hide]There is an infinity of such functions. Every function with the property that $ 3 \inf f \geq \sup f $ is a good one. I wonder if there is a way to find all the solutions. It seems very strange.[/hide]

Write the number $1000$ as the sum of at least two consecutive positive integers. How many (different) ways are there to write it?

Let $d$ be a positive integer. Define the sequence $a_1, a_2, a_3, \dots$ such that \[\begin{cases} a_1 = 1 \\ a_{n+1} = n\left\lfloor\frac{a_n}{n}\right\rfloor + d, \quad n \ge 1.\end{cases}\] Prove that there exists a positive integer $M$ such that $a_M, a_{M+1}, a_{M+2}, \dots$ is an arithmetic sequence.

The polynomial $P(x)=x^3+ax+1$ has exactly one solution on the interval $[-2,0)$ and has exactly one solution on the interval $(0,1]$ where $a$ is a real number. Which of the followings cannot be equal to $P(2)$? $ \textbf{(A)}\ \sqrt{17} \qquad\textbf{(B)}\ \sqrt[3]{30} \qquad\textbf{(C)}\ \sqrt{26}-1 \qquad\textbf{(D)}\ \sqrt {30} \qquad\textbf{(E)}\ \sqrt [3]{10} $

Let $p$ be an odd prime number such that $p\equiv 2\pmod{3}.$ Define a permutation $\pi$ of the residue classes modulo $p$ by $\pi(x)\equiv x^3\pmod{p}.$ Show that $\pi$ is an even permutation if and only if $p\equiv 3\pmod{4}.$

Carl chooses a [i]functional expression[/i]* $E$ which is a finite nonempty string formed from a set $x_1, x_2, \dots$ of variables and applications of a function $f$, together with addition, subtraction, multiplication (but not division), and fixed real constants. He then considers the equation $E = 0$, and lets $S$ denote the set of functions $f \colon \mathbb R \to \mathbb R$ such that the equation holds for any choices of real numbers $x_1, x_2, \dots$. (For example, if Carl chooses the functional equation $$ f(2f(x_1)+x_2) - 2f(x_1)-x_2 = 0, $$ then $S$ consists of one function, the identity function. (a) Let $X$ denote the set of functions with domain $\mathbb R$ and image exactly $\mathbb Z$. Show that Carl can choose his functional equation such that $S$ is nonempty but $S \subseteq X$. (b) Can Carl choose his functional equation such that $|S|=1$ and $S \subseteq X$? *These can be defined formally in the following way: the set of functional expressions is the minimal one (by inclusion) such that (i) any fixed real constant is a functional expression, (ii) for any positive integer $i$, the variable $x_i$ is a functional expression, and (iii) if $V$ and $W$ are functional expressions, then so are $f(V)$, $V+W$, $V-W$, and $V \cdot W$. [i]Proposed by Carl Schildkraut[/i]

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

Given that $\{a_n\}$ is a sequence of integers satisfying the following condition for all positive integral values of $n$: $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2016$. Find all possible values of $a_1$ and $a_2$

let $x,y$ be non-zero reals such that : $x^3+y^3+3x^2y^2=x^3y^3$ find all values of $\frac{1}{x}+\frac{1}{y}$

Let $n>1$ be a given integer. The Mint issues coins of $n$ different values $a_1, a_2, ..., a_n$, where each $a_i$ is a positive integer (the number of coins of each value is unlimited). A set of values $\{a_1, a_2,..., a_n\}$ is called [i]lucky[/i], if the sum $a_1+ a_2+...+ a_n$ can be collected in a unique way (namely, by taking one coin of each value). (a) Prove that there exists a lucky set of values $\{a_1, a_2, ..., a_n\}$ with $$a_1+ a_2+...+ a_n < n \cdot 2^n.$$ (b) Prove that every lucky set of values $\{a_1, a_2,..., a_n\}$ satisfies $$a_1+ a_2+...+ a_n >n \cdot 2^{n-1}.$$ Proposed by Ilya Bogdanov

Determine the largest positive integer $n$ such that the following statement is true: There exists $n$ real polynomials, $P_1(x),\ldots,P_n(x)$ such that the sum of any two of them have no real roots but the sum of any three does.

A set of positive integers is called [i]fragrant[/i] if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$ is fragrant?

Determine whether there exists a function $f: \mathbb{Z}_{> 0} \rightarrow \mathbb{Z}_{> 0}$ such that for all positive integers $m$ and $n$, \[f(m+nf(m))=f(n)^m+2024! \cdot m.\] [i]Jaedon Whyte[/i]

Find all $ x\in \mathbb{Z} $ and $ a\in \mathbb{R} $ satisfying \[\sqrt{x^2-4}+\sqrt{x+2} = \sqrt{x-a}+a \]