Let $a_0, a_1, a_2,\ldots $ be a sequence of real numbers satisfying $a_0=1$ and $a_n=a_{\lfloor 7n/9\rfloor}+a_{\lfloor n/9\rfloor}$ for $n=1, 2,\ldots $ Prove that there exists a positive integer $k$ with $a_k<\frac{k}{2001!}$.

Let $n$ be a positive square free integer, $S$ is a subset of $[n]:=\{1,2,\ldots ,n\}$ such that $|S|\ge n/2.$ Prove that there exists three elements $a,b,c\in S$ (can be same), satisfy $ab\equiv c\pmod n.$ [i]Created by Zhenhua Qu[/i]

Each square of a $33\times 33$ square grid is colored in one of the three colors: red, yellow or blue, such that the numbers of squares in each color are the same. If two squares sharing a common edge are in different colors, call that common edge a separating edge. Find the minimal number of separating edges in the grid.

Find all prime number $p$ such that the number of positive integer pair $(x,y)$ satisfy the following is not $29$. [list] [*]$1\le x,y\le 29$ [*]$29\mid y^2-x^p-26$ [/list]

J has several cheetahs in his dresser, which has $7$ drawers, such that each drawer has the same number of cheetahs. He notices that he can take out one drawer, and redistribute all of the cheetahs (including those in the removed drawer) in the remaining $6$ drawers such that each drawer still has an equal number of cheetahs as the other drawers. If he has at least one cheetah, what is the smallest number of cheetahs that he can have?

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

Let $D$ be a point on the side $BC$ of $\triangle ABC$. If $AB=8$, $AC=7$, $BD=2$, and $CD=1$, find $AD$.

Prove that if $x>0$ and $n\in\mathbb N,$ then we have \[\frac{x^n(x^{n+1}+1)}{x^n+1}\leq\left(\frac {x+1}{2}\right)^{2n+1}.\]

Consider an integrable function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $af(a)+bf(b)=0$ when $ab=1$. Find the value of the following integration: $$ \int_{0}^{\infty} f(x) \,dx $$

A jeweler can get an alloy that is $40\%$ gold for $200$ dollars per ounce, an alloy that is $60\%$ gold for $300$ dollar per ounce, and an alloy that is $90\%$ gold for $400$ dollars per ounce. The jeweler will purchase some of these gold alloy products, melt them down, and combine them to get an alloy that is $50\%$ gold. Find the minimum number of dollars the jeweler will need to spend for each ounce of the alloy she makes.

Let the roots $a,b,c$ of $$f(x)=x^3 +p x^2 + qx+r$$ be real, and let $a\leq b\leq c$. Prove that $f'(x)$ has a root in the interval $\left[\frac{b+c}{2}, \frac{b+2c}{3}\right]$. What will be the form of $f(x)$ if the root in question falls at either end of the interval?

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $x,y>0$ and $xy\ge1.$ Prove that $x^3+y^3+4xy\ge x^2+y^2+x+y+2.$ Let $x,y>0$ and $xy\ge1.$ Prove that $2(x^3+y^3+xy+x+y)\ge5(x^2+y^2).$

Let $a,b,c,d$ be positive integers such that $ab\equal{}cd$. Prove that $a\plus{}b\plus{}c\plus{}d$ is a composite number.

You have some cyan, magenta, and yellow beads on a non-reorientable circle, and you can perform only the following operations: 1. Move a cyan bead right (clockwise) past a yellow bead, and turn the yellow bead magenta. 2. Move a magenta bead left of a cyan bead, and insert a yellow bead left of where the magenta bead ends up. 3. Do either of the above, switching the roles of the words ``magenta'' and ``left'' with those of ``yellow'' and ``right'', respectively. 4. Pick any two disjoint consecutive pairs of beads, each either yellow-magenta or magenta-yellow, appearing somewhere in the circle, and swap the orders of each pair. 5. Remove four consecutive beads of one color. Starting with the circle: ``yellow, yellow, magenta, magenta, cyan, cyan, cyan'', determine whether or not you can reach a) ``yellow, magenta, yellow, magenta, cyan, cyan, cyan'', b) ``cyan, yellow, cyan, magenta, cyan'', c) ``magenta, magenta, cyan, cyan, cyan'', d) ``yellow, cyan, cyan, cyan''. [i]Proposed by Sammy Luo[/i]

An acute-angled triangle $ABC$ is inscribed in a circle with center $O$. The bisector of $\angle A$ meets $BC$ at $D$, and the perpendicular to $AO$ through $D$ meets the segment $AC$ in a point $P$. Show that $AB = AP$.

Let $x$ be a real number such that $0<x<\frac{\pi}{2}$. Prove that \[\cos^2(x)\cot (x)+\sin^2(x)\tan (x)\ge 1\]

Let $a$, $b$, $c$ be positive real numbers such that $abc=1$. Prove that \[\frac a{a^{2}+2}+\frac b{b^{2}+2}+\frac c{c^{2}+2}\leq 1 \]

We consider $5 \times 5$ tables containing a real number in each of the $25$ cells. The same number may occur in different cells, but no row or column contains five equal numbers. Such a table is [i]balanced [/i] if the number in the middle cell of every row and column is the average of the numbers in that row or column. A cell is called [i]small [/i] if the number in that cell is strictly smaller than the number in the cell in the very middle of the table. What is the least number of small cells that a balanced table can have?

Positive reals $a$, $b$, and $c$ obey $\frac{a^2+b^2+c^2}{ab+bc+ca} = \frac{ab+bc+ca+1}{2}$. Prove that \[ \sqrt{a^2+b^2+c^2} \le 1 + \frac{\lvert a-b \rvert + \lvert b-c \rvert + \lvert c-a \rvert}{2}. \][i]Proposed by Evan Chen[/i]

Let $n \in N^*$ . Let $a_1, a_2..., a_n$ be real such that $a_1 + a_2 +...+ a_n \ge 0$. Prove the inequality $\sqrt{a_1^2+1}+\sqrt{a_2^2+1}+...+\sqrt{a_1^2+1}\ge \sqrt{2n(a_1 + a_2 +...+ a_n )}$.

Let $ABC$ be an acute, scalene triangle, and let $P$ be an arbitrary point in its interior. Let $\mathcal{P}_A$ be the parabola with focus $P$ and directrix $BC$, and define $\mathcal{P}_B$ and $\mathcal{P}_C$ similarly. (a) Show that if $Q$ is an intersection point of $\mathcal{P}_B$ and $\mathcal{P}_C$, then $P$ and $Q$ are on the same side of $AB$, and $P$ and $Q$ are on the same side of $AC$. (b) You are given that $\mathcal{P}_B$ and $\mathcal{P}_C$ intersect at exactly two points. Let $\ell_A$ be the line between these points, and define $\ell_B$ and $\ell_C$ similarly. Show that $\ell_A$, $\ell_B$, and $\ell_C$ concur. [i]Note: A parabola with focus point $X$ and directrix line $\ell$ is the set of all points $Z$ that are the same distance from $X$ and $\ell$.[/i]

Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. [i]Proposed by David Stoner[/i]

Prove that there are only two real numbers $x$ such that \[(x-1)(x-2)(x-3)(x-4)(x-5)(x-6) = 720\]

Let $ ABC$ be a right triangle with $ \angle A \equal{} 90^\circ$. Let $ D$ be the midpoint of $ AB$ and let $ E$ be a point on segment $ AC$ such that $ AD \equal{} AE$. Let $ BE$ meet $ CD$ at $ F$. If $ \angle BFC \equal{} 135^\circ$, determine $ BC / AB$.

Let $P(n)=(n-1^3)(n-2^3)\ldots (n-40^3)$ for positive integers $n$. Let $d$ be the largest positive integer such that $d \mid P(n)$ for any $n>2023$. If $d$ is product of $m$ not necessarily distinct primes, find $m$.