Trapezoid $ABCD$ has sides $AB=92,BC=50,CD=19,AD=70$ $AB$ is parallel to $CD$ A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$.Given that $AP=\dfrac mn$ (Where $m,n$ are relatively prime).What is $m+n$?

Find all $x;y\in\mathbb{Z}$ satisfying the following condition: $$x^3=y^4+9x^2$$

Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is $(m + n \pi)/p$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p$.

Let $f(x)$ be a nonconstant real polynomial. The sequence $\{a_i\}_{i=1}^{\infty}$ of real numbers is strictly increasing and unbounded, as $$a_{i+1}<a_i+2020.$$ The integers $\lfloor{|f(a_1)|}\rfloor$ , $\lfloor{|f(a_2)|}\rfloor$ , $\lfloor{|f(a_3)|}\rfloor$ , $\dots$ are written consecutively in such a way that their digits form an infinite sequence of digits $\{s_k\}_{k=1}^{\infty}$ (here $s_k\in\{0, 1, \dots, 9\}$). $\quad$If $n\in\mathbb{N}$ , prove that among the numbers $\overline{s_{n(k-1)+1}s_{n(k-1)+2}\cdots s_{nk}}$ , where $k\in\mathbb{N}$ , all $n$-digit numbers appear.

In a town there are four pubs, $A,B,C,D$, and any two of them are connected to each other except $A$ and $D$. A drunkard wanders about the pubs starting with $A$ and, after having a drink, goes to any of the pubs directly connected, with equal probability. (a) What is the probability that the drunkard is at $C$ at its fifth drink? (b) Where is the drunkard most likely to be after $n$ drinks ($n > 5$)?

Consider a $12$-gon with sidelengths $1$, $2$, $3$, $4$, ..., $12$. Prove that there are three consecutive sides in this $12$-gon, whose lengths have a sum $> 20$.

Each of the sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ is divided into three equal parts, $|AE| = |EF| = |F B|$ , $|DP| = |P Q| = |QC|$. The diagonals of $AEPD$ and $FBCQ$ intersect at $M$ and $N$, respectively. Prove that the sum of the areas of $\vartriangle AMD$ and $\vartriangle BNC$ is equal to the sum of the areas of $\vartriangle EPM$ and $\vartriangle FNQ$.

The polynomial $ x^3\minus{}ax^2\plus{}bx\minus{}2010$ has three positive integer zeros. What is the smallest possible value of $ a$? $ \textbf{(A)}\ 78 \qquad \textbf{(B)}\ 88 \qquad \textbf{(C)}\ 98 \qquad \textbf{(D)}\ 108 \qquad \textbf{(E)}\ 118$

Let $n$ and $k$ be integers with $k\leq n-2$. The absolute value of the sum of elements of any $k$-element subset of $\{a_1,a_2,\cdots,a_n\}$ is less than or equal to 1. Show that: If $|a_1|\geq1$, then for any $2\leq i \leq n$, we have: $$|a_1|+|a_i|\leq2$$

Let $ABC$ be a triangle. The internal bisector of $\angle B$ meets $AC$ in $P$ and $I$ is the incenter of $ABC$. Prove that if $AP+AB = CB$, then $API$ is an isosceles triangle.

Two dueling wizards are at an altitude of $100$ above the sea. They cast spells in turn, and each spell is of the form "decrease the altitude by $a$ for me and by $b$ for my rival" where $a$ and $b$ are real numbers such that $0 < a < b$. Different spells have different values for $a$ and $b$. The set of spells is the same for both wizards, the spells may be cast in any order, and the same spell may be cast many times. A wizard wins if after some spell, he is still above water but his rival is not. Does there exist a set of spells such that the second wizard has a guaranteed win, if the number of spells is $(a)$ finite; $(b)$ infinite?

Let $I$ and $H$ be the incentre and orthocentre of triangle $ABC$ respectively. Let $P,Q$ be the midpoints of $AB,AC$. The rays $PI,QI$ intersect $AC,AB$ at $R,S$ respectively. Suppose that $T$ is the circumcentre of triangle $BHC$. Let $RS$ intersect $BC$ at $K$. Prove that $A,I$ and $T$ are collinear if and only if $[BKS]=[CKR]$. [i]Shen Wunxuan[/i]

Shiori places seven books, numbered from $1$ to $7$, on a bookshelf in some order. Later, she discovers that she can place two dividers between the books, separating the books into left, middle, and right sections, such that: $\bullet$ The left section is numbered in increasing order from left to right, or has at most one book. $\bullet$ The middle section is numbered in decreasing order from left to right, or has at most one book. $\bullet$ The right section is numbered in increasing order from left to right, or has at most one book. In how many possible orderings could Shiori have placed the books? For example, $(2, 3, 5, 4, 1, 6, 7)$ and $(2, 3, 4, 1, 5, 6, 7)$ are possible orderings with the partitions $2, 3, 5|4, 1|6, 7$ and $2, 3, 4|1|5, 6, 7$, but $(5, 6, 2, 4, 1, 3, 7)$ is not.

Find the sum of all positive integers $n$ such that \[ \frac{2n+1}{n(n-1)} \] has a terminating decimal representation. [i]Proposed by Evan Chen[/i]

The integer $ 9$ can be written as a sum of two consecutive integers: $ 9 \equal{} 4\plus{}5.$ Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: $ 9 \equal{} 4\plus{}5 \equal{} 2\plus{}3\plus{}4.$ Is there an integer that can be written as a sum of $ 1990$ consecutive integers and that can be written as a sum of (more than one) consecutive positive integers in exactly $ 1990$ ways?

A convex quadrilateral $ABCD$ is inscribed into a circle $\omega$ . Suppose that there is a point $X$ on the segment $AC$ such that the $XB$ and $XD$ tangents to the circle $\omega$ . Tangent of $\omega$ at $C$, intersect $XD$ at $Q$. Let $E$ ($E\ne A$) be the intersection of the line $AQ$ with $\omega$ . Prove that $AD, BE$, and $CQ$ are concurrent.

From an equilateral triangle with side 2014 we cut off another equilateral triangle with side 214, such that both triangles have one common vertex and two of the side of the smaller triangles lie on two of the side of the bigger triangle. Is it possible to cover this figure with figures in the picture without overlapping (rotation is allowed) if all figures are made of equilateral triangles with side 1? Explain the answer! [asy] import olympiad; unitsize(20); pair A,B,C,D,E,F,G,H; A=(0,0); B=(1,0); C=rotate(60)*B; D=rotate(60)*C; E=rotate(60)*D; F=rotate(60)*E; G=rotate(60)*F; draw(A--B); draw(A--C); draw(A--D); draw(A--E); draw(A--F); draw(A--G); draw(B--C--D--E--F--G--B); A=(2,0); B=A+(1,0); C=A+rotate(60)*(B-A); D=A+rotate(60)*(C-A); E=A+rotate(120)*(D-A); F=A+rotate(60)*(E-A); G=2*F-E; H=2*C-D; draw(A--D--C--A--B--C--H--B--G--F--E--A--F--B); A=(4,0); B=A+(1,0); C=A+rotate(-60)*(B-A); D=B+rotate(60)*(C-B); E=B+rotate(60)*(D-B); F=B+rotate(60)*(E-B); G=E+rotate(60)*(D-E); H=E+rotate(60)*(G-E); draw(A--B--C--A); draw(C--D--B); draw(D--E--B); draw(B--F--E); draw(E--G--D); draw(E--H--G); A=(8.5,0.5); B=A+(1,0); C=A+rotate(60)*(B-A); D=A+rotate(60)*(C-A); E=A+rotate(60)*(D-A); F=A+rotate(60)*(E-A); G=A+rotate(60)*(F-A); H=G+rotate(60)*(F-G); draw(A--B); draw(A--C); draw(A--D); draw(A--E); draw(A--F); draw(A--G); draw(B--C); draw(D--E--F--G--B); draw(G--H--F);[/asy]

Inscribed hexagon $AB_1CA_1BC_1$ is given. Circle $\omega$ is inscribed in both triangles $ABC$ and $A_1B_1C_1$ and touches segments $AB$ and $A_1B_1$ at points $D$ and $D_1$ respectively. Prove that if $\angle ACD = \angle BCD_1$, then $\angle A_1C_1D_1 = \angle B_1C_1D$.

Suppose $a$ is a real number such that $\sin(\pi \cdot \cos a) = \cos(\pi \cdot \sin a)$. Evaluate $35 \sin^2(2a) + 84 \cos^2(4a)$.

Find all functions $f : (0, +\infty) \to \mathbb{R}$ satisfying the following conditions: $(i)$ $f(x) + f(\frac{1}{x}) = 1$ for all $x> 0$; $(ii)$ $f(xy + x + y) = f(x)f(y)$ for all $x, y> 0$.

In the Martian language every finite sequence of letters of the Latin alphabet letters is a word. The publisher “Martian Words” makes a collection of all words in many volumes. In the first volume there are only one-letter words, in the second, two-letter words, etc., and the numeration of the words in each of the volumes continues the numeration of the previous volume. Find the word whose numeration is equal to the sum of numerations of the words Prague, Olympiad, Mathematics.

Does there exist a function $f:\mathbb{Z}\to\mathbb{Z}$ such that $f(x+f(y))=f(x)-y$ for all integers $x$ and $y$?

Triangles $ABC$ and $DEF$ have pairwise parallel sides: $EF \| BC, FD \| CA$, and $DE \| AB$. The line $m_A$ is the reflection of $EF$ through $BC$, similarly $m_B$ is the reflection of $FD$ through $CA$, and $m_C$ the reflection of $DE$ through $AB$. Assume that the lines $m_A, m_B$, and $m_C$ meet in a common point. What is the ratio between the areas of triangles $ABC$ and $DEF$?

Find all polynomials $f(x)$ with integer coefficients such that $f(n)$ and $f(2^{n})$ are co-prime for all natural numbers $n$.

In triangle $ABC$, $I$ is the incenter. We have chosen points $P,Q,R$ on segments $IA,IB,IC$ respectively such that $IP\cdot IA=IQ \cdot IB=IR\cdot IC$. Prove that the points $I$ and $O$ belong to Euler line of triangle $PQR$ where $O$ is circumcenter of $ABC$.