[b]p1.[/b] Evaluate $1+3+5+··· +2019$. [b]p2.[/b] Evaluate $1^2 -2^2 +3^2 -4^2 +...· +99^2 -100^2$. [b]p3. [/b]Find the sum of all solutions to $|2018+|x -2018|| = 2018$. [b]p4.[/b] The angles in a triangle form a geometric series with common ratio $\frac12$ . Find the smallest angle in the triangle. [b]p5.[/b] Compute the number of ordered pairs $(a,b,c,d)$ of positive integers $1 \le a,b,c,d \le 6$ such that $ab +cd$ is a multiple of seven. [b]p6.[/b] How many ways are there to arrange three birch trees, four maple, and five oak trees in a row if trees of the same species are considered indistinguishable. [b]p7.[/b] How many ways are there for Mr. Paul to climb a flight of 9 stairs, taking steps of either two or three at a time? [b]p8.[/b] Find the largest natural number $x$ for which $x^x$ divides $17!$ [b]p9.[/b] How many positive integers less than or equal to $2018$ have an odd number of factors? [b]p10.[/b] Square $MAIL$ and equilateral triangle $LIT$ share side $IL$ and point $T$ is on the interior of the square. What is the measure of angle $LMT$? [b]p11.[/b] The product of all divisors of $2018^3$ can be written in the form $2^a \cdot 2018^b$ for positive integers $a$ and $b$. Find $a +b$. [b]p12.[/b] Find the sum all four digit palindromes. (A number is said to be palindromic if its digits read the same forwards and backwards. [b]p13.[/b] How ways are there for an ant to travel from point $(0,0)$ to $(5,5)$ in the coordinate plane if it may only move one unit in the positive x or y directions each step, and may not pass through the point $(1, 1)$ or $(4, 4)$? [b]p14.[/b] A certain square has area $6$. A triangle is constructed such that each vertex is a point on the perimeter of the square. What is the maximum possible area of the triangle? [b]p15.[/b] Find the value of ab if positive integers $a,b$ satisfy $9a^2 -12ab +2b^2 +36b = 162$. [b]p16.[/b] $\vartriangle ABC$ is an equilateral triangle with side length $3$. Point $D$ lies on the segment $BC$ such that $BD = 1$ and $E$ lies on $AC$ such that $AE = AD$. Compute the area of $\vartriangle ADE$. [b]p17[/b]. Let $A_1, A_2,..., A_{10}$ be $10$ points evenly spaced out on a line, in that order. Points $B_1$ and $B_2$ lie on opposite sides of the perpendicular bisector of $A_1A_{10}$ and are equidistant to $l$. Lines $B_1A_1,...,B_1A_{10}$ and $B_2A_1,...· ,B_2A_{10}$ are drawn. How many triangles of any size are present? [b]p18.[/b] Let $T_n = 1+2+3··· +n$ be the $n$th triangular number. Determine the value of the infinite sum $\sum_{k\ge 1} \frac{T_k}{2^k}$. [b]p19.[/b] An infinitely large bag of coins is such that for every $0.5 < p \le 1$, there is exactly one coin in the bag with probability $p$ of landing on heads and probability $1- p$ of landing on tails. There are no other coins besides these in the bag. A coin is pulled out of the bag at random and when flipped lands on heads. Find the probability that the coin lands on heads when flipped again. [b]p20.[/b] The sequence $\{x_n\}_{n\ge 1}$ satisfies $x1 = 1$ and $(4+ x_1 + x_2 +··· + x_n)(x_1 + x_2 +··· + x_{n+1}) = 1$ for all $n \ge 1$. Compute $\left \lfloor \frac{x_{2018}}{x_{2019}} \right \rfloor$. PS. You had better use hide for answers.

Consider the sets $M = \{0,1,2,, 2019\}$ and $$A=\left\{ x\in M\,\, | \frac{x^3-x}{24} \in N\right\} $$ a) How many elements does the set $A$ have? b) Determine the smallest natural number $n$, $n \ge 2$, which has the property that any $n$-element subset of the set $A $contains two distinct elements whose difference is divisible by $40$.

In a cube $ABCDA'B'C'D' $two points are considered, $M \in (CD')$ and $N \in (DA')$. Show that the $MN$ is common perpendicular to the lines $CD'$ and $DA'$ if and only if $$\frac{D'M}{D'C}=\frac{DN}{DA'} =\frac{1}{3}.$$

Different points $A$ and $D$ are on the same side of the line $BC$, with $|AB| = | BC|= |CD|$ and lines $AD$ and $BC$ are perpendicular. Let $E$ be the intersection point of lines $AD$ and $BC$. Prove that $||BE| - |CE|| < |AD| \sqrt3$

Let $M$ be the set of the integer numbers from the range $[-n, n]$. The subset $P$ of $M$ is called a [i]base subset[/i] if every number from $M$ can be expressed as a sum of some different numbers from $P$. Find the smallest natural number $k$ such that every $k$ numbers that belongs to $M$ form a base subset.

In a social network with a fixed finite setback of users, each user had a fixed set of [i]followers[/i] among the other users. Each user has an initial positive integer rating (not necessarily the same for all users). Every midnight, the rating of every user increases by the sum of the ratings that his followers had just before midnight. Let $m$ be a positive integer. A hacker, who is not a user of the social network, wants all the users to have ratings divisible by $m$. Every day, he can either choose a user and increase his rating by 1, or do nothing. Prove that the hacker can achieve his goal after some number of days. [i]Vladislav Novikov[/i]

On a unit circle with center $O$, $AB$ is an arc with the central angle $\alpha<90^\circ$. Point $H$ is the foot of the perpendicular from $A$ to $OB$, $T$ is a point on arc $AB$, and $l$ is the tangent to the circle at $T$. The line $l$ and the angle $AHB$ form a triangle $\Delta$. (a) Prove that the area of $\Delta$ is minimal when $T$ is the midpoint of arc $AB$. (b) Prove that if $S_\alpha$ is the minimal area of $\Delta$ then the function $\frac{S_\alpha}\alpha$ has a limit when $\alpha\to0$ and find this limit.

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ meet at $O$. Let $m$ be the measure of the acute angle formed by these diagonals. A variable angle $xOy$ of measure $m$ intersects the quadrilateral by a convex quadrilateral of constant area. Prove that $ABCD$ is a square.

Jeff writes down the two-digit base-$10$ prime $\overline{ab_{10}}$. He realizes that if he misinterprets the number as the base $11$ number $\overline{ab_{11}}$ or the base $12$ number $\overline{ab_{12}}$, it is still a prime. What is the least possible value of Jeff’s number (in base $10$)? [i]Proposed byMuztaba Syed[/i]

Show that for each positive integer n, \[n!=\prod_{i=1}^n \; \text{lcm} \; \{1, 2, \ldots, \left\lfloor\frac{n}{i} \right\rfloor\}\] (Here lcm denotes the least common multiple, and $\lfloor x\rfloor$ denotes the greatest integer $\le x$.)

Rectangle $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The inscribed circle of triangle $BEF$ is tangent to $EF$ at point $P,$ and the inscribed circle of triangle $DEF$ is tangent to $EF$ at point $Q.$ Find $PQ.$

How many positive integers $n$ satisfy the inequality \[ \left\lceil \frac{n}{101} \right\rceil + 1 > \frac{n}{100} \, ? \] Recall that $\lceil a \rceil$ is the least integer that is greater than or equal to $a$.

Given the set $S$ whose elements are zero and the even integers, positive and negative. Of the five operations applied to any pair of elements: (1) addition (2) subtraction (3) multiplication (4) division (5) finding the arithmetic mean (average), those elements that only yield elements of $S$ are: $ \textbf{(A)}\ \text{all} \qquad\textbf{(B)}\ 1,2,3,4\qquad\textbf{(C)}\ 1,2,3,5\qquad\textbf{(D)}\ 1,2,3\qquad\textbf{(E)}\ 1,3,5 $

Show that for every integer $n\geq2$ there are two distinct powers of $n$ such that their sum is greater than $10^{2023}$ and their positive difference is divisible with $2023$.

Find all fractions which can be written simultaneously in the forms $\frac{7k- 5}{5k - 3}$ and $\frac{6l - 1}{4l - 3}$ , for some integers $k, l$.

Let $n$ be a positive integer. A [i]Japanese triangle[/i] consists of $1 + 2 + \dots + n$ circles arranged in an equilateral triangular shape such that for each $i = 1$, $2$, $\dots$, $n$, the $i^{th}$ row contains exactly $i$ circles, exactly one of which is coloured red. A [i]ninja path[/i] in a Japanese triangle is a sequence of $n$ circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with $n = 6$, along with a ninja path in that triangle containing two red circles. [asy] // credit to vEnhance for the diagram (which was better than my original asy): size(4cm); pair X = dir(240); pair Y = dir(0); path c = scale(0.5)*unitcircle; int[] t = {0,0,2,2,3,0}; for (int i=0; i<=5; ++i) { for (int j=0; j<=i; ++j) { filldraw(shift(i*X+j*Y)*c, (t[i]==j) ? lightred : white); draw(shift(i*X+j*Y)*c); } } draw((0,0)--(X+Y)--(2*X+Y)--(3*X+2*Y)--(4*X+2*Y)--(5*X+2*Y),linewidth(1.5)); path q = (3,-3sqrt(3))--(-3,-3sqrt(3)); draw(q,Arrows(TeXHead, 1)); label("$n = 6$", q, S); label("$n = 6$", q, S); [/asy] In terms of $n$, find the greatest $k$ such that in each Japanese triangle there is a ninja path containing at least $k$ red circles.

The sequence $\{a_n\}$ satisfies $a_0=1, a_1=2011,$ and $a_n=2a_{n-1}+a_{n-2}$ for all $n \geq 2$. Let \[ S = \sum_{i=1}^{\infty} \frac{a_{i-1}}{a_i^2-a_{i-1}^2} \] What is $\frac{1}{S}$? [i]Author: Ray Li[/i]

Let $n$ and $k$ be positive integers. Two infinite sequences $\{s_i\}_{i\geq 1}$ and $\{t_i\}_{i\geq 1}$ are [i]equivalent[/i] if, for all positive integers $i$ and $j$, $s_i = s_j$ if and only if $t_i = t_j$. A sequence $\{r_i\}_{i\geq 1}$ has [i]equi-period[/i] $k$ if $r_1, r_2, \ldots $ and $r_{k+1}, r_{k+2}, \ldots$ are equivalent. Suppose $M$ infinite sequences with equi-period $k$ whose terms are in the set $\{1, \ldots, n\}$ can be chosen such that no two chosen sequences are equivalent to each other. Determine the largest possible value of $M$ in terms of $n$ and $k$.

Solve the equation $(2015x -2014)^3 = 8(x-1)^3 + (2013x -2012)^3$

[b]p1.[/b] The final Big $10$ standings for the $1996$ Women's Softball season were 1. Michigan 2. Minnesota З. Iowa 4. Indiana 5. Michigan State 6. Purdue 7. Northwestern 8. Ohio State 9. Penn State 10. Wisconsin (Illinois does not participate in Women's Softball.) When you compare the $1996$ final standings (above) to the final standings for the $1999$ season, you find that the following pairs of teams changed order relative to each other from $1996$ to $1999$ (there are no ties, and no other pairs changed places): (Iowa, Michigan State) (Indiana, Penn State) (Purdue, Wisconsin) (Iowa, Penn State) (Indiana, Wisconsin) (Northwestern, Penn State) (Indiana, Michigan State) (Michigan State, Penn State) (Northwestern, Wisconsin) (Indiana, Purdue) (Purdue, Northwestern) (Ohio State, Penn State) (Indiana, Northwestern) (Purdue, Penn State) (Ohio State, Penn State) (Indiana, Ohio State) Determine as much as you can about the final Big $10$ standings for the $1999$ Women's Softball season. If you cannot determine the standings, explain why you do not have enough information. You must justify your answer. [b]p2.[/b] a) Take as a given that any expression of the form $A \sin t + B \cos t$ ($A>0$) can be put in the form $C \sin (t + D)$, where $C>0$ and $-\pi /2 <D <\pi /2 $. Determine $C$ and $D$ in terms of $A$ and $B$. b) For the values of $C$ and $D$ found in part a), prove that $A \sin t + B \cos t = C \sin (t + D)$. c) Find the maximum value of $3 \sin t +2 \cos t$. [b]pЗ.[/b] А $6$-bу-$6$ checkerboard is completelу filled with $18$ dominoes (blocks of size $1$-bу-$2$). Prove that some horizontal or vertical line cuts the board in two parts but does not cut anу of the dominoes. [b]p4.[/b] a) The midpoints of the sides of a regular hexagon are the vertices of a new hexagon. What is the ratio of the area of the new hexagon to the area of the original hexagon? Justify your answer and simplify as much as possible. b) The midpoints of the sides of a regular $n$-gon ($n >2$) are the vertices of a new $n$-gon. What is the ratio of the area of the new $n$-gon to that of the old? Justify your answer and simplify as much as possible. [b]p5. [/b] You run a boarding house that has $90$ rooms. You have $100$ guests registered, but on any given night only $90$ of these guests actually stay in the boarding house. Each evening a different random set of $90$ guests will show up. You don't know which $90$ it will be, but they all arrive for dinner before you have to assign rooms for the night. You want to give out keys to your guests so that for any set of $90$ guests, you can assign each to a private room without any switching of keys. a) You could give every guest a key to every room. But this requires $9000$ keys. Find a way to hand out fewer than $9000$ keys so that each guest will have a key to a private room. b) What is the smallest number of keys necessary so that each guest will have a key to a private room? Describe how you would distribute these keys and assign the rooms. Prove that this number of keys is as small as possible. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ S(n)$ be the sum of decimal digits of a natural number $ n$. Find the least value of $ S(m)$ if $ m$ is an integral multiple of $ 2003$.

Find the smallest positive integer $n$ such that $n^4 + (n+1)^4$ is composite.

[b]p7[/b] Cindy cuts regular hexagon $ABCDEF$ out of a sheet of paper. She folds $B$ over $AC$, resulting in a pentagon. Then, she folds $A$ over $CF$, resulting in a quadrilateral. The area of $ABCDEF$ is $k$ times the area of the resulting folded shape. Find $k$. [b]p8[/b] Call a sequence $\{a_n\} = a_1, a_2, a_3, . . .$ of positive integers [i]Fib-o’nacci[/i] if it satisfies $a_n = a_{n-1}+a_{n-2}$ for all $n \ge 3$. Suppose that $m$ is the largest even positive integer such that exactly one [i]Fib-o’nacci[/i] sequence satisfies $a_5 = m$, and suppose that $n$ is the largest odd positive integer such that exactly one [i]Fib-o’nacci[/i] sequence satisfies $a_5 = n$. Find $mn$. [b]p9[/b] Compute the number of ways there are to pick three non-empty subsets $A$, $B$, and $C$ of $\{1, 2, 3, 4, 5, 6\}$, such that $|A| = |B| = |C|$ and the following property holds: $$A \cap B \cap C = A \cap B = B \cap C = C \cap A.$$

Three points of a triangle are among 8 vertex of a cube. So the number of such acute triangles is $\text{(A)}0\qquad\text{(B)}6\qquad\text{(C)}8\qquad\text{(D)}24$

In a non equilateral triangle $ABC$ the sides $a,b,c$ form an arithmetic progression. Let $I$ be the incentre and $O$ the circumcentre of the triangle $ABC$. Prove that (1) $IO$ is perpendicular to $BI$; (2) If $BI$ meets $AC$ in $K$, and $D$, $E$ are the midpoints of $BC$, $BA$ respectively then $I$ is the circumcentre of triangle $DKE$.