Let a triangle $ABC$ be given. For any point $X$ of the triangle denote $m(X)=\min\{XA,XB,XC\}.$ Find all points $X$ (of triangle $ABC$) such that $m(X)$ is maximal.

Let $ ABC$ be an isoceles triangle with $ AC\equal{}BC$. A point $ P$ lies inside $ ABC$ such that \[ \angle PAB \equal{} \angle PBC, \angle PAC \equal{} \angle PCB.\] Let $ M$ be the midpoint of $ AB$ and $ K$ be the intersection of $ BP$ and $ AC$. Prove that $ AP$ and $ PK$ trisect $ \angle MPC$.

In a regular hexagon, segments with lengths from $1$ to $6$ were drawn as shown in the right figure (the segments go sequentially in increasing length, all the angles between them are right). Find the side length of this hexagon. [img]https://cdn.artofproblemsolving.com/attachments/3/1/82e4225b56d984e897a43ba1f403d89e5f4736.png[/img]

Let $ABC$ be a triangle such that $AC\not= BC,AB<AC$ and let $K$ be it's circumcircle. The tangent to $K$ at the point $A$ intersects the line $BC$ at the point $D$. Let $K_1$ be the circle tangent to $K$ and to the segments $(AD),(BD)$. We denote by $M$ the point where $K_1$ touches $(BD)$. Show that $AC=MC$ if and only if $AM$ is the bisector of the $\angle DAB$. [i]Neculai Roman[/i]

Let $ABC$ be an acute triangle (with $AB \ne AC$) and $M$ be the midpoint of $BC$. The circle of diameter $AM$ cuts $AC$ at $N$ and $BC$ again at $H$. A point $K$ is taken on $AC$ (between $A$ and $N$) such that $CN = NK$. Segments $AH$ and $BK$ intersect at $L$. The circle that passes through $A$,$K$ and $L$ cuts $AB$ at $P$. Prove that $C$,$L$ and $P$ are collinear.

In the convex quadrilateral $ABCD$, the bisectors of angles $A$ and $C$ intersect in $I$. Prove that $ABCD$ is circumscriptible if and only if $$S[AIB] + S[CID] =S[AID]+S[BIC]$$ ( $S[XYZ]$ denotes the area of the triangle $XYZ$)

Prove that the parallelogram $ABCD$ with relation $\angle ABD + \angle DAC = 90^o$, is either a rectangle or a rhombus.

A right cone in $xyz$-space has its apex at $(0,0,0)$, and the endpoints of a diameter on its base are $(12,13,-9)$ and $(12,-5,15)$. The volume of the cone can be expressed as $a\pi$. What is $a$?

Isabella is making sushi. She slices a piece of salmon into the shape of a solid triangular prism. The prism is $2$ cm thick, and its triangular faces have side lengths $7$ cm, $ 24$cm, and $25$ cm. Find the volume of this piece of salmon in cm$^3$. [i]Proposed by Isabella Li[/i]

It is given circle with center in center of coordinate center with radius of $2016$. On circle and inside it are $540$ points with integer coordinates such that no three of them are collinear. Prove that there exist two triangles with vertices in given points such that they have same area

Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\mathcal{P}$ and $\mathcal{Q}$. The intersection of planes $\mathcal{P}$ and $\mathcal{Q}$ is the line $\ell$. The distance from line $\ell$ to the point where the sphere with radius $13$ is tangent to plane $\mathcal{P}$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://imgur.com/1mfBNNL.png[/img]

Let $ABC$ be an equilateral triangle and a point M inside the triangle such that $MA^2 = MB^2 +MC^2$. Draw an equilateral triangle $ACD$ where $D \ne B$. Let the point $N$ inside $\vartriangle ACD$ such that $AMN$ is an equilateral triangle. Determine $\angle BMC$.

Let $ABCD$ be a trapezoid in which $AB \parallel CD$ and $AB = 2CD$. A line $\ell$ perpendicular to $CD$ was drawn through point $C$. A circle with center at point $D$ and radius $DA$ intersects line $\ell$ at points $P$ and $Q$. Prove that $AP \perp BQ$.

The base $A_{1}A_{2}\ldots A_{n}$ of the pyramid $VA_{1}A_{2}\ldots A_{n}$ is a regular polygon. Prove that if \[\angle VA_{1}A_{2}\equiv \angle VA_{2}A_{3}\equiv \cdots \equiv \angle VA_{n-1}A_{n}\equiv \angle VA_{n}A_{1},\] then the pyramid is regular.

The lines $y=x$, $y=2x$, and $y=3x$ are the three medians of a triangle with perimeter $1$. Find the length of the longest side of the triangle.

Two circumferences of radius $1$ that do not intersect, $c_1$ and $c_2$, are placed inside an angle whose vertex is $O$. $c_1$ is tangent to one of the rays of the angle, while $c_2$ is tangent to the other ray. One of the common internal tangents of $c_1$ and $c_2$ passes through $O$, and the other one intersects the rays of the angle at points $A$ and $B$, with $AO=BO$. Find the distance of point $A$ to the line $OB$.

In the non-isosceles triangle $ABC$,$D$ is the midpoint of side $BC$,$E$ is the midpoint of side $CA$,$F$ is the midpoint of side $AB$.The line(different from line $BC$) that is tangent to the inscribed circle of triangle $ABC$ and passing through point $D$ intersect line $EF$ at $X$.Define $Y,Z$ similarly.Prove that $X,Y,Z$ are collinear.

[b]p1.[/b] If $a \diamond b = ab - a + b$, find $(3 \diamond 4) \diamond 5$ [b]p2.[/b] If $5$ chickens lay $5$ eggs in $5$ days, how many chickens are needed to lay $10$ eggs in $10$ days? [b]p3.[/b] As Alissa left her house to go to work one hour away, she noticed that her odometer read $16261$ miles. This number is a "special" number for Alissa because it is a palindrome and it contains exactly $1$ prime digit. When she got home that evening, it had changed to the next greatest "special" number. What was Alissa's average speed, in miles per hour, during her two hour trip? [b]p4.[/b] How many $1$ in by $3$ in by $8$ in blocks can be placed in a $4$ in by $4$ in by $9$ in box? [b]p5.[/b] Apple loves eating bananas, but she prefers unripe ones. There are $12$ bananas in each bunch sold. Given any bunch, if there is a $\frac13$ probability that there are $4$ ripe bananas, a $\frac16$ probability that there are $6$ ripe bananas, and a $\frac12$ probability that there are $10$ ripe bananas, what is the expected number of unripe bananas in $12$ bunches of bananas? [b]p6.[/b] The sum of the digits of a $3$-digit number $n$ is equal to the same number without the hundreds digit. What is the tens digit of $n$? [b]p7.[/b] How many ordered pairs of positive integers $(a, b)$ satisfy $a \le 20$, $b \le 20$, $ab > 15$? [b]p8.[/b] Let $z(n)$ represent the number of trailing zeroes of $n!$. What is $z(z(6!))?$ (Note: $n! = n\cdot (n-1) \cdot\cdot\cdot 2 \cdot 1$) [b]p9.[/b] On the Cartesian plane, points $A = (-1, 3)$, $B = (1, 8)$, and $C = (0, 10)$ are marked. $\vartriangle ABC$ is reflected over the line $y = 2x + 3$ to obtain $\vartriangle A'B'C'$. The sum of the $x$-coordinates of the vertices of $\vartriangle A'B'C'$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$. Compute $a + b$. [b]p10.[/b] How many ways can Bill pick three distinct points from the figure so that the points form a non-degenerate triangle? [img]https://cdn.artofproblemsolving.com/attachments/6/a/8b06f70d474a071b75556823f70a2535317944.png[/img] [b]p11.[/b] Say piece $A$ is attacking piece $B$ if the piece $B$ is on a square that piece $A$ can move to. How many ways are there to place a king and a rook on an $8\times 8$ chessboard such that the rook isn't attacking the king, and the king isn't attacking the rook? Consider rotations of the board to be indistinguishable. (Note: rooks move horizontally or vertically by any number of squares, while kings move $1$ square adjacent horizontally, vertically, or diagonally). [b]p12.[/b] Let the remainder when $P(x) = x^{2020} - x^{2017} - 1$ is divided by $S(x) = x^3 - 7$ be the polynomial $R(x) = ax^2 + bx + c$ for integers $a$, $b$, $c$. Find the remainder when $R(1)$ is divided by $1000$. [b]p13.[/b] Let $S(x) = \left \lfloor \frac{2020}{x} \right\rfloor + \left \lfloor \frac{2020}{x + 1} \right\rfloor$. Find the number of distinct values $S(x)$ achieves for integers $x$ in the interval $[1, 2020]$. [b]p14.[/b] Triangle $\vartriangle ABC$ is inscribed in a circle with center $O$ and has sides $AB = 24$, $BC = 25$, $CA = 26$. Let $M$ be the midpoint of $\overline{AB}$. Points $K$ and $L$ are chosen on sides $\overline{BC}$ and $\overline{CA}$, respectively such that $BK < KC$ and $CL < LA$. Given that $OM = OL = OK$, the area of triangle $\vartriangle MLK$ can be expressed as $\frac{a\sqrt{b}}{c}$ where $a, b, c$ are positive integers, $gcd(a, c) = 1$ and $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p15.[/b] Euler's totient function, $\phi (n)$, is defined as the number of positive integers less than $n$ that are relatively prime to $n$. Let $S(n)$ be the set of composite divisors of $n$. Evaluate $$\sum^{50}_{k=1}\left( k - \sum_{d\in S(k)} \phi (d) \right)$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A cube of edge length $s>0$ has the property that its surface area is equal to the sum of its volume and five times its edge length. Compute all possible values of $s$.

About pentagon $ABCDE$ is known that angle $A$ and angle $C$ are right and that the sides $| AB | = 4$, $| BC | = 5$, $| CD | = 10$, $| DE | = 6$. Furthermore, the point $C'$ that appears by mirroring $C$ in the line $BD$, lies on the line segment $AE$. Find angle $E$.

Let $ABC$ be a triangle. Let $x_1$ and $x_2$ be two congruent circles, which touch each other and the segment $BC$, and which both lie within triangle $ABC$, and for which it also holds that $x_1$ touches the segment $CA$, and that $x_2$ is the segment $AB$. Let $X$ be the contact point of these two circles $x_1$ and $x_2$. Let $y_1$ and $y_2$ two congruent circles that touch each other and the segment $CA$, and both within of triangle $ABC$, and for which it also holds that $y_1$ touches the segment $AB$, and that $y_2$ the segment $BC$. Let $Y$ be the contact point of these two circles $y_1$ and $y_2$. Let $z_1$ and $z_2$ be two congruent circles that touch each other and the segment $AB$, and both within triangle $ABC$, and for which it also holds that $z_1$ touches the segment $BC$, and that $z_2$ the segment $CA$. Let $Z$ be the contact point of these two circles $z_1$ and $z_2$. Prove that the straight lines $AX, BY$ and $CZ$ intersect at a point.

The altitudes $AA_1$ and $BB_1$ of an acute-angled triangle ABC meet at point $O$. Let $A_1A_2$ and $B_1B_2$ be the altitudes of triangles $OBA_1$ and $OAB_1$ respectively. Prove that $A_2B_2$ is parallel to $AB$. (L.Steingarts)

An equilateral triangle of altitude $1$ is considered. For every point $P$ on the interior of the triangle, denote by $x, y , z$ the distances from the point $P$ to the sides of the triangle. a) Prove that for every point $P$ inside the triangle it is true that $x + y + z = 1$. b) For which points of the triangle does it hold that the distance to one side is greater than the sum of the distances to the other two? c) We have a bar of length $1$ and we break it into three pieces. find the probability that with these pieces a triangle can be formed.

Consider the set of all rectangles with a given area $S$. Find the largest value o $ M = \frac{S}{2S+p + 2}$ where $p$ is the perimeter of the rectangle.

The faces of a convex polyhedron are congruent parallelograms. Prove that they are all rhombuses.