Let $ a $ and $ b $ be different real numbers. Prove that for any real numbers $ c_1, c_2, \ldots,c_n $ there exists a sequence of $ n $-elements $ (x_i) $, each term of which is equal to one of the numbers $ a $ or $ b $ such that $$ |x_1c_1 + x_2c_2 + \ldots + x_nc_n| \geq \frac{|b-a|}{2}(|c_1|+|c_2|+\ldots+|c_n|).$$

A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$. [i]Proposed by Aaron Lin[/i]

Real numbers $a, b, c$ are such that $$a^2+c-bc = b^2+a-ca = c^2+b-ab$$ Does it follow that $a=b=c$? [i]Proposed by Mykhailo Shtandenko[/i]

Given a positive integer $c$, we construct a sequence of fractions $a_1, a_2, a_3,...$ as follows: $\bullet$ $a_1 =\frac{c}{c+1} $ $\bullet$ to get $a_n$, we take $a_{n-1}$ (in its most simplified form, with both the numerator and denominator chosen to be positive) and we add $2$ to the numerator and $3$ to the denominator. Then we simplify the result again as much as possible, with positive numerator and denominator. For example, if we take $c = 20$, then $a_1 =\frac{20}{21}$ and $a_2 =\frac{22}{24} = \frac{11}{12}$ . Then we find that $a_3 =\frac{13}{15}$ (which is already simplified) and $a_4 =\frac{15}{18} =\frac{5}{6}$. (a) Let $c = 10$, hence $a_1 =\frac{10}{11}$ . Determine the largest $n$ for which a simplification is needed in the construction of $a_n$. (b) Let $c = 99$, hence $a_1 =\frac{99}{100}$ . Determine whether a simplification is needed somewhere in the sequence. (c) Find two values of $c$ for which in the first step of the construction of $a_5$ (before simplification) the numerator and denominator are divisible by $5$.

Does there exist a second-degree polynomial $p(x, y)$ in two variables such that every non-negative integer $ n $ equals $p(k,m)$ for one and only one ordered pair $(k,m)$ of non-negative integers? [i]Proposed by Finland.[/i]

Let $ n \equal{} 2k \minus{} 1$ where $ k \geq 6$ is an integer. Let $ T$ be the set of all $ n\minus{}$tuples $ (x_1, x_2, \ldots, x_n)$ where $ x_i \in \{0,1\}$ $ \forall i \equal{} \{1,2, \ldots, n\}$ For $ x \equal{} (x_1, x_2, \ldots, x_n) \in T$ and $ y \equal{} (y_1, y_2, \ldots, y_n) \in T$ let $ d(x,y)$ denote the number of integers $ j$ with $ 1 \leq j \leq n$ such that $ x_i \neq x_j$, in particular $ d(x,x) \equal{} 0.$ Suppose that there exists a subset $ S$ of $ T$ with $ 2^k$ elements that has the following property: Given any element $ x \in T,$ there is a unique element $ y \in S$ with $ d(x, y) \leq 3.$ Prove that $ n \equal{} 23.$

Prove that if $ a, b, c $ are non-negative numbers, then $$ a^3 + b^3 + c^3 + 3abc \geq a^2(b + c) + b^2(a + c) + c^2(a + b).$$

A positive number $\dfrac{m}{n}$ has the property that it is equal to the ratio of $7$ plus the number’s reciprocal and $65$ minus the number’s reciprocal. Given that $m$ and $n$ are relatively prime positive integers, find $2m + n$.

Find all pairs of integers, $n$ and $k$, $2 < k < n$, such that the binomial coefficients \[\binom{n}{k-1}, \binom{n}{k}, \binom{n}{k+1}\] form an increasing arithmetic series.

What is the sum of the reciprocals of the roots of the equation \[ \frac {2003}{2004}x \plus{} 1 \plus{} \frac {1}{x} \equal{} 0? \] $ \textbf{(A)}\ \minus{}\! \frac {2004}{2003} \qquad \textbf{(B)}\ \minus{} \!1 \qquad \textbf{(C)}\ \frac {2003}{2004} \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \frac {2004}{2003}$

Let \[f_1(x) = \frac{1}{x}\quad\text{and}\quad f_2(x) = 1 - x\] Let $H$ be the set of all compositions of the form $h_1 \circ h_2 \circ \ldots \circ h_k$, where each $h_i$ is either $f_1$ or $f_2$. For all $h$ in $H$, let $h^{(n)}$ denote $h$ composed with itself $n$ times. Find the greatest integer $N$ such that $\pi, h(\pi), \ldots, h^{(N)}(\pi)$ are all distinct for some $h$ in $H$.

For which real numbers $a$ is the set of all solutions of the inequality $$(x^2 + ax + 4)(x^2 - 5x + 6) < 0$$ an interval?

Prove the following assertion: If $c_1,c_2,\ldots ,c_n\ (n\ge 2)$ are real numbers such that \[ (n-1)(c_1^2+c_2^2+\cdots +c_n^2)=(c_1+c_2+\cdots + c_n)^2,\] then either all these numbers are nonnegative or all these numbers are nonpositive.

Determine the number of distinct values which appear in the sequence \[\left\lfloor\frac{2025}{1}\right\rfloor,\left\lfloor\frac{2025}{2}\right\rfloor,\left\lfloor\frac{2025}{3}\right\rfloor,\dots,\left\lfloor\frac{2025}{2024}\right\rfloor,\left\lfloor\frac{2025}{2025}\right\rfloor.\]

Find all functions $f: \mathbb{Q}[x] \to \mathbb{R}$ such that: (a) for all $P, Q \in \mathbb{Q}[x]$, $f(P \circ Q) = f(Q \circ P);$ (b) for all $P, Q \in \mathbb{Q}[x]$ with $PQ \neq 0$, $f(P\cdot Q) = f(P) + f(Q).$ ($P \circ Q$ indicates $P(Q(x))$.)

Let $P(x)$ be a quadratic polynomial satisfying the following conditions: [list] [*] $P(x)$ has leading coefficient $1$. [*] $P(x)$ has nonnegative integer roots that are at most $2022$. [*] the set of the roots of $P(x)$ is a subset of the set of the roots of $P(P(x))$. [/list] Let $S$ be the set of all such possible $P(x)$, and let $Q(x)$ be the polynomial obtained upon summing all the elements of $S$. Find the sum of the roots of $Q(x)$.

Investigate the boundary of the domain of stability ($\max \text{Re }\lambda_j < 0$) in the space of coefficients of the equation $\dddot{x} + a\ddot{x} + b\dot{x} + cx = 0$.

Evin’s calculator is broken and can only perform $3$ operations: Operation $1$: Given a number $x$, output $2x$. Operation $2$: Given a number $x$, output $4x +1$. Operation $3$: Given a number $x$, output $8x +3$. After initially given the number $0$, how many numbers at most $128$ can he make?

Prove that $$\frac{2}{2!}+\frac{7}{3!}+\frac{14}{4!}+\frac{23}{5!}+...+\frac{k^2-2}{k!}+...+\frac{9998}{100!}<3$$ where $n! = 1 \times 2 \times ... \times n.$ (V Senderov)

Let be given the sequence $(x_n)$ defined by $x_1 = 1$ and $x_{n+1} = 3x_n + \lfloor x_n \sqrt5 \rfloor$ for all $n = 1,2,3,...,$ where $\lfloor x \rfloor$ denotes the greatest integer that does not exceed $x$. Prove that for any positive integer $n$ we have $$x_nx_{n+2} - x^2_{n+1} = 4^{n-1}$$ Trần Nam Dũng

Suppose that $a, b$ are natural numbers so that all the roots of $x^2 + ax - b$ and $x^2 - ax + b$ are integers. Show that exists a right triangle with integer sides, with $a$ the length of the hypotenuse and $b$ the area .

Find all $(m,n)$ in $\mathbb{N}^2$ such that $m\mid n^2+1$ and $n\mid m^2+1$.

Let $\lfloor z \rfloor$ denote the greatest integer less than or equal to $z.$ Compute $$\sum_{j=-1000}^{1000} \left\lfloor \frac{2025}{j+0.5}\right\rfloor.$$

Let $a \in \mathbb{R}$ be a parameter. (1) Prove that the curves of $y = x^2 + (a + 2)x - 2a + 1$ pass through a fixed point; also, the vertices of these parabolas all lie on the curve of a certain parabola. (2) If the function $x^2 + (a + 2)x - 2a + 1 = 0$ has two distinct real roots, find the value range of the larger root.

i) Find all infinite arithmetic progressions formed with positive integers such that there exists a number $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, the $p$-th term of the progression is also prime. ii) Find all polynomials $f(X) \in \mathbb{Z}[X]$, such that there exist $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, $| f(p) |$ is also prime. [i]Dan Schwarz[/i]