Given positive odd number $m$ and integer ${a}.$ Proof: For any real number $c,$ $$\#\left\{x\in\mathbb Z\cap [c,c+\sqrt m]\mid x^2\equiv a\pmod m\right\}\le 2+\log_2m.$$ [i]Proposed by Yinghua Ai[/i]

For every positive integer $n{}$ denote $a_n$ as the last digit of the sum of the number from $1$ to $n{}$. For example $a_5=5, a_6=1.$ a) Find $a_{21}.$ b) Compute the sum $a_1+a_2+\ldots+a_{2015}.$

Color the vertexes of a quadrangular pyramid in one color, satisfying that two end points of any edge are in different colors. We have only 5 colors, then the number of ways coloring the quadrangular pyramid is________.

A permutation $\pi : \{1,2,\ldots,N\} \rightarrow \{1,2, \ldots,N\}$ is [i]very odd[/i] if the smallest positive integer $k$ such that $\pi^k(a) = a$ for all $1 \le a \le N$ is odd, where $\pi^k$ denotes $\pi$ composed with itself $k$ times. Let $X_0 = 1,$ and for $i \ge 1,$ let $X_i$ be the fraction of all permutations of $\{1,2,\ldots,i\}$ that are very odd. Let $S$ denote the set of all ordered $4$-tuples $(A,B,C,D)$ of nonnegative integers such that $A+B +C +D =2023.$ Find the last three digits of the integer $$2023\sum_{(A,B,C,D) \in S} X_AX_BX_CX_D.$$

When Troy writes his digits, his $0$, $1$, and $8$ look the same right-side-up and upside-down as seen in the figure below. His $6$ and $9$ look like upside-down images of each other. None of his other digits look like digits when they are inverted. How many different five-digit numbers (which do not begin with the digit zero) can Troy write which read the same right-side-up and upside-down? [asy] frame l; label(l,"\textsf{0}\qquad \textsf{l}\qquad\textsf{2}\qquad\textsf{3}\qquad\textsf{4}\qquad\textsf{5}\qquad\textsf{6}\qquad\textsf{7}\qquad\textsf{8}\qquad\textsf{9}"); add(rotate(180)*l); label("\textsf{0}\qquad\textsf{l}\qquad\textsf{2}\qquad\textsf{3}\qquad\textsf{4}\qquad\textsf{5}\qquad\textsf{6}\qquad\textsf{7}\qquad\textsf{8}\qquad\textsf{9}",(0,20)); [/asy]

Let $O$ be the circumcenter and $H$ be the orthocenter of an acute-angled triangle $ABC$. Point $T$ is the midpoint of the segment $AO$. The perpendicular bisector of $AO$ intersects the line $BC$ at point $S$. Prove that the circumcircle of the triangle $AST$ bisects the segment $OH$. [i](M. Berindeanu, RMC 2018 book)[/i]

A frog trainer places one frog at each vertex of an equilateral triangle $ABC$ of unit sidelength. The trainer can make one frog jump over another along the line joining the two, so that the total length of the jump is an even multiple of the distance between the two frogs just before the jump. Let $M$ and $N$ be two points on the rays $AB$ and $AC$, respectively, emanating from $A$, such that $AM = AN = \ell$, where $\ell$ is a positive integer. After a finite number of jumps, the three frogs all lie in the triangle $AMN$ (inside or on the boundary), and no more jumps are performed. Determine the number of final positions the three frogs may reach in the triangle $AMN$. (During the process, the frogs may leave the triangle $AMN$, only their nal positions are to be in that triangle.)

Find the maximum number of elements which can be chosen from the set $ \{1,2,3,\ldots,2003\}$ such that the sum of any two chosen elements is not divisible by 3.

Let $a$, $b$, $c$ be real positive numbers. Show that \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq \frac{a+b}{b+c}+\frac{b+c}{a+b}+1\]

Let $\mathcal S$ be the sphere with center $(0,0,1)$ and radius $1$ in $\mathbb R^3$. A plane $\mathcal P$ is tangent to $\mathcal S$ at the point $(x_0,y_0,z_0)$, where $x_0$, $y_0$, and $z_0$ are all positive. Suppose the intersection of plane $\mathcal P$ with the $xy$-plane is the line with equation $2x+y=10$ in $xy$-space. What is $z_0$?

[b]p1.[/b] Compute $$\sqrt{(\sqrt{63} +\sqrt{112} +\sqrt{175})(-\sqrt{63} +\sqrt{112} +\sqrt{175})(\sqrt{63}-\sqrt{112} +\sqrt{175})(\sqrt{63} +\sqrt{112} -\sqrt{175})}$$ [b]p2.[/b] Consider the set $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many distinct $3$-element subsets are there such that the sum of the elements in each subset is divisible by $3$? [b]p3.[/b] Let $a^2$ and $b^2$ be two integers. Consider the triangle with one vertex at the origin, and the other two at the intersections of the circle $x^2 + y^2 = a^2 + b^2$ with the graph $ay = b|x|$. If the area of the triangle is numerically equal to the radius of the circle, what is this area? [b]p4.[/b] Suppose $f(x) = x^3 + x - 1$ has roots $a$, $b$ and $c$. What is $$\frac{a^3}{1-a}+\frac{b^3}{1-b}+\frac{c^3}{1-c} ?$$ [b]p5.[/b] Lisa has a $2D$ rectangular box that is $48$ units long and $126$ units wide. She shines a laser beam into the box through one of the corners such that the beam is at a $45^o$ angle with respect to the sides of the box. Whenever the laser beam hits a side of the box, it is reflected perfectly, again at a $45^o$ angle. Compute the distance the laser beam travels until it hits one of the four corners of the box. [b]p6.[/b] How many ways can we form a group with an odd number of members (plural) from $99$ people total? Express your answer in the form $a^b + c$, where $a$, $b$, and $c$ are integers, and $a$ is prime. [b]p7.[/b] Let $$S = \log_2 9 \log_3 16 \log_4 25 ...\log_{999} 1000000.$$ Compute the greatest integer less than or equal to $\log_2 S$. [b]p8.[/b] A prison, housing exactly four hundred prisoners in four hundred cells numbered $1$-$400$, has a really messed-up warden. One night, when all the prisoners are asleep and all of their doors are locked, the warden toggles the locks on all of their doors (that is, if the door is locked, he unlocks the door, and if the door is unlocked, he locks it again), starting at door $1$ and ending at door $400$. The warden then toggles the lock on every other door starting at door $2$ ($2$, $4$, $6$, etc). After he has toggled the lock on every other door, the warden then toggles every third door (doors $3$, $6$, $9$, etc.), then every fourth door, etc., finishing by toggling every $400$th door (consisting of only the $400$th door). He then collapses in exhaustion. Compute the number of prisoners who go free (that is, the number of unlocked doors) when they wake up the next morning. [b]p9.[/b] Let $A$ and $B$ be fixed points on a $2$-dimensional plane with distance $AB = 1$. An ant walks on a straight line from point $A$ to some point $C$ on the same plane and finds that the distance from itself to $B$ always decreases at any time during this walk. Compute the area of the locus of points where point $C$ could possibly be located. [b]p10.[/b] A robot starts in the bottom left corner of a $4 \times 4$ grid of squares. How many ways can it travel to each square exactly once and then return to its start if it is only allowed to move to an adjacent (not diagonal) square at each step? [b]p11.[/b] Assuming real values for $p$, $q$, $r$, and $s$, the equation $$x^4 + px^3 + qx^2 + rx + s$$ has four non-real roots. The sum of two of these roots is $4 + 7i$, and the product of the other two roots is $3 - 4i$. Find $q$. [b]p12.[/b] A cube is inscribed in a right circular cone such that one face of the cube lies on the base of the cone. If the ratio of the height of the cone to the radius of the cone is $2 : 1$, what fraction of the cone's volume does the cube take up? Express your answer in simplest radical form. [b]p13.[/b] Let $$y =\dfrac{1}{1 +\dfrac{1}{9 +\dfrac{1}{5 +\dfrac{1}{9 +\dfrac{1}{5 +...}}}}}$$ If $y$ can be represented as $\frac{a\sqrt{b} + c}{d}$, where $b$ is not divisible by the square of any prime, and the greatest common divisor of $a$ and $d$ is $1$, find the sum $a + b + c + d$. [b]p14.[/b] Alice wants to paint each face of an octahedron either red or blue. She can paint any number of faces a particular color, including zero. Compute the number of ways in which she can do this. Two ways of painting the octahedron are considered the same if you can rotate the octahedron to get from one to the other. [b]p15.[/b] Find $n$ in the equation $$133^5 + 110^5 + 84^5 + 27^5 = n^5,$$ where $n$ is an integer less than $170$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a circumference $\Gamma$ a chord $PQ$ is considered such that the segment that joins the midpoint of the smallest arc $PQ$ and the midpoint of the segment $PQ$ measures $1$. Let $\Gamma_1, \Gamma_2$ and $\Gamma_3$ be three tangent circumferences to the chord $PQ$ that are in the same half plane than the center of $\Gamma$ with respect to the line $PQ$. Furthermore, $\Gamma_1$ and $\Gamma_3$ are internally tangent to $\Gamma$ and externally tangent to$ \Gamma_2$, and the centers of $\Gamma_1$ and $\Gamma_3$ are on different halfplanes with respect to the line determined by the centers of $\Gamma$ and $\Gamma_2$. If the sum of the radii of $\Gamma_1, \Gamma_2$ and $\Gamma_3$ is equal to the radius of $\Gamma$, calculate the radius of $\Gamma_2$.

On a board are written some positive reals (not necessarily distinct). For every two numbers in the frame $a$ and $b$ distinct such that $$\frac{1}{2}<\frac{a}{b}<2,$$ an allowed operation is to delete $a$ and $b$ and write $2a-b$ and $2b-a$ in their place. Show that we can do the operation only a finite number of times.

Find all positive integers $n$ such that there are $k \geq 2$ positive rational numbers $a_1, a_2, \ldots, a_k$ satisfying $a_1 + a_2 + \ldots + a_k = a_1 \cdot a_2 \cdots a_k = n.$

Define the distance between two triangles to be the closest distance between two vertices, one from each triangle. Is it possible to draw five triangles in the plane such that for any two of them, their distance equals the sum of their circumradii?

In triangle $ABC$, angle $C$ is right and the two catheti are both length $1$. For one given the choice of the point $P$ on the cathetus $BC$, the point $Q$ on the hypotenuse and the point $R$ are plotted on the second cathetus so that $PQ$ is parallel to $AC$ and $QR$ is parallel to $BC$. Thereby the triangle is divided into three parts. Determine the locations of point $P$ for which the rectangular part has a larger area than each of the other two parts.

Evaluate $$\frac{(2 + 2)^2}{2^2} \cdot \frac{(3 + 3 + 3 + 3)^3}{(3 + 3 + 3)^3} \cdot \frac{(6 + 6 + 6 + 6 + 6 + 6)^6}{(6 + 6 + 6 + 6)^6}$$

For a positive integer $n$, let $\tau(n)$ and $\sigma(n)$ be the number of positive divisors of $n$ and the sum of positive divisors of $n$, respectively. let $a$ and $b$ be positive integers such that $\sigma(a^n)$ divides $\sigma(b^n)$ for all $n\in \mathbb{N}$. Prove that each prime factor of $\tau(a)$ divides $\tau(b)$. Proposed by MohammadAmin Sharifi

Adeline, Bonnie, and Cathy are walking along a long flat path, with their initial distances shown below. [asy] size(10cm); draw((0,0)--(12,0)--(28,0)); label("Adeline",(0,1)); label("Bonnie",(12,1)); label("Cathy",(28,1)); label("12 miles",(6,-1)); label("16 miles",(20,-1)); dot((0,0)); dot((12,0)); dot((28,0)); [/asy] Adeline and Bonnie walk towards each other at constant speeds of $1$ and $2$ miles per hour, respectively. Cathy walks in the same direction as Bonnie. If all three girls meet each other at the same time, what is Cathy's walking speed, in miles per hour? $\textbf{(A) } 4~\text{mph}\qquad\textbf{(B) } 4.5~\text{mph}\qquad\textbf{(C) } 5~\text{mph}\qquad\textbf{(D) } 5.5~\text{mph}\qquad\textbf{(E) } 6~\text{mph}$

The sequence $ (x_n)$ of real numbers is defined by $ x_1\equal{}\frac{29}{10}$ and $ x_{n\plus{}1}\equal{}\frac{x_n}{\sqrt{x_n^2\minus{}1}}\plus{}\sqrt{3}$ for all $ n\ge 1$. Find a real number $ a$ (if exists) such that $ x_{2k\minus{}1}>a>x_{2k}$.

Does there exist a sequence of positive integers $a_1,a_2,...$ such that every positive integer occurs exactly once and that the number $\tau (na_{n+1}^n+(n+1)a_n^{n+1})$ is divisible by $n$ for all positive integer. Here $\tau (n)$ denotes the number of positive divisor of $n$.

There are $9$ points arranged in a $3\times 3$ square grid. Let two points be adjacent if the distance between them is half the side length of the grid. (There should be $12$ pairs of adjacent points). Suppose that we wanted to connect $8$ pairs of adjacent points, such that all points are connected to each other. In how many ways is this possible? [i]Proposed by Kevin You[/i]

Given a prime number $n \ge 5$. Prove that for any natural number $a \le \frac{n}{2} $, we can search for natural number $b \le \frac{n}{2}$ so the number of non-negative integer solutions $(x, y)$ of the equation $ax+by=n$ to be odd*. Clarification: * For example when $n = 7, a = 3$, we can choose$ b = 1$ so that there number of solutions og $3x + y = 7$ to be $3$ (odd), namely: $(0, 7), (1, 4), (2, 1)$

A box contains $900$ cards, labeled from $100$ to $999$. Cards are removed one at a time without replacement. What is the smallest number of cards that must be removed to guarantee that the labels of at least three removed cards have equal sums of digits?

Let $ABC$ be a right triangle with right angle at $B$. Let $ACDE$ be a square drawn exterior to triangle $ABC$. If $M$ is the center of this square, find the measure of $\angle MBC$.