A chord $CD$ of a circle with center $O$ is perpendicular to a diameter $AB$. A chord $AE$ bisects the radius $OC$. Show that the line $DE$ bisects the chord $BC$ [i]V. Gordon[/i]

Each face of a $7 \times 7 \times 7$ cube is divided into unit squares. What is the maximum number of squares that can be chosen so that no two chosen squares have a common point? [i]A. Chukhnov[/i]

Let $ABC$ be a triangle, and $O$ the center of its circumcircle. Let a line through the point $O$ intersect the lines $AB$ and $AC$ at the points $M$ and $N$, respectively. Denote by $S$ and $R$ the midpoints of the segments $BN$ and $CM$, respectively. Prove that $\measuredangle ROS=\measuredangle BAC$.

Triangle $ABC$ is not isosceles. The incircle of $\triangle ABC$ touches the sides $BC$, $CA$, $AB$ in the points $K$, $L$, $M$. The parallel with $LM$ through $B$ meets $KL$ at $D$, the parallel with $LM$ through $C$ meets $KM$ at $E$. Prove that $DE$ passes through the midpoint of $\overline{LM}$.

Let $ABC$ be a triangle with $AB = 15$, $AC = 20$, and right angle at $A$. Let $D$ be the point on $\overline{BC}$ such that $\overline{AD}$ is perpendicular to $\overline{BC}$, and let $E$ be the midpoint of $\overline{AC}$. If $F$ is the point on $\overline{BC}$ such that $\overline{AD} \parallel \overline{EF}$, what is the area of quadrilateral $ADFE$?

In a triangle $ABC, P$ and $P'$ are points on side $BC, Q$ on side $CA$, and $R $ on side $AB$, such that $\frac{AR}{RB}=\frac{BP}{PC}=\frac{CQ}{QA}=\frac{CP'}{P'B}$ . Let $G$ be the centroid of triangle $ABC$ and $K$ be the intersection point of $AP'$ and $RQ$. Prove that points $P,G,K$ are collinear.

In $\triangle ABC$, $AC = BC$, and point $D$ is on $\overline{BC}$ so that $CD = 3 \cdot BD$. Let $E$ be the midpoint of $\overline{AD}$. Given that $CE = \sqrt{7}$ and $BE = 3$, the area of $\triangle ABC$ can be expressed in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n$.

A trapezoid $ABCD$ ($AB || CF$,$AB > CD$) is circumscribed.The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$,respectively.Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.

[b]p1.[/b] If $x$ is a real number that satisfies $\frac{48}{x} = 16$, find the value of $x$. [b]p2.[/b] If $ABC$ is a right triangle with hypotenuse $BC$ such that $\angle ABC = 35^o$, what is $\angle BCA$ in degrees? [img]https://cdn.artofproblemsolving.com/attachments/a/b/0f83dc34fb7934281e0e3f988ac34f653cc3f1.png[/img] [b]p3.[/b] If $a\vartriangle b = a + b - ab$, find $4\vartriangle 9$. [b]p4.[/b] Grizzly is $6$ feet tall. He measures his shadow to be $4$ feet long. At the same time, his friend Panda helps him measure the shadow of a nearby lamp post, and it is $6$ feet long. How tall is the lamp post in feet? [b]p5.[/b] Jerry is currently twice as old as Tom was $7$ years ago. Tom is $6$ years younger than Jerry. How many years old is Tom? [b]p6.[/b] Out of the $10, 000$ possible four-digit passcodes on a phone, how many of them contain only prime digits? [b]p7.[/b] It started snowing, which means Moor needs to buy snow shoes for his $6$ cows and $7$ sky bison. A cow has $4$ legs, and a sky bison has $6$ legs. If Moor has 36 snow shoes already, how many more shoes does he need to buy? Assume cows and sky bison wear the same type of shoe and each leg gets one shoe. [b]p8.[/b] How many integers $n$ with $1 \le n \le 100$ have exactly $3$ positive divisors? [b]p9.[/b] James has three $3$ candies and $3$ green candies. $3$ people come in and each randomly take $2$ candies. What is the probability that no one got $2$ candies of the same color? Express your answer as a decimal or a fraction in lowest terms. [b]p10.[/b] When Box flips a strange coin, the coin can land heads, tails, or on the side. It has a $\frac{1}{10}$probability of landing on the side, and the probability of landing heads equals the probability of landing tails. If Box flips a strange coin $3$ times, what is the probability that the number of heads flipped is equal to the number of tails flipped? Express your answer as a decimal or a fraction in lowest terms. [b]p11.[/b] James is travelling on a river. His canoe goes $4$ miles per hour upstream and $6$ miles per hour downstream. He travels $8$ miles upstream and then $8$ miles downstream (to where he started). What is his average speed, in miles per hour? Express your answer as a decimal or a fraction in lowest terms. [b]p12.[/b] Four boxes of cookies and one bag of chips cost exactly $1000$ jelly beans. Five bags of chips and one box of cookies cost less than $1000$ jelly beans. If both chips and cookies cost a whole number of jelly beans, what is the maximum possible cost of a bag of chips? [b]p13.[/b] June is making a pumpkin pie, which takes the shape of a truncated cone, as shown below. The pie tin is $18$ inches wide at the top, $16$ inches wide at the bottom, and $1$ inch high. How many cubic inches of pumpkin filling are needed to fill the pie? [img]https://cdn.artofproblemsolving.com/attachments/7/0/22c38dd6bc42d15ad9352817b25143f0e4729b.png[/img] [b]p14.[/b] For two real numbers $a$ and $b$, let $a\# b = ab - 2a - 2b + 6$. Find a positive real number $x$ such that $(x\#7) \#x = 82$. [b]p15.[/b] Find the sum of all positive integers $n$ such that $\frac{n^2 + 20n + 51}{n^2 + 4n + 3}$ is an integer. [b]p16.[/b] Let $ABC$ be a right triangle with hypotenuse $AB$ such that $AC = 36$ and $BC = 15$. A semicircle is inscribed in $ABC$ as shown, such that the diameter $XC$ of the semicircle lies on side $AC$ and that the semicircle is tangent to $AB$. What is the radius of the semicircle? [img]https://cdn.artofproblemsolving.com/attachments/4/2/714f7dfd09f6da1d61a8f910b5052e60dcd2fb.png[/img] [b]p17.[/b] Let $a$ and $b$ be relatively prime positive integers such that the product $ab$ is equal to the least common multiple of $16500$ and $990$. If $\frac{16500}{a}$ and $\frac{990}{b}$ are both integers, what is the minimum value of $a + b$? [b]p18.[/b] Let $x$ be a positive real number so that $x - \frac{1}{x} = 1$. Compute $x^8 - \frac{1}{x^8}$ . [b]p19.[/b] Six people sit around a round table. Each person rolls a standard $6$-sided die. If no two people sitting next to each other rolled the same number, we will say that the roll is valid. How many dierent rolls are valid? [b]p20.[/b] Given that $\frac{1}{31} = 0.\overline{a_1a_2a_3a_4a_5... a_n}$ (that is, $\frac{1}{31}$ can be written as the repeating decimal expansion $0.a_1a_2... a_na_1a_2... a_na_1a_2...$ ), what is the minimum value of $n$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine whether there exist $100$ distinct lines in the plane having exactly $1985$ distinct points of intersection

Let $ABC$ be an acute-angled triangle and $F$ a point in its interior such that $$ \angle AFB = \angle BFC = \angle CFA = 120^{\circ}.$$ Prove that the Euler lines of the triangles $AFB, BFC$ and $CFA$ are concurrent.

Frist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall that is located north and east of Frist can be expressed in the form $\frac{a}{b} \pi - c$, where $a, b, c$ are positive integers and $a$ and $b$ are relatively prime. Find $a + b + c$.

Find inside the triangle $ABC$, points $G$ and $H$ for which, respectively, the geometric mean and the harmonic mean of the distances to the sides of the triangle acquire maximum values. In which cases is the segment $GH$ parallel to one of the sides of the triangle? Find the length of such a segment $GH$.

The circle inscribed in the triangle $ABC$ with center $I$ touches the side $BC$ at the point $D$. The line $DI$ intersects the side $AC$ at the point $M$. The tangent from $M$ to the inscribed circle, different from $AC$, intersects the side $AB$ at the point $N$. The line $NI$ intersects the side $BC$ at the point $P$. Prove that $AB = BP$.

Let $ABC$ be an equilateral triangle, with sidelength equal to $a$. Let $P$ be a point in the interior of triangle $ABC$, and let $D,E$ and $F$ be the feet of the altitudes from $P$ on $AB, BC$ and $CA$, respectively. Prove that $\frac{|PD|+|PE|+|PF|}{3a}=\frac{\sqrt{3}}{6}$

Faith has four different integer length segments. It turned out that any three of them can form a triangle. What is the smallest total length of this set of segments?

[b]p1.[/b] Consider a normal $8 \times 8$ chessboard, where each square is labelled with either $1$ or $-1$. Let $a_k$ be the product of the numbers in the $k$th row, and let $b_k$ be the product of the numbers in the $k$th column. Find, with proof, all possible values of $\sum^8_{k=1}(a_kb_k)$. [b]p2.[/b] Let $\overline{AB}$ be a line segment with $AB = 1$, and $P$ be a point on $\overline{AB}$ with $AP = x$, for some $0 < x < 1$. Draw circles $C_1$ and $C_2$ with $\overline{AP}$, $\overline{PB}$ as diameters, respectively. Let $\overline{AB_1}$, $\overline{AB_2}$ be tangent to $C_2$ at $B_1$ and $B_2$, and let $\overline{BA_1}$;$\overline{BA_2}$ be tangent to $C_1$ at $A_1$ and $A_2$. Now $C_3$ is a circle tangent to $C_2$, $\overline{AB_1}$, and $\overline{AB_2}$; $C_4$ is a circle tangent to $C_1$, $\overline{BA_1}$, and $\overline{BA_2}$. (a) Express the radius of $C_3$ as a function of $x$. (b) Prove that $C_3$ and $C_4$ are congruent. [img]https://cdn.artofproblemsolving.com/attachments/c/a/fd28ad91ed0a4893608b92f5ccbd01088ae424.png[/img] [b]p3.[/b] Suppose that the graphs of $y = (x + a)^2$ and $x = (y + a)^2$ are tangent to one another at a point on the line $y = x$. Find all possible values of $a$. [b]p4.[/b] You may assume without proof or justification that the infinite radical expressions $\sqrt{a-\sqrt{a-\sqrt{a-\sqrt{a-...}}}}$ and $\sqrt{a-\sqrt{a+\sqrt{a-\sqrt{a+...}}}}$ represent unique values for $a > 2$. (a) Find a real number $a$ such that $$\sqrt{a-\sqrt{a-\sqrt{a-\sqrt{a+...}}}}= 2017$$ (b) Show that $$\sqrt{2018-\sqrt{2018+\sqrt{2018-\sqrt{2018+...}}}}=\sqrt{2017-\sqrt{2017-\sqrt{2017-\sqrt{2017-...}}}}$$ [b]p5.[/b] (a) Suppose that $m, n$ are positive integers such that $7n^2 - m^2 > 0$. Prove that, in fact, $7n^2 - m^2 \ge 3$. (b) Suppose that $m, n$ are positive integers such that $\frac{m}{n} <\sqrt7$. Prove that, in fact, $\frac{m}{n}+\frac{1}{mn} <\sqrt7$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On the sides $ BC, CA $ and $ AB $ (extremities excluded) of the triangle $ ABC, $ consider the arbitrary points $ P,Q,R $ and the circumcenters $ O_1,O_2,O_3 $ of $ AQR,BRP,CPQ. $ Show that $ O_1O_2O_3\sim ABC. $

[b]p1.[/b] A right triangle has hypotenuse of length $12$ cm. The height corresponding to the right angle has length $7$ cm. Is this possible? [img]https://cdn.artofproblemsolving.com/attachments/0/e/3a0c82dc59097b814a68e1063a8570358222a6.png[/img] [b]p2.[/b] Prove that from any $5$ integers one can choose $3$ such that their sum is divisible by $3$. [b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a knight on an arbitrary square. Then the second player can put another knight on a free square that is not controlled by the first knight. Then the first player can put a new knight on a free square that is not controlled by the knights on the board. Then the second player can do the same, etc. A player who cannot put a new knight on the board loses the game. Who has a winning strategy? [b]p4.[/b] Consider a regular octagon $ABCDEGH$ (i.e., all sides of the octagon are equal and all angles of the octagon are equal). Show that the area of the rectangle $ABEF$ is one half of the area of the octagon. [img]https://cdn.artofproblemsolving.com/attachments/d/1/674034f0b045c0bcde3d03172b01aae337fba7.png[/img] [b]p5.[/b] Can you find a positive whole number such that after deleting the first digit and the zeros following it (if they are) the number becomes $24$ times smaller? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Rectangle $ABCD$ has an area of 30. Four circles of radius $r_1 = 2$, $r_2 = 3$, $r_3 = 5$, and $r_4 = 4$ are centered on the four vertices $A$, $B$, $C$, and $D$ respectively. Two pairs of external tangents are drawn for the circles at A and $C$ and for the circles at $B$ and $D$. These four tangents intersect to form a quadrilateral $W XY Z$ where $\overline{W X}$ and $\overline{Y Z}$ lie on the tangents through the circles on $A$ and $C$. If $\overline{W X} + \overline{Y Z} = 20$, find the area of quadrilateral $W XY Z$. [img]https://cdn.artofproblemsolving.com/attachments/5/a/cb3b3457f588a15ffb4c875b1646ef2aec8d11.png[/img]

Consider a chess board, with the numbers $1$ through $64$ placed in the squares as in the diagram below. \[\begin{tabular}{| c | c | c | c | c | c | c | c |} \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ \hline 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 \\ \hline 25 & 26 & 27 & 28 & 29 & 30 & 31 & 32 \\ \hline 33 & 34 & 35 & 36 & 37 & 38 & 39 & 40 \\ \hline 41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 \\ \hline 49 & 50 & 51 & 52 & 53 & 54 & 55 & 56 \\ \hline 57 & 58 & 59 & 60 & 61 & 62 & 63 & 64 \\ \hline \end{tabular}\] Assume we have an infinite supply of knights. We place knights in the chess board squares such that no two knights attack one another and compute the sum of the numbers of the cells on which the knights are placed. What is the maximum sum that we can attain? Note. For any $2\times3$ or $3\times2$ rectangle that has the knight in its corner square, the knight can attack the square in the opposite corner.

Let $a_1, a_2, a_3, b_1, b_2, b_3$ be positive real numbers. Prove that \[(a_1b_2 + a_2b_1 + a_1b_3 + a_3b_1 + a_2b_3 + a_3b_2)^2 \geq 4(a_1a_2 + a_2a_3 + a_3a_1)(b_1b_2 + b_2b_3 + b_3b_1)\] and show that the two sides of the inequality are equal if and only if $\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}.$

Line $PQ$ is tangent to the incircle of triangle $ABC$ in such a way that the points $P$ and $Q$ lie on the sides $AB$ and $AC$, respectively. On the sides $AB$ and $AC$ are selected points $M$ and $N$, respectively, so that $AM = BP$ and $AN = CQ$. Prove that all lines constructed in this manner $MN$ pass through one point

Given a hexagon $ABCDEF$ such that $AB=BC$, $CD=DE$ , $EF=FA$ and $\angle A = \angle C = \angle E $ Prove that $AD, BE, CF$ are concurrent.

Let a circle through $B$ and $C$ of a triangle $ABC$ intersect $AB$ and $AC$ in $Y$ and $Z$ , respectively. Let $P$ be the intersection of $BZ$ and $CY$ , and let $X$ be the intersection of $AP$ and $BC$ . Let $M$ be the point that is distinct from $X$ and on the intersection of the circumcircle of the triangle $XYZ$ with $BC$. Prove that $M$ is the midpoint of $BC$