Let $P$ and $Q$ be the midpoints of non-parallel chords $k_1$ and $k_2$ of a circle $\omega$, respectively. Let the tangent lines of $\omega$ passing through the endpoints of $k_1$ intersect at $A$ and the tangent lines passing through the endpoints of $k_2$ intersect at $B$. Let the symmetric point of the orthocenter of triangle $ABP$ with respect to the line $AB$ be $R$ and let the feet of the perpendiculars from $R$ to the lines $AP, BP, AQ, BQ$ be $R_1, R_2, R_3, R_4$, respectively. Prove that \[ \frac{AR_1}{PR_1} \cdot \frac{PR_2}{BR_2} = \frac{AR_3}{QR_3} \cdot \frac{QR_4}{BR_4} \]

Call a polygon [i]normal[/i] if it can be inscribed in a unit circle. How many non-congruent normal polygons are there such that the square of each side length is a positive integer?

The ratio of $w$ to $x$ is $4 : 3$, the ratio of $y$ to $z$ is $3 : 2$, and the ratio of $z$ to $x$ is $1 : 6$. What is the ratio of $w$ to $y$? $\textbf{(A) }4:3 \qquad \textbf{(B) }3:2 \qquad \textbf{(C) } 8:3 \qquad \textbf{(D) } 4:1 \qquad \textbf{(E) } 16:3 $

In circle $ O$, $ G$ is a moving point on diameter $ \overline{AB}$. $ \overline{AA'}$ is drawn perpendicular to $ \overline{AB}$ and equal to $ \overline{AG}$. $ \overline{BB'}$ is drawn perpendicular to $ \overline{AB}$, on the same side of diameter $ \overline{AB}$ as $ \overline{AA'}$, and equal to $ BG$. Let $ O'$ be the midpoint of $ \overline{A'B'}$. Then, as $ G$ moves from $ A$ to $ B$, point $ O'$: $ \textbf{(A)}\ \text{moves on a straight line parallel to }{AB}\qquad \\ \textbf{(B)}\ \text{remains stationary}\qquad \\ \textbf{(C)}\ \text{moves on a straight line perpendicular to }{AB}\qquad \\ \textbf{(D)}\ \text{moves in a small circle intersecting the given circle}\qquad \\ \textbf{(E)}\ \text{follows a path which is neither a circle nor a straight line}$

Let $ f$ be a function from the non-negative integers to the non-negative integers such that $ f(nm) \equal{} n f(m) \plus{} m f(n), f(10) \equal{} 19, f(12) \equal{} 52,$ and $ f(15) \equal{} 26.$ What is $ f(8)$? A. 12 B. 24 C. 36 D. 48 E. 60

Let $\mathbb{N}_{\geqslant 1}$ be the set of positive integers. Find all functions $f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}$ such that, for all positive integers $m$ and $n$: \[\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) = \mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left(f(m),n\right).\] Note: if $a$ and $b$ are positive integers, $\mathrm{GCD}(a,b)$ is the largest positive integer that divides both $a$ and $b$, and $\mathrm{LCM}(a,b)$ is the smallest positive integer that is a multiple of both $a$ and $b$.

A positive integer is called [i]pretty[/i] if it is equal to the sum of the fourth powers of five distinct divisors. [list=a] [*]Prove that every pretty number is divisible by $5$. [*]Determine if there are infinitely many beautiful numbers. [/list]

The integers from $1$ to $1993$ are written in a line in some order. The following operation is performed with this line: if the first number is $k$ then the first $k$ numbers are rewritten in reverse order. Prove that after some finite number of these operations, the first number in the line of numbers will be $1$.

[b]p1.[/b] An animal farm has geese and pigs with a total of $30$ heads and $84$ legs. Find the number of pigs and geese on this farm. [b]p2.[/b] What is the maximum number of $1 \times 1$ squares of a $7 \times 7$ board that can be colored black in such a way that the black squares don’t touch each other even at their corners? Show your answer on the figure below and explain why it is not possible to get more black squares satisfying the given conditions. [img]https://cdn.artofproblemsolving.com/attachments/d/5/2a0528428f4a5811565b94061486699df0577c.png[/img] [b]p3.[/b] Decide whether it is possible to divide a regular hexagon into three equal not necessarily regular hexagons? A regular hexagon is a hexagon with equal sides and equal angles. [img]https://cdn.artofproblemsolving.com/attachments/3/7/5d941b599a90e13a2e8ada635e1f1f3f234703.png[/img] [b]p4.[/b] A rectangle is subdivided into a number of smaller rectangles. One observes that perimeters of all smaller rectangles are whole numbers. Is it possible that the perimeter of the original rectangle is not a whole number? [b]p5.[/b] Place parentheses on the left hand side of the following equality to make it correct. $$ 4 \times 12 + 18 : 6 + 3 = 50$$ [b]p6.[/b] Is it possible to cut a $16\times 9$ rectangle into two equal parts which can be assembled into a square? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a isosceles triangle with $AB=AC$ an $D$ be the foot of perpendicular of $A$. $P$ be an interior point of triangle $ADC$ such that $m(APB)>90$ and $m(PBD)+m(PAD)=m(PCB)$. $CP$ and $AD$ intersects at $Q$, $BP$ and $AD$ intersects at $R$. Let $T$ be a point on $[AB]$ and $S$ be a point on $[AP$ and not belongs to $[AP]$ satisfying $m(TRB)=m(DQC)$ and $m(PSR)=2m(PAR)$. Show that $RS=RT$

Let $ABC$ be a triangle, let $I$ be its incentre and let $D$ be the orthogonal projection of $I$ on $BC.$ The circle $\odot(ABC)$ crosses the line $AI$ again at $M,$ and the line $DM$ again at $N.$ Prove that the lines $AN$ and $IN$ are perpendicular. [i]Freddie Illingworth & Dominic Yeo[/i]

In an abandoned chemistry lab Gerome found a two-pan balance scale and three 1-gram weights, three 5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile. Find the number of different positive weights of chemicals that Gerome could measure.

In hexagon $ABCDEF$, which is nonconvex but not self-intersecting, no pair of opposite sides are parallel. The internal angles satisfy $\angle A=3\angle D$, $\angle C=3\angle F$, and $\angle E=3\angle B$. Furthermore $AB=DE$, $BC=EF$, and $CD=FA$. Prove that diagonals $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ are concurrent.

An ant walks across the floor of a circular path of radius $r$ and moves in a straight line, but sometimes stops. Each time it stops, before resuming the march, it rotates $60^o$ alternating the direction (if the last time it turned $60^o$ to its right, the next one does it $60^o$ to its left, and vice versa). Find the maximum possible length of the path the ant goes through. Prove that the length found is, in fact, as long as possible. Figure: turn $60^o$ to the right .

Let $\phi = \frac{1+\sqrt{5}}{2}$. A [i]base-$\phi$ number[/i] $(a_n a_{n-1} ... a_1 {a_0})_{\phi}$, where $0 \le a_n, a_{n-1}, ..., a_0 \le 1$ are integers, is defined by \[ (a_n a_{n-1} ... a_1 {a_0})_{\phi} = a_n \cdot \phi^n + a_{n-1} \cdot \phi^{n-1} + ... + a_1 \cdot \phi^1 + a_0. \] Compute the number of base-$\phi$ numbers $(b_jb_{j-1}... b_1{b_0})_\phi$ which satisfy $b_j \ne 0$ and \[ (b_jb_{j-1}... b_1{b_0})_\phi = \underbrace{(100 ... 100)_\phi}_{\text{Twenty}\ 100's}. \][i]Proposed by Yang Liu[/i]

Zelda played the [i]Adventures of Math[/I] game on August 1 and scored $1700$ points. She continued to play daily over the next $5$ days. The bar chart below shows the daily change in her score compared to the day before. (For example, Zelda's score on August 2 was $1700 + 80 = 1780$ points.) What was Zelda's average score in points over the $6$ days? [img]https://cdn.artofproblemsolving.com/attachments/5/c/d246d9bf4002bfe23f859bd21605f882d8b7bc.png[/img] $\textbf{(A) }1700\qquad\textbf{(B) }1702\qquad\textbf{(C) }1703\qquad\textbf{(D) }1713\qquad\textbf{(E) }1715$

Solve the equation \[\int_0^1(x+y)^2u(x)dx=\lambda u(y)+1\]

For coprime integers $m > n > 1$ consider the polynomials $f(x) = x^{m+n} -x^{m+1} -x+1$ and $g(x) = x^{m+n} +x^{n+1} -x+1$. If $f$ and $g$ have a common divisor of degree greater than $1$, find this divisor.

Prove that for any $n$ there exists a positive integer $k$ such that all the numbers $k\cdot2^s+1~(s=1,\ldots,n)$ are composite.

Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$. Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$. [i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$. Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.

Let $ABC$ be a triangle such that $AB = 13$, $BC = 14$, $AC = 15$. Let $M$ be the midpoint of $BC$ and define $P \ne B$ to be a point on the circumcircle of $ABC$ such that $BP \perp PM$. Furthermore, let $H$ be the orthocenter of $ABM$ and define $Q$ to be the intersection of $BP$ and $AC$. If $R$ is a point on $HQ$ such that $RB \perp BC$, find the length of $RB$.

How many non-congruent triangles with integer sides and perimeter 1999 can be constructed?

Let $n$ be a positive integer. Prove that $a_1!a_2! ... a_n! < k!$, where $k$ is an integer which is greater than the sum of the positive integers $a_1, a_2,.., a_n$.

Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1).$ During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using up $t_n$ minutes. If the total elapsed time has exceeded $1$ minute during the $n$th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most $3$ steps in all. What is the probability that Amelia’s position when she stops will be greater than $1$? $\textbf{(A) }\frac{1}{3} \qquad \textbf{(B) }\frac{1}{2} \qquad \textbf{(C) }\frac{2}{3} \qquad \textbf{(D) }\frac{3}{4} \qquad \textbf{(E) }\frac{5}{6}$

In a right-angled and isosceles triangle, the two catheti are both length $1$. Find the length of the shortest line segment dividing the triangle into two figures with the same area, and specify the location of this line segment