[b]p1.[/b] Given a group of $n$ people. An $A$-list celebrity is one that is known by everybody else (that is, $n - 1$ of them) but does not know anybody. A $B$-list celebrity is one that is known by exactly $n - 2$ people but knows at most one person. (a) What is the maximum number of $A$-list celebrities? You must prove that this number is attainable. (b) What is the maximum number of $B$-list celebrities? You must prove that this number is attainable. [b]p2.[/b] A polynomial $p(x)$ has a remainder of $2$, $-13$ and $5$ respectively when divided by $x+1$, $x-4$ and $x-2$. What is the remainder when $p(x)$ is divided by $(x + 1)(x - 4)(x - 2)$? [b]p3.[/b] (a) Let $x$ and y be positive integers satisfying $x^2 + y = 4p$ and $y^2 + x = 2p$, where $p$ is an odd prime number. Prove: $x + y = p + 1$. (b) Find all values of $x, y$ and $p$ that satisfy the conditions of part (a). You will need to prove that you have found all such solutions. [b]p4.[/b] Let function $f(x, y, z)$ be defined as following: $$f(x, y, z) = \cos^2(x - y) + \cos^2(y - z) + \cos^2(z - x), x, y, z \in R.$$ Find the minimum value and prove the result. [b]p5.[/b] In the diagram below, $ABC$ is a triangle with side lengths $a = 5$, $b = 12$,$ c = 13$. Let $P$ and $Q$ be points on $AB$ and $AC$, respectively, chosen so that the segment $PQ$ bisects the area of $\vartriangle ABC$. Find the minimum possible value for the length $PQ$. [img]https://cdn.artofproblemsolving.com/attachments/b/2/91a09dd3d831b299b844b07cd695ddf51cb12b.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url]. Thanks to gauss202 for sending the problems.

$10$ rectangles have their vertices lie on a circle. The vertices divide the circle into $40$ equal arcs. Prove that two of the rectangles are congruent. [i]Proposed by Evan Chang (squareman), USA[/i]

Let $M$ be the midpoint of the side $BC$ of the $\triangle ABC$. The perpendicular at $A$ to $AM$ meets $(ABC)$ at $K$. The altitudes $BE,CF$ of the triangle $ABC$ meet $AK$ at $P, Q$. Show that the radical axis of the circumcircles of the triangles $PKE, QKF$ is perpendicular to $BC$.

A triangle with side lengths in the ratio $ 3: 4: 5$ is inscribed in a circle of radius $ 3$. What is the area of the triangle? $ \textbf{(A)}\ 8.64 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 5\pi \qquad \textbf{(D)}\ 17.28 \qquad \textbf{(E)}\ 18$

Convex quadrilateral $ABCD$ has side-lengths $AB=7$, $BC=9$, $CD=15$, and there exists a circle, lying inside the quadrilateral and having center $I$, that is tangent to all four sides of the quadrilateral. Points $M$ and $N$ are the midpoints of $AC$ and $BD$ respectively. It can be proven that point $I$ always lies on segment $MN$. Supposing further that $I$ is the midpoint of $MN$, the area of quadrilateral $ABCD$ may be expressed as $p\sqrt q$, where $p$ and $q$ are positive integers and $q$ is not divisible by the square of any prime. Compute $p\cdot q$.

Let $I$ be the incenter of triangle $ABC$ and let $D$ the foot of altitude from $I$ to $BC$. Suppose the reflection point $D’$ of $D$ with respect to $I$ satisfying $\overline{AD’} = \overline{ID’}$. Let $\Gamma$ be the circle centered at $D’$ that passing through $A$ and $I$, and let $X$, $Y\neq A$ be the intersection of $\Gamma$ and $AB$, $AC$, respectively. Suppose $Z$ is a point on $\Gamma$ so that $AZ$ is perpendicular to $BC$. Prove that $AD$, $D’Z$, $XY$ concurrent at a point.

Let $ABC$ be an isosceles triangle with apex at $C.$ Let $D$ and $E$ be two points on the sides $AC$ and $BC$ such that the angle bisectors $\angle DEB$ and $\angle ADE$ meet at $F,$ which lies on segment $AB.$ Prove that $F$ is the midpoint of $AB.$

A version of billiards is played on a right triangular table, with a pocket in each of the three corners, and one of the acute angles being $30^o$. A ball is played from just in front of the pocket at the $30^o$. vertex toward the midpoint of the opposite side. Prove that if the ball is played hard enough, it will land in the pocket of the $60^o$ vertex after $8$ reflections.

Three circles, $\omega_1$, $\omega_2$, $\omega_3$ are drawn, with $\omega_3$ externally tangent to $\omega_1$ at $C$ and internally tangent to $\omega_2$ at $D$. Say also that $\omega_1$, $\omega_2$ intersect at points $A, B$. Suppose the radius of $\omega_1$ is $20$, the radius of $\omega_2$ is $15$, and the radius of $\omega_3$ is $6$. Draw line $CD$, and suppose it meets $AB$ at point $X$. If $AB = 24$, then $CX$ can be written in the form $\frac{a \sqrt{b}}{c}$, where$ a, b, c$ are positive integers where $b$ is square-free, and $a, c$ are relatively prime. Find $a + b + c$.

Let $ABC$ be a triangle with $AB = 5$, $BC = 6$, and $AC = 7$. Let its orthocenter be $H$ and the feet of the altitudes from $A, B, C$ to the opposite sides be $D, E, F$ respectively. Let the line $DF$ intersect the circumcircle of $AHF$ again at $X$. Find the length of $EX$.

Through the midpoint $ S $ of the segment $ MN $ with endpoints lying on the legs of an isosceles triangle, a straight line is drawn parallel to the base of the triangle, intersecting its legs at points $ K $ and $ L $. Prove that the orthogonal projection of the segment $ MN $ onto the base of the triangle is equal to the segment $ KL $.

Let $ABC$ be an acute triangle with orthocenter $H$, and let $BE$ and $CF$ be the altitudes of the triangle. Choose two points $P$ and $Q$ on rays $BH$ and $CH$ respectively, such that: $\bullet$ $PQ$ is parallel to $BC$; $\bullet$ The quadrilateral $APHQ$ is cyclic. Suppose the circumcircles of triangles $APF$ and $AQE$ meet again at $X\neq A$. Prove that $AX$ is parallel to $BC$. [i]Proposed by Ivan Chan Kai Chin[/i]

$ ABCD$ is a convex quadrilateral for which $ AB$ is the longest side. Points $ M$ and $ N$ are located on sides $ AB$ and $ BC$ respectively, so that each of the segments $ AN$ and $ CM$ divides the quadrilateral into two parts of equal area. Prove that the segment $ MN$ bisects the diagonal $ BD$.

[b]p1.[/b] Suppose that Yunseo wants to order a pizza that is cut into $4$ identical slices. For each slice, there are $2$ toppings to choose from: pineapples and apples. Each slice must have exactly one topping. How many distinct pizzas can Yunseo order? Pizzas that can be obtained by rotating one pizza are considered the same. [b]p2.[/b] How many triples of distinct positive integers $(E, M, C)$ are there such that $E = MC^2$ and $E \le 50$? [b]p3.[/b] Given that the cubic polynomial $p(x)$ has leading coefficient $1$ and satisfies $p(0) = 0$, $p(1) = 1$, and $p(2) = 2$. Find $p(3)$. [b]p4.[/b] Olaf asks Anna to guess a two-digit number and tells her that it’s a multiple of $7$ with two distinct digits. Anna makes her first guess. Olaf says one digit is right but in the wrong place. Anna adjusts her guess based on Olaf’s comment, but Olaf answers with the same comment again. Anna now knows what the number is. What is the sum of all the numbers that Olaf could have picked? [b]p5.[/b] Vincent the Bug draws all the diagonals of a regular hexagon with area $720$, splitting it into many pieces. Compute the area of the smallest piece. [b]p6.[/b] Given that $y - \frac{1}{y} = 7 + \frac{1}{7}$, compute the least integer greater than $y^4 + \frac{1}{y^4}$. [b]p7.[/b] At $9:00$ A.M., Joe sees three clouds in the sky. Each hour afterwards, a new cloud appears in the sky, while each old cloud has a $40\%$ chance of disappearing. Given that the expected number of clouds that Joe will see right after $1:00$ P.M. can be written in the form $p/q$ , where $p$ and $q$ are relatively prime positive integers, what is $p + q$? [b]p8.[/b] Compute the unique three-digit integer with the largest number of divisors. [b]p9.[/b] Jo has a collection of $101$ books, which she reads one each evening for $101$ evenings in a predetermined order. In the morning of each day that Jo reads a book, Amy chooses a random book from Jo’s collection and burns one page in it. What is the expected number of pages that Jo misses? [b]p10.[/b] Given that $x, y, z$ are positive real numbers satisfying $2x + y = 14 - xy$, $3y + 2z = 30 - yz$, and $z + 3x = 69 - zx$, the expression $x + y + z$ can be written as $p\sqrt{q} - r$, where $p, q, r$ are positive integers and $q$ is not divisible by the square of any prime. Compute $p + q + r$. [b]p11.[/b] In rectangle $TRIG$, points $A$ and $L$ lie on sides $TG$ and $TR$ respectively such that $TA = AG$ and $TL = 2LR$. Diagonal $GR$ intersects segments $IL$ and $IA$ at $B$ and $E$ respectively. Suppose that the area of the convex pentagon with vertices $TABLE$ is equal to $21$. What is the area of $TRIG$? [b]p12.[/b] Call a number nice if it can be written in the form $2^m \cdot 3^n$, where $m$ and $n$ are nonnegative integers. Vincent the Bug fills in a $3$ by $3$ grid with distinct nice numbers, such that the product of the numbers in each row and each column are the same. What is the smallest possible value of the largest number Vincent wrote? [b]p13.[/b] Let $s(n)$ denote the sum of digits of positive integer $n$ and define $f(n) = s(202n) - s(22n)$. Given that $M$ is the greatest possible value of $f(n)$ for $0 < n < 350$ and $N$ is the least value such that $f(N) = M$, compute $M + N$. [b]p14.[/b] In triangle $ABC$, let M be the midpoint of $BC$ and let $E, F$ be points on $AB, AC$, respectively, such that $\angle MEF = 30^o$ and $\angle MFE = 60^o$. Given that $\angle A = 60^o$, $AE = 10$, and $EB = 6$,compute $AB + AC$. [b]p15.[/b] A unit cube moves on top of a $6 \times 6$ checkerboard whose squares are unit squares. Beginning in the bottom left corner, the cube is allowed to roll up or right, rolling about its bottom edges to travel from square to square, until it reaches the top right corner. Given that the side of the cube facing upwards in the beginning is also facing upwards after the cube reaches the top right corner, how many total paths are possible? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In an acute-angled triangle $ABC$, the point $O$ is the center of the circumcircle, $CH$ is the height of the triangle, and the point $T$ is the foot of the perpendicular dropped from the vertex $C$ on the line $AO$. Prove that the line $TH$ passes through the midpoint of the side $BC$ .

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Let $ ABCD $ be a cyclic quadrilateral and let $ M $ be the set of incenters and excenters of the triangles $ BCD $, $ CDA $, $ DAB $, $ ABC $ (so 16 points in total). Prove that there exist two sets $ \mathcal{K} $ and $ \mathcal{L} $ of four parallel lines each, such that every line in $ \mathcal{K} \cup \mathcal{L} $ contains exactly four points of $ M $.

Let $ABCD$ be a cyclic quadrilateral with center $O$. Suppose the circumcircles of triangles $AOB$ and $COD$ meet again at $G$, while the circumcircles of triangles $AOD$ and $BOC$ meet again at $H$. Let $\omega_1$ denote the circle passing through $G$ as well as the feet of the perpendiculars from $G$ to $AB$ and $CD$. Define $\omega_2$ analogously as the circle passing through $H$ and the feet of the perpendiculars from $H$ to $BC$ and $DA$. Show that the midpoint of $GH$ lies on the radical axis of $\omega_1$ and $\omega_2$. [i]Proposed by Yang Liu[/i]

Points $A$ and $B$ on the rays $CX$ and $CY$ respectively of an acute angle $XCY$ are given so that $CX < CA = CB < CY$. Construct a line meeting the ray $CX$ and the segments $AB,BC$ at $K,L,M$, respectively, such that $KA \cdot YB = XA \cdot MB = LA\cdot LB \ne 0$.

Given a real number $t$ with $0 < t < 1$, define the real-valued function $f(t, \theta) = \sum^{\infty}_{n=-\infty} t^{|n|}\omega^n$, where $\omega = e^{i \theta} = \cos \theta + i\sin \theta$. For $\theta \in [0, 2\pi)$, the polar curve $r(\theta) = f(t, \theta)$ traces out an ellipse $E_t$ with a horizontal major axis whose left focus is at the origin. Let $A(t)$ be the area of the ellipse $E_t$. Let $A\left( \frac12 \right) = \frac{a\pi}{b}$ , where $a, b$ are relatively prime positive integers. Find $100a +b$ .

Let $ABC$ be an isosceles triangle $(AB=AC)$ so that $\angle A< 2 \angle B$ . Let $D,Z $ points on the extension of height $AM$ so that $\angle CBD = \angle A$ and $\angle ZBA = 90^\circ$. Let $E$ the orthogonal projection of $M$ on height $BF$, and let $K$ the orthogonal projection of $Z$ on $AE$. Prove that $ \angle KDZ = \angle KDB = \angle KZB$.

The circumcenter and incenter of a given tetrahedron coincide. Prove that all its faces are congruent.

How many tangents to the curve $y = x^3-3x\:\: (y = x^3 + px)$ can be drawn from different points in the plane?

Let $ABC$ be a triangle, and $M$ be the midpoint of $BC$, Let $N$ be the point on the segment $AM$ with $AN = 3NM$, and $P$ be the intersection point of the lines $BN$ and $AC$. What is the area in cm$^2$ of the triangle $ANP$ if the area of the triangle $ABC$ is $40$ cm$^2$?

For any integer $n\ge 4$, prove that there exists a $n$-degree polynomial $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ satisfying the two following properties: [b](1)[/b] $a_i$ is a positive integer for any $i=0,1,\ldots,n-1$, and [b](2)[/b] For any two positive integers $m$ and $k$ ($k\ge 2$) there exist distinct positive integers $r_1,r_2,...,r_k$, such that $f(m)\ne\prod_{i=1}^{k}f(r_i)$.