In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.

Let $p$ be a prime number such that $p\equiv 1\pmod 9$. Show that there exist an integer $n$ such that $n^3-3n+1$ is divisible by $p$.

Of $4n$ points in a row, $2n$ are colored white and $2n$ are colored black. Swot tha tthere are $2n$ consecutive points of which exactly $n$ are white and $n$ are black.

Find all pairs of two positive integers $(m,n)$ satisfying $ 3^m - 7^n = 2 $.

For every positive real pair $(x,y)$ satisfying the equation $x^3+y^4 = x^2y$, if the greatest value of $x$ is $A$, and the greatest value of $y$ is $B$, then $A/B = ?$ $ \textbf{(A)}\ \frac{2}{3} \qquad \textbf{(B)}\ \frac{512}{729} \qquad \textbf{(C)}\ \frac{729}{1024} \qquad \textbf{(D)}\ \frac{3}{4} \qquad \textbf{(E)}\ \frac{243}{256}$

A positive integer $n$ is called nice if it has at least $3$ proper divisors and it is equal to the sum of its three largest proper divisors. For example, $6$ is nice because its largest proper divisors are $3,2,1$ and $6=3+2+1$. Find the number of nice integers not greater than $3000$.

[b]p1.[/b] The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were on each bus? [b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $99$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a $6$ cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $500$ coins? [b]p3.[/b] Find all solutions $a, b, c, d, e, f, g, h$ if these letters represent distinct digits and the following multiplication is correct: $\begin{tabular}{ccccc} & & a & b & c \\ + & & & d & e \\ \hline & f & a & g & c \\ x & b & b & h & \\ \hline f & f & e & g & c \\ \end{tabular}$ [b]p4.[/b] Is it possible to find a rectangle of perimeter $10$ m and cut it in rectangles (as many as you want) so that the sum of the perimeters is $500$ m? [b]p5.[/b] The picture shows a maze with chambers (shown as circles) and passageways (shown as segments). A cat located in chamber $C$ tries to catch a mouse that was originally in the chamber $M$. The cat makes the first move, moving from chamber $C$ to one of the neighboring chambers. Then the mouse moves, then the cat, and so forth. At each step, the cat and the mouse can move to any neighboring chamber or not move at all. The cat catches the mouse by moving into the chamber currently occupied by the mouse. Can the cat get the mouse? [img]https://cdn.artofproblemsolving.com/attachments/9/9/25f61e1499ff1cfeea591cb436d33eb2cdd682.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Over a period of $k$ consecutive days, a total of $2014$ babies were born in a certain city, with at least one baby being born each day. Show that: (a) If $1014 < k \le 2014$, there must be a period of consecutive days during which exactly $100$ babies were born. (b) By contrast, if $k = 1014$, such a period might not exist.

Find all pairs $(m,n)$ of integers, $m ,n \ge 2$ such that $mn - 1$ divides $n^3 - 1$.

Let $C$ be a finite set of chords in a circle such that each chord passes through the midpoint of some other chord. Prove that any two of these chords intersect inside the circle.

Let right $\triangle ABC$ have $AC = 3$, $BC = 4$, and right angle at $C$. Let $D$ be the projection from $C$ to $\overline{AB}$. Let $\omega$ be a circle with center $D$ and radius $\overline{CD}$, and let $E$ be a variable point on the circumference of $\omega$. Let $F$ be the reflection of $E$ over point $D$, and let $O$ be the center of the circumcircle of $\triangle ABE$. Let $H$ be the intersection of the altitudes of $\triangle EFO$. As $E$ varies, the path of $H$ traces a region $\mathcal R$. The area of $\mathcal R$ can be written as $\tfrac{m\pi}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $\sqrt{m}+\sqrt{n}$. [i]Proposed by ApraTrip[/i]

Let $B$ and $C$ be fixed points, and let $A$ be a variable point such that $\angle BAC$ is fixed. The midpoints of $AB$ and $AC$ are $D$ and $E$ respectively, and $F,G$ are points such that $DF\perp AB$, $EG\perp AC$ and $BF$ and $CG$ are perpendicular to $BC$. Prove that $BF\cdot CG$ remains constant as $A$ varies.

On a plane $\Pi$ is given a straight line $\ell$ and on the line $\ell$ are given two different points $A_1, A_2$. We consider on the plane $\Pi$, outside the line $\ell$, two different points $A_3, A_4$. Examine if it is possible to put points $A_3$ and $A_4$ on such positions such the four points $A_1, A_2, A_3, A_4$ form the maximal number of possible isosceles triangles, in the following cases: (a) when the points $A_3, A_4$ belong to dierent semi-planes with respect to $\ell$; (b) when the points $A_3, A_4$ belong to the same semi-planes with respect to $\ell$. Give all possible cases and explain how is possible to construct in each case the points $A_3$ and $A_4$.

Two circles of equal radius can tightly fit inside right triangle $ABC$, which has $AB=13$, $BC=12$, and $CA=5$, in the three positions illustrated below. Determine the radii of the circles in each case. [asy] size(400); defaultpen(linewidth(0.7)+fontsize(12)); picture p = new picture; pair s1 = (20,0), s2 = (40,0); real r1 = 1.5, r2 = 10/9, r3 = 26/7; pair A=(12,5), B=(0,0), C=(12,0); draw(p,A--B--C--cycle); label(p,"$B$",B,SW); label(p,"$A$",A,NE); label(p,"$C$",C,SE); add(p); add(shift(s1)*p); add(shift(s2)*p); draw(circle(C+(-r1,r1),r1)); draw(circle(C+(-3*r1,r1),r1)); draw(circle(s1+C+(-r2,r2),r2)); draw(circle(s1+C+(-r2,3*r2),r2)); pair D=s2+156/17*(A-B)/abs(A-B), E=s2+(169/17,0), F=extension(D,E,s2+A,s2+C); draw(incircle(s2+B,D,E)); draw(incircle(s2+A,D,F)); label("Case (i)",(6,-3)); label("Case (ii)",s1+(6,-3)); label("Case (iii)",s2+(6,-3));[/asy]

Given an integer $n\geq2$, let $x_1<x_2<\cdots<x_n$ and $y_1<y_2<\cdots<y_n$ be positive reals. Prove that for every value $C\in (-2,2)$ (by taking $y_{n+1}=y_1$) it holds that $\hspace{122px}\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_i+y_i^2}<\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_{i+1}+y_{i+1}^2}$. [i]Proposed by Mirko Petrusevski[/i]

Let $S$ be a shape in the plane which is obtained as a union of finitely many unit squares. Prove that the ratio of the perimeter and the area of $S$ is at most $8$.

Let $P$ and $Q$ be points selected uniformly and independently at random inside a regular hexagon $ABCDEF.$ Compute the probability that segment $\overline{PQ}$ is entirely contained in at least one of the quadrilaterals $ABCD,$ $BCDE,$ $CDEF,$ $DEFA,$ $EFAB,$ or $FABC.$

For which positive integers $n \ge 2$ it is possible to represent the number $n^2$ as a sum of several distinct positive integers not exceeding $2n$?

Constant points $B$ and $C$ lie on the circle $\omega$. The point middle of $BC$ is named $M$ by us. Assume that $A$ is a variable point on the $\omega$ and $H$ is the orthocenter of the triangle $ABC$. From the point $H$ we drop a perpendicular line to $MH$ to intersect the lines $AB$ and $AC$ at $X$ and $Y$ respectively. Prove that with the movement of $A$ on the $\omega$, the orthocenter of the triangle $AXY$ also moves on a circle.

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Rays $AO$, $CO$ intersect sides $BC, BA$ in points $A_1, C_1$ respectively, $K$ is the projection of $O$ onto the segment $A_1C_1$, $M$ is the midpoint of $AC$. Prove that $\angle HMA = \angle BKC_1$. [i]Proposed by Anton Trygub[/i]

Let $B$ be a bounded closed convex symmetric (with respect to the origin) set in $\mathbb{R}^{2}$ with boundary $\Gamma$. Let $B$ have the property that the ellipse of maximal area contained in $B$ is the disc $D$ of radius $1$ centered at the origin with boundary $C$. Prove that $A \cap \Gamma \ne \emptyset$ for any arc $A$ of $C$ of length $l(A)\geq \frac{\pi}{2}$.

Jane tells you that she is thinking of a three-digit number that is greater than $500$ that has exactly $20$ positive divisors. If Jane tells you the sum of the positive divisors of her number, you would not be able to figure out her number. If, instead, Jane had told you the sum of the \textit{prime} divisors of her number, then you also would not have been able to figure out her number. What is Jane's number? (Note: the sum of the prime divisors of $12$ is $2 + 3 = 5$, not $2 + 2 + 3 = 7$.)

If we divide number $19250$ with one number, we get remainder $11$. If we divide number $20302$ with the same number, we get the reamainder $3$. Which number is that?

On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widgets, and on each succeeding day, she sold two more widgets than she had sold on the previous day. How many widgets in total had Janabel sold after working $20$ days? $\textbf{(A) }39\qquad\textbf{(B) }40\qquad\textbf{(C) }210\qquad\textbf{(D) }400\qquad \textbf{(E) }401$

Find all positive integers $a, b, c, d$ with $a \le b$ and $c \le d$ such that $\begin{cases} a + b = cd \\ c + d = ab \end{cases}$ .