A real number $a$ is given. The sequence $n_{1}< n_{2}< n_{3}< ...$ consists of all the positive integral $n$ such that $\{na\}< \frac{1}{10}$. Prove that there are at most three different numbers among the numbers $n_{2}-n_{1}$, $n_{3}-n_{2}$, $n_{4}-n_{3}$, $\ldots$. [i]A corollary of a theorem from ergodic theory[/i]

Given a triangle $ABC$ inscribed in circle $c(O,R)$ (with center $O$ and radius $R$) with $AB<AC<BC$ and let $BD$ be a diameter of the circle $c$. The perpendicular bisector of $BD$ intersects line $AC$ at point $M$ and line $AB$ at point $N$. Line $ND$ intersects the circle $c$ at point $T$. Let $S$ be the second intersection point of cicumcircles $c_1$ of triangle $OCM$, and $c_2$ of triangle $OAD$. Prove that lines $AD, CT$ and $OS$ pass through the same point.

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

A positive real number sequence $a_1, a_2, a_3,\dots $ and a positive integer \(s\) is given. Let $f_n(0) = \frac{a_n+\dots+a_1}{n}$ and for each $0<k<n$ \[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\] Then for every integer $n\geq s,$ the condition \[a_{n+1}=\max_{0\leq k<n}(f_n(k))\] is satisfied. Prove that this sequence must be eventually constant.

Solve the equation,$$ \sin (\pi \log x) + \cos (\pi \log x) = 1$$

Given integer $d>1,m$,prove that there exists integer $k>l>0$, such that $$(2^{2^k}+d,2^{2^l}+d)>m.$$

Prove that there exists an infinite sequence of positive integers $a_1,a_2,a_3,\dots$ such that for any positive integer $k$, $a_k^2+a_k+2023$ has at least $k$ distinct positive divisors.

A subset of $X$ of $\{1,2,3, \ldots 10000 \}$ has the following property: If $a,b$ are distinct elements of $X$, then $ab\not\in X$. What is the maximal number of elements in $X$?

[Help me] Suppose that the equation $x^3+px^2+qx+r = 0$ has 3 real roots $x_1; x_2; x_3$; where p; q; r are integer numbers. Put $S_n = x_1^n+x_2^n+x_3^n$ ; n = 1; 2; : : : Prove that $S_{2012}$ is an integer.

Integers $ a_1,a_2,\ldots a_9$ satisfy the relations $ a_{k\plus{}1}\equal{}a_k^3\plus{}a_k^2\plus{}a_k\plus{}2$ for $ k\equal{}1,2,...,8$. Prove that among these numbers there exist three with a common divisor greater than $ 1$.

Define the sequences $x_0,x_1,x_2,\ldots$ and $y_0,y_1,y_2,\ldots$ such that $x_0=1$, $y_0=2021$, and for all nonnegative integers $n$, we have $x_{n+1}=\sqrt{x_ny_n}$ and $y_{n+1}=\frac{x_n+y_n}{2}.$ There is some constant $X$ such that as $n$ grows large, $x_n-X$ and $y_n-X$ both approach $0$. Estimate $X$. An estimate of $E$ earns $\max(0,2-0.02|A-E|)$ points, where $A$ is the actual answer. [i]2021 CCA Math Bonanza Lightning Round #5.2[/i]

Given isosceles triangle $ABC$ ($AB = AC$). A straight line $\ell$ is drawn through its vertex $B$ at a right angle with $AB$ . On the straight line $AC$, an arbitrary point $D$ is taken, different from the vertices, and a straight line is drawn through it at a right angle with $AC$, intersecting $\ell$ at the point $F$. Prove that the center of the circle circumscribed around the triangle $BCD$ lies on the circumscribed circle of triangle $ABD$.

Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$. [i]Proposed by Jaroslaw Wroblewski, Poland[/i]

Calculate the sum of vector products $[[x, y], z] + [[y, z], x] + [[z, x], y]$

Let $ n$ be a positive integer. Find the number of pairs $ P,Q$ of polynomials with real coefficients such that \[ (P(X))^2\plus{}(Q(X))^2\equal{}X^{2n}\plus{}1\] and $ \text{deg}P<\text{deg}{Q}.$

Determine all solutions of \[ x + y^2 = p^m \] \[ x^2 + y = p^n \] For $x,y,m,n$ positive integers and $p$ being a prime.

Let $f:[0,\infty )\rightarrow\mathbb{R}$ be a periodical function, with period $1$, integrable on $[0,1]$. For a strictly increasing and unbounded sequence $(x_n)_{n\ge 0},\, x_0=0,$ with $\lim_{n\rightarrow\infty} (x_{n+1}-x_n)=0$, we denote $r(n)=\max \{ k\mid x_k\le n\}$. a) Show that: \[\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{r(n)}(x_k-x_{k+1})f(x_k)=\int_0^1 f(x)\, dx\] b) Show that: \[ \lim_{n\rightarrow\infty} \frac{1}{\ln n}\sum_{k=1}^{r(n)}\frac{f(\ln k)}{k}=\int_0^1f(x)\, dx\]

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for any pair of naturals $m,n$, $$\gcd(f(m),n) = \gcd(m,f(n)).$$

Determine all pairs $(a,b)$ of positive integers satisfying \[a^2+b\mid a^2b+a\quad\text{and}\quad b^2-a\mid ab^2+b.\]

Susan wants to place 35.5 kg of sugar in small bags. If each bag holds 0.5 kg, how many bags are needed? $\textbf{(A)}\ 36 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 53 \qquad \textbf{(D)}\ 70 \qquad \textbf{(E)}\ 71$

Points $P$ and $Q$ lies inside triangle $ABC$ such that $\angle ACP =\angle BCQ$ and $\angle CAP = \angle BAQ$. Denote by $D,E$, and $F$ the feet of perpendiculars from $P$ to lines $BC,CA$, and $AB$, respectively. Prove that if $\angle DEF = 90^o$, then $Q$ is the orthocenter of triangle $BDF$.

There are $1996$ points marked on a straight line at regular intervals. Petya colors half of them red and the rest blue. Then Vasya divides them into pairs ''red'' - ''blue'' so that the sum distances between points in pairs was maximum. Prove that this maximum does not depend on what coloring Petya made.

Let $s_n$ be the number of solutions to $a_1 + a_2 + a_3 +a _4 + b_1 + b_2 = n$, where $a_1,a_2,a_3$ and $a_4$ are elements of the set $\{2, 3, 5, 7\}$ and $b_1$ and $b_2$ are elements of the set $\{ 1, 2, 3, 4\}$. Find the number of $n$ for which $s_n$ is odd. [i]Author: Alex Zhu[/i] [hide="Clarification"]$s_n$ is the number of [i]ordered[/i] solutions $(a_1, a_2, a_3, a_4, b_1, b_2)$ to the equation, where each $a_i$ lies in $\{2, 3, 5, 7\}$ and each $b_i$ lies in $\{1, 2, 3, 4\}$. [/hide]

The two polynomials $(x) =x^4+ax^3+bx^2+cx+d$ and $Q(x) = x^2+px+q$ take negative values on an interval $I$ of length greater than $2$, and nonnegative values outside of $I$. Prove that there exists $x_0 \in \mathbb R$ such that $P(x_0) < Q(x_0)$.

Given a positive integer $n$, does there exist a planar polygon and a point in its plane such that every line through that point meets the boundary of the polygon at exactly $2n$ points?