Let $ r$ and $ s$ two orthogonal lines that does not lay on the same plane. Let $ AB$ be their common perpendicular, where $ A\in{}r$ and $ B\in{}s$(*).Consider the sphere of diameter $ AB$. The points $ M\in{r}$ and $ N\in{s}$ varies with the condition that $ MN$ is tangent to the sphere on the point $ T$. Find the locus of $ T$. Note: The plane that contains $ B$ and $ r$ is perpendicular to $ s$.

Let $ABC$ be a triangle and $ P, Q ( \ne A, B, C ) $ are the points lying on segments $ BC , CA $. Let $ I, J, K $ be the incenters of triangle $ ABP, APQ, CPQ $. Prove that $ PIJK $ is a convex quadrilateral.

Let $ ABC $ be a triangle, and let $M$ be midpoint of $BC$. Let $ I_b $ and $ I_c $ be incenters of $ AMB $ and $ AMC $. Prove that the second intersection of circumcircles of $ ABI_b $ and $ ACI_c $ distinct from $A$ lies on line $AM$.

In triangle $ABC$, $A'$, $B'$, and $C'$ are on the sides $BC$, $AC$, and $AB$, respectively. Given that $AA'$, $BB'$, and $CC'$ are concurrent at the point $O$, and that \[\frac{AO}{OA'}+\frac{BO}{OB'}+\frac{CO}{OC'}=92,\] find \[\frac{AO}{OA'}\cdot \frac{BO}{OB'}\cdot \frac{CO}{OC'}.\]

Let $ABC$ be an non- isosceles triangle, $H_a$, $H_b$, and $H_c$ be the feet of the altitudes drawn from the vertices $A, B$, and $C$, respectively, and $M_a$, $M_b$, and $M_c$ be the midpoints of the sides $BC$, $CA$, and $AB$, respectively. The circumscribed circles of triangles $AH_bH_c$ and $AM_bM_c$ intersect for second time at point $A'$. The circumscribed circles of triangles $BH_cH_a$ and $BM_cM_a$ intersect for second time at point $B'$. The circumscribed circles of triangles $CH_aH_b$ and $CM_aM_b$ intersect for second time at point $C'$. Prove that points $A', B'$ and $C'$ lie on the same line.

$n$ is a natural number that $\frac{x^{n}+1}{x+1}$ is irreducible over $\mathbb Z_{2}[x]$. Consider a vector in $\mathbb Z_{2}^{n}$ that it has odd number of $1$'s (as entries) and at least one of its entries are $0$. Prove that these vector and its translations are a basis for $\mathbb Z_{2}^{n}$

In the triangle $ABC$, $AC^2$ is the arithmetic mean of $BC^2$ and $AB^2$. Show that $\cot^2B\ge \cot A\cdot\cot C$.

[u]Round 5[/u] [b]p13.[/b] Initially, the three numbers $20$, $201$, and $2016$ are written on a blackboard. Each minute, Zhuo selects two of the numbers on the board and adds $1$ to each. Find the minimum $n$ for which Zhuo can make all three numbers equal to $n$. [b]p14.[/b] Call a three-letter string rearrangeable if, when the first letter is moved to the end, the resulting string comes later alphabetically than the original string. For example, $AAA$ and $BAA$ are not rearrangeable, while $ABB$ is rearrangeable. How many three-letters strings with (not necessarily distinct) uppercase letters are rearrangeable? [b]p15.[/b] Triangle $ABC$ is an isosceles right triangle with $\angle C = 90^o$ and $AC = 1$. Points $D$, $E$ and $F$ are chosen on sides $BC$,$CA$ and $AB$, respectively, such that $AEF$, $BFD$, $CDE$, and $DEF$ are isosceles right triangles. Find the sum of all distinct possible lengths of segment $DE$. [u]Round 6[/u] [b]p16.[/b] Let $p, q$, and $r$ be prime numbers such that $pqr = 17(p + q + r)$. Find the value of the product $pqr$. [b]p17.[/b] A cylindrical cup containing some water is tilted $45$ degrees from the vertical. The point on the surface of the water closest to the bottom of the cup is $6$ units away. The point on the surface of the water farthest from the bottom of the cup is $10$ units away. Compute the volume of the water in the cup. [b]p18.[/b] Each dot in an equilateral triangular grid with $63$ rows and $2016 = \frac12 \cdot 63 \cdot 64$ dots is colored black or white. Every unit equilateral triangle with three dots has the property that exactly one of its vertices is colored black. Find all possible values of the number of black dots in the grid. [u]Round 7[/u] [b]p19.[/b] Tomasz starts with the number $2$. Each minute, he either adds $2$ to his number, subtracts $2$ from his number, multiplies his number by $2$, or divides his number by $2$. Find the minimum number of minutes he will need in order to make his number equal $2016$. [b]p20.[/b] The edges of a regular octahedron $ABCDEF$ are painted with $3$ distinct colors such that no two edges with the same color lie on the same face. In how many ways can the octahedron be painted? Colorings are considered different under rotation or reflection. [b]p21.[/b] Jacob is trapped inside an equilateral triangle $ABC$ and must visit each edge of triangle $ABC$ at least once. (Visiting an edge means reaching a point on the edge.) His distances to sides $AB$, $BC$, and $CA$ are currently $3$, $4$, and $5$, respectively. If he does not need to return to his starting point, compute the least possible distance that Jacob must travel. [u]Round 8[/u] [b]p22.[/b] Four integers $a, b, c$, and $d$ with a $\le b \le c \le d$ satisfy the property that the product of any two of them is equal to the sum of the other two. Given that the four numbers are not all equal, determine the $4$-tuple $(a, b, c, d)$. [b]p23.[/b] In equilateral triangle $ABC$, points $D$,$E$, and $F$ lie on sides $BC$,$CA$ and $AB$, respectively, such that $BD = 4$ and $CD = 5$. If $DEF$ is an isosceles right triangle with right angle at $D$, compute $EA + FA$. [b]p24.[/b] On each edge of a regular tetrahedron, four points that separate the edge into five equal segments are marked. There are sixteen planes that are parallel to a face of the tetrahedron and pass through exactly three of the marked points. When the tetrahedron is cut along each of these sixteen planes, how many new tetrahedrons are produced? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2934049p26256220]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 5[/u] [b]p13.[/b] Mandy is baking cookies. Her recipe calls for $N$ grams of flour, where $N$ is the number of perfect square divisors of $20! + 24!$. Find $N$. [b]p14.[/b] Consider a circular table with center $R$. Beef-loving Bryan places a steak at point $I$ on the circumference of the table. Then he places a bowl of rice at points $C$ and $E$ on the circumference of the table such that $CE \parallel IR$ and $\angle ICE = 25^o$. Find $\angle CIE$. [b]p15.[/b] Enya writes the $4$-letter words $LEEK$, $BEAN$, $SOUP$, $PEAS$, $HAMS$, and $TACO$ on the board. She then thinks of one of these words and gives Daria, Ava, Harini, and Tiffany a slip of paper containing exactly one letter from that word such that if they ordered the letters on their slips correctly, they would form the word. Each person announces at the same time whether they know the word or not. Ava, Harini, and Tiffany all say they do not know the word, while Daria says she knows the word. After hearing this, Ava, Harini, and Tiffany all know the word. Assuming all four girls are perfect logicians and they all thought of the same correct word, determine Daria’s letter. [u]Round 6[/u] [b]p16.[/b] Michael receives a cheese cube and a chocolate octahedron for his 5th birthday. On every day after, he slices off each corner of his cheese and chocolate with a knife. Each slice cuts off exactly one corner. He then eats each corner sliced off. Find the difference between the total number of cheese and chocolate pieces he has eaten by the end of his $6$th birthday. (Michael’s $5$th and $6$th birthdays do not occur on leap years.) [b]p17.[/b] Let $D$ be the average of all positive integers n satisfying $$lcm (gcd (n, 2000), gcd (n, 24)) = gcd (lcm (n, 2000), lcm (n, 24)).$$ Find $3D$. [b]p18.[/b] The base $\vartriangle ABC$ of the triangular pyramid $PABC$ is an equilateral triangle with a side length of $3$. Given that $PA = 3$, $PB = 4$, and $PC = 5$, find the circumradius of $PABC$. [u]Round 7[/u] [b]p19.[/b] $2049300$ points are arranged in an equilateral triangle point grid, a smaller version of which is shown below, such that the sides contain $2024$ points each. Peter starts at the topmost point of the grid. At $9:00$ am each day, he moves to an adjacent point in the row below him. Derrick wants to prevent Peter from reaching the bottom row, so at $12:00$ pm each day, he selects a point on the bottom row and places a rock at that point. Peter stops moving as soon as he is guaranteed to end up at a point with a rock on it. At least how many moves will Peter complete, no matter how Derrick places the rocks? [img]https://cdn.artofproblemsolving.com/attachments/f/a/346d25a5d7bb7a5fbefae7edad727965312b25.png[/img] [b]p20.[/b] There are $N$ stones in a pile, where $N$ is a positive integer. Ava and Anika take turns playing a game, with Ava moving first. If there are n stones in the pile, a move consists of removing $x$ stones, where $1 < gcd(x, n) \le x < n$. Whoever first has no possible moves on their turn wins. Both Ava and Anika play optimally. Find the $2024$th smallest value of $N$ for which Ava wins. [b]p21.[/b] Alan is bored and alone, so he plays a fun game with himself. He writes down all quadratic polynomials with leading coefficient $1$ whose coefficients are integers between $-10$ and $10$, inclusive, on a blackboard. He then erases all polynomials which have a non-integer root. Alan defines the size of a polynomial $P(x)$ to be $P(1)$ and spends an hour adding up the sizes of all the polynomials remaining on the blackboard. Assuming Alan does computation perfectly, find the sum Alan obtains. [u]Round 8[/u] [b]p22.[/b] A prime number is a positive integer with exactly two distinct divisors. You must submit a prime number for this problem. If you do not submit a prime number, you gain $0$ points, and your submission will not be considered valid. The median of all valid submitted numbers is $M$ (duplicates are counted). Estimate $2M$. If your team’s absolute difference between $2M$ and your submission is the $i$th smallest absolute difference among all teams, you gain max$(23 - 2i, 0)$ points. All teams who did not submit any number gain $0$ points. (In the case of a tie, all teams that tied gain the same amount of points.) [b]p23.[/b] Ribbotson the Frog is at the point $(0, 0)$ and wants to reach the point $(18, 18)$ in $36$ steps. Each step, he either moves one unit in the $+x$ direction or one unit in the $+y$ direction. However, Ribbotson hates turning, so he must make at least two steps in any direction before switching directions. If $m$ is the number of different paths Ribbotson the Frog can make, estimate $m$. If $N$ is your team’s submitted number, your team earns points equal to the closest integer to $21\left(1 -\left|\log_{10}\frac{N}{m} \right|^2\right)$. [b]p24.[/b] Let $M = \pi^{\pi^{\pi^{\pi}}}$. Estimate $k$, where $M = 10^{10^{k}}$. If $N$ is your team’s submitted number, your team earns points equal to the closest integer to $21 \cdot 1.01^{(-|N-k|^3)}$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3248729p29808138]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a convex quadrilateral inscribed in a circle of radius $1$. Prove that \[ 0< (AB+BC+CD+AD)-(AC+BD) < 4. \]

$S$ is a sphere center $O. G$ and $G'$ are two perpendicular great circles on $S$. Take $A, B, C$ on $G$ and $D$ on $G'$ such that the altitudes of the tetrahedron $ABCD$ intersect at a point. Find the locus of the intersection.

In a quadrilateral $ABCD$ let $A', B', C'$ and $D'$ be the circumcenters of the triangles $BCD, CDA, DAB$ and $ABC$, respectively. Denote by $S(X, YZ)$ the plane which passes through the point $X$ and is perpendicular to the line $YZ.$ Prove that if $A', B', C'$ and $D'$ don't lie in a plane, then four planes $S(A, C'D'), S(B, A'D'), S(C, A'B')$ and $S(D, B'C')$ pass through a common point.

Let $ABC$ be triangle with $A$-excenter $J$. The reflection of $J$ in $BC$ is $K$. The points $E$ and $F$ are on $BJ, CJ$ such that $\angle EAB=\angle CAF=90^{\circ}$. Prove that $\angle FKE+\angle FJE=180^{\circ}$.

Consider a convex quadrilateral $ABCD$. Let $O$ be the intersection of $AC$ and $BD$; $M, N$ be the centroid of $\Delta AOB$ and $\Delta COD$ and $P, Q$ be orthocenter of $\Delta BOC$ and $\Delta DOA$, respectively. Prove that $MN\bot PQ$.

Let $ABC$ be an acute triangle with $AB<AC$. The angle bisector of $BAC$ intersects the side $BC$ and the circumcircle of $ABC$ at $D$ and $M\neq A$, respectively. Points $X$ and $Y$ are chosen so that $MX \perp AB$, $BX \perp MB$, $MY \perp AC$, and $CY \perp MC$. Prove that the points $X,D,Y$ are collinear.

The diagram below shows a $12$ by $20$ rectangle split into four strips of equal widths all surrounding an isosceles triangle. Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/9/e/ed6be5110d923965c64887a2ca8e858c977700.png[/img]

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

$(HUN 3)$ In the plane $4000$ points are given such that each line passes through at most $2$ of these points. Prove that there exist $1000$ disjoint quadrilaterals in the plane with vertices at these points.

Two triangles $ ABC$and $ A'B'C'$ are positioned in the space such that the length of every side of $ \triangle ABC$ is not less than $ a$, and the length of every side of $ \triangle A'B'C'$ is not less than $ a'$. Prove that one can select a vertex of $ \triangle ABC$ and a vertex of $ \triangle A'B'C'$ so that the distance between the two selected vertices is not less than $ \sqrt {\frac {a^2 \plus{} a'^2}{3}}$.

In rectangle $ABCD$, $AB = 40$ and $AD = 30$. Let $C' $ be the reflection of $C$ over $BD$. Find the length of $AC'$.

$H$ is the orthocentre of an acute–angled triangle $ABC$. A point $E$ is taken on the line segment $CH$ such that $ABE$ is a right–angled triangle. Prove that the area of the triangle $ABE$ is the geometric mean of the areas of triangles $ABC$ and $ABH$.

A disc is divided into $30$ segments which are labelled by $50,100,150,\ldots,1500$ in some order. Show that there always exist three successive segments, the sum of whose labels is at least $2350$.

The base of the pyramid is a convex quadrangle. Is there necessarily a section of this pyramid that does not intersect the base and is an inscribed quadrangle? (M. Volchkevich)

A triangle $ABC$ is given with barycentre $G$ and circumcentre $O$. The perpendicular bisectors of $GA, GB$ meet at $C_1$,of $GB,GC$ meet at $A _1$, and $GC,GA$ meet at $B_1$. Prove that $O$ is the barycenter of the triangle $A_1B_1C_1$.

Given a triangle $ \triangle{ABC} $ whose incenter is $ I $ and $ A $-excenter is $ J $. $ A' $ is point so that $ AA' $ is a diameter of $ \odot\left(\triangle{ABC}\right) $. Define $ H_{1}, H_{2} $ to be the orthocenters of $ \triangle{BIA'} $ and $ \triangle{CJA'} $. Show that $ H_{1}H_{2} \parallel BC $