Find all the positive perfect cubes that are not divisible by $10$ such that the number obtained by erasing the last three digits is also a perfect cube.

Let $S$ be a set of positive integers satisfying the following two conditions: • For each positive integer $n$, at least one of $n, 2n, \dots, 100n$ is in $S$. • If $a_1, a_2, b_1, b_2$ are positive integers such that $\gcd(a_1a_2, b_1b_2) = 1$ and $a_1b_1, a_2b_2 \in S,$ then $a_2b_1, a_1b_2 \in S.$ Suppose that $S$ has natural density $r$. Compute the minimum possible value of $\lfloor 10^5r\rfloor$. Note: $S$ has natural density $r$ if $\tfrac{1}{n}|S \cap {1, \dots, n}|$ approaches $r$ as $n$ approaches $\infty$.

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

suppose that $O$ is the circumcenter of acute triangle $ABC$. we have circle with center $O$ that is tangent too $BC$ that named $w$ suppose that $X$ and $Y$ are the points of intersection of the tangent from $A$ to $w$ with line $BC$($X$ and $B$ are in the same side of $AO$) $T$ is the intersection of the line tangent to circumcirle of $ABC$ in $B$ and the line from $X$ parallel to $AC$. $S$ is the intersection of the line tangent to circumcirle of $ABC$ in $C$ and the line from $Y$ parallel to $AB$. prove that $ST$ is tangent $ABC$.

Can the number obtained by writing the numbers from 1 to $n$ in order ($n > 1$) be the same when read left-to-right and right-to-left? [i]N. Agakhanov[/i]

Let $\{a_1,a_2,...,a_{100}\}$ be a sequence of $100$ distinct real numbers. Show that there exists either an increasing subsequence $a_{i_1}<a_{i_2}<...<a_{i_{10}}$ $(i_1<i_2<...<i_{10})$ of $10$ numbers, or a decreasing subsequence $ a_{j_1}>a_{j_2}>...>a_{j_{12}}$ $(j_1<j_2<...<j_{12})$ of $12$ numbers, or both.

Let $f: \mathbb{R} \to (0, \infty)$ be a continuously differentiable function. Prove that there exists $\xi \in (0,1)$ such that $$e^{f'(\xi)} \cdot f(0)^{f(\xi)} = f(1)^{f(\xi)}$$

There are $n$ cards such that for each $i=1,2, \cdots n$, there are exactly one card labeled $i$. Initially the cards are piled with increasing order from top to bottom. There are two operations: [list] [*] $A$ : One can take the top card of the pile and move it to the bottom; [*] $B$ : One can remove the top card from the pile. [/list] The operation $ABBABBABBABB \cdots $ is repeated until only one card gets left. Let $L(n)$ be the labeled number on the final pile. Find all integers $k$ such that $L(3k)=k$.

For $p\in (0;\infty)$ find the area of the region bounded by the curves $y^{2}=4px$ and $16py^{2}=5(x-p)^{3}$

Find the number of functions $f$ from $\{0,1,2,3,4,5,6\}$ to the integers such that $f(0)=0, f(6)=12$, and \[|x-y| \le |f(x)-f(y)| \le 3 |x-y| \]for all $x$ and $y$ in $\{0,1,2,3,4,5,6\}$.

There are given points with integer coordinate $(m,n)$ such that $1\leq m,n\leq 4$. Two players, Ana and Ben, are playing a game: First Ana color one of the coordinates with red one, then she pass the turn to Ben who color one of the remaining coordinates with yellow one, then this process they repeate again one after other. The game win the first player who can create a rectangle with same color of vertices and the length of sides are positive integer numbers, otherwise the game is a tie. Does there exist a strategy for any of the player to win the game?

A quadrilateral $ABCD$ with $AB \perp BC$ , $AD \perp DC$, $E$ is a point that is on the line $BD$ with $EC=CA$ , $F$, $G$ is on the line $AB$ $AD$ such that $EF\perp AC $ and $EG\perp AC$ ,let $X Y$ be the midpoint of segment $AF AG $ , let $Z W$ be the midpoint of segment $BE DE $ , try to proof that $(WBX)$ is tangent to $(ZDY)$

Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number $$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$ is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$. [i]Proposed by Bayarmagnai Gombodorj, Mongolia[/i]

[b]p1.[/b] There are $4$ mirrors facing the inside of a $5\times 7$ rectangle as shown in the figure. A ray of light comes into the inside of a rectangle through $A$ with an angle of $45^o$. When it hits the sides of the rectangle, it bounces off at the same angle, as shown in the diagram. How many times will the ray of light bounce before it reaches any one of the corners $A$, $B$, $C$, $D$? A bounce is a time when the ray hit a mirror and reflects off it. [img]https://cdn.artofproblemsolving.com/attachments/1/e/d6ea83941cdb4b2dab187d09a0c45782af1691.png[/img] [b]p2.[/b] Jerry cuts $4$ unit squares out from the corners of a $45\times 45$ square and folds it into a $43\times 43\times 1$ tray. He then divides the bottom of the tray into a $43\times 43$ grid and drops a unit cube, which lands in precisely one of the squares on the grid with uniform probability. Suppose that the average number of sides of the cube that are in contact with the tray is given by $\frac{m}{n}$ where $m, n$ are positive integers that are relatively prime. Find $m + n$. [b]p3.[/b] Compute $2021^4 - 4 \cdot 2023^4 + 6 \cdot 2025^4 - 4 \cdot 2027^4 + 2029^4$. [b]p4.[/b] Find the number of distinct subsets $S \subseteq \{1, 2,..., 20\}$, such that the sum of elements in $S$ leaves a remainder of $10$ when divided by $32$. [b]p5.[/b] Some $k$ consecutive integers have the sum $45$. What is the maximum value of $k$? [b]p6.[/b] Jerry picks $4$ distinct diagonals from a regular nonagon (a regular polygon with $9$-sides). A diagonal is a segment connecting two vertices of the nonagon that is not a side. Let the probability that no two of these diagonals are parallel be $\frac{m}{n}$ where $m, n$ are positive integers that are relatively prime. Find $m + n$. [b]p7.[/b] The Olympic logo is made of $5$ circles of radius $1$, as shown in the figure [img]https://cdn.artofproblemsolving.com/attachments/1/7/9dafe6b72aa8471234afbaf4c51e3e97c49ee5.png[/img] Suppose that the total area covered by these $5$ circles is $a+b\pi$ where $a, b$ are rational numbers. Find $10a + 20b$. [b]p8.[/b] Let $P(x)$ be an integer polynomial (polynomial with integer coefficients) with $P(-5) = 3$ and $P(5) = 23$. Find the minimum possible value of $|P(-2) + P(2)|$. [b]p9. [/b]There exists a unique tuple of rational numbers $(a, b, c)$ such that the equation $$a \log 10 + b \log 12 + c \log 90 = \log 2025.$$ What is the value of $a + b + c$? [b]p10.[/b] Each grid of a board $7\times 7$ is filled with a natural number smaller than $7$ such that the number in the grid at the $i$th row and $j$th column is congruent to $i + j$ modulo $7$. Now, we can choose any two different columns or two different rows, and swap them. How many different boards can we obtain from a finite number of swaps? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the curves with equations $x^n+y^n=1$ for $n=2,4,6,8,\dots$. Denote $L_{2k}$ the length of the curve with $n=2k$. Find $\lim_{k\rightarrow\infty}L_{2k}$.

(a) Show that for any angle $\theta$ and for any natural number $m$: $$| \sin m\theta| \le m| \sin \theta|$$ (b) Show that for all angles $\theta_1$ and $\theta_2$ and for all even natural numbers $m$: $$| \sin m \theta_2 - \sin m \theta_1| \le m| \sin (\theta_2 - \theta_1)|$$ (c) Show that for every odd natural number $m$ there are two angles, resp. $\theta_1$ and $\theta_2$, exist for which the inequality in (b) is not valid.

An $n$-tuple $(x_1, x_2,\dots, x_n)$ is called unearthly if $q_1x_1 +q_2x_2 +\dots+q_nx_n$ is irrational for any non-negative rational coefficients $q_1, q_2, \dots, q_n$ where $q_i$’s are not all zero. Prove that it is possible to select an unearthly $n$-tuple from any $2n-1$ distinct irrational numbers.

Prove that the sum: \[ S_n=\binom{n}{1}+\binom{n}{3}\cdot 2005+\binom{n}{5}\cdot 2005^2+...=\sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor}\binom{n}{2k+1}\cdot 2005^k \] is divisible by $2^{n-1}$ for any positive integer $n$.

An isosceles trapezoid $ABCD$ $(AB \parallel CD)$ is given. Points $E$ and $F$ lie on the sides $BC$ and $AD$, and the points $M$ and $N$ lie on the segment $EF$ such that $DF = BE$ and $FM = NE$. Let $K$ and $L$ be the foot of perpendicular lines from $M$ and $N$ to $AB$ and $CD$, respectively. Prove that $EKFL$ is a parallelogram. [i]Proposed by Mahdi Etesamifard[/i]

For a given positive integer $n >2$, let $C_{1},C_{2},C_{3}$ be the boundaries of three convex $n-$ gons in the plane , such that $C_{1}\cap C_{2}, C_{2}\cap C_{3},C_{1}\cap C_{3}$ are finite. Find the maximum number of points of the sets $C_{1}\cap C_{2}\cap C_{3}$.

In the plane there are $2020$ points, some of which are black and the rest are green. For every black point, the following applies: [i]There are exactly two green points that represent the distance $2020$ from that black point. [/i] Find the smallest possible number of green dots. (Walther Janous)

Let the class of functions $f_n$ be defined such that $f_1(x)=|x^3-x^2|$ and $f_{k+1}(x)=|f_k(x)-x^3|$ for all $k\ge1$. Denote by $S_n$ the sum of all $y$-values of $f_n(x)$'s "sharp" points in the First Quadrant. (A "sharp" point is a point for which the derivative is not defined.) Find the ratio of odd to even terms, $$\lim_{k\to\infty}\frac{S_{2k+1}}{S_{2k}}$$

Let the decimal representations of numbers $A$ and $B$ be given as: $A = 0.a_1a_2\cdots a_k > 0$, $B = 0.b_1b_2\cdots b_k > 0$ (where $a_k, b_k$ can be 0), and let $S$ be the count of numbers $0.c_1c_2\cdots c_k$ such that $0.c_1c_2\cdots c_k < A$ and $0.c_kc_{k-1}\cdots c_1 < B$ ($c_k, c_1$ can also be 0). (Here, $0.c_1c_2\cdots c_r (c_r \neq 0)$ is considered the same as $0.c_1c_2\cdots c_r0\cdots0$). Prove: $\left| S - 10^k AB \right| \leq 9k.$

A set $S$ of unit cells of an $n\times n$ array, $n\geq 2$, is said [i]full[/i] if each row and each column of the array contain at least one element of $S$, but which has this property no more when any of its elements is removed. A full set having maximum cardinality is said [i]fat[/i], while a full set of minimum cardinality is said [i]meagre[/i]. i) Determine the cardinality $m(n)$ of the meagre sets, describe all meagre sets and give their count. ii) Determine the cardinality $M(n)$ of the fat sets, describe all fat sets and give their count. [i](Dan Schwarz)[/i]

There are 10 cards, labeled from 1 to 10. Three cards denoted by $ a,\ b,\ c\ (a > b > c)$ are drawn from the cards at the same time. Find the probability such that $ \int_0^a (x^2 \minus{} 2bx \plus{} 3c)\ dx \equal{} 0$.