A natural number $n$ is said to be $good$ if $n$ is the sum or $r$ consecutive positive integers, for some $r \geq 2 $. Find the number of good numbers in the set $\{1,2 \dots , 100\}$.

25. Chris and Paul each rent a different room of a hotel from rooms $1 - 60$. However, the hotel manager mistakes them for one person and gives ”Chris Paul” a room with Chris’s and Paul’s room concatenated. For example, if Chris had $15$ and Paul had $9$, ”Chris Paul” has $159$. If there are $360$ rooms in the hotel, what is the probability that ”Chris Paul” has a valid room? 26. Find the number of ways to choose two nonempty subsets $X$ and $Y$ of $\{1, 2, \ldots , 2001\}$, such that $|Y| = 1001$ and the smallest element of $Y$ is equal to the largest element of $X$. 27. Let $r_1, r_2, r_3, r_4$ be the four roots of the polynomial $x^4 - 4x^3 + 8x^2 - 7x + 3$. Find the value of $$\frac{r_1^2}{r_2^2+r_3^2+r_4^2}+\frac{r_2^2}{r_1^2+r_3^2+r_4^2}+\frac{r_3^2}{r_1^2+r_2^2+r_4^2}+\frac{r_4^2}{r_1^2+r_2^2+r_3^2}$$

Let $ABCD$ be a rectangle with $AB=10$ and $AD=5.$ Suppose points $P$ and $Q$ are on segments $CD$ and $BC,$ respectively, such that the following conditions hold: [list] [*] $BD \parallel PQ$ [*] $\angle APQ=90^{\circ}.$ [/list] What is the area of $\triangle CPQ?$ [i]Proposed by Kyle Lee[/i]

Lennart and Eddy are playing a betting game. Lennart starts with $7$ dollars and Eddy starts with $3$ dollars. Each round, both Lennart and Eddy bet an amount equal to the amount of the player with the least money. For example, on the first round, both players bet $3$ dollars. A fair coin is then tossed. If it lands heads, Lennart wins all the money bet; if it lands tails, Eddy wins all the money bet. They continue playing this game until one person has no money. What is the probability that Eddy ends with $10$ dollars?

The positive integers are partitioned into 2 sequences $a_1<a_2<\dots$ and $b_1<b_2<\dots$ such that $b_n=a_n+n$ for every positive integer $n$. Show that $a_n+b_n=a_{b_n}$.

Given a convex polygon M invariant under a $90^\circ$ rotation, show that there exist two circles, the ratio of whose radii is $\sqrt2$, one containing M and the other contained in M. [i]A. Khrabrov[/i]

[b]p1.[/b] Solve for $x$ and $y$ if $\frac{1}{x^2}+\frac{1}{xy}=\frac{1}{9}$ and $\frac{1}{y^2}+\frac{1}{xy}=\frac{1}{16}$ [b]p2.[/b] Find positive integers $p$ and $q$, with $q$ as small as possible, such that $\frac{7}{10} <\frac{p}{q} <\frac{11}{15}$. [b]p3.[/b] Define $a_1 = 2$ and $a_{n+1} = a^2_n -a_n + 1$ for all positive integers $n$. If $i > j$, prove that $a_i$ and $a_j$ have no common prime factor. [b]p4.[/b] A number of points are given in the interior of a triangle. Connect these points, as well as the vertices of the triangle, by segments that do not cross each other until the interior is subdivided into smaller disjoint regions that are all triangles. It is required that each of the givien points is always a vertex of any triangle containing it. Prove that the number of these smaller triangular regions is always odd. [b]p5.[/b] In triangle $ABC$, let $\angle ABC=\angle ACB=40^o$ is extended to $D$ such that $AD=BC$. Prove that $\angle BCD=10^o$. [img]https://cdn.artofproblemsolving.com/attachments/6/c/8abfbf0dc38b76f017b12fa3ec040849e7b2cd.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a$ and $d$ be two positive integers. Prove that there exists a constant $K$ such that every set of $K$ consecutive elements of the arithmetic progression $\{a+nd\}_{n=1}^\infty$ contains at least one number which is not prime.

Equilateral triangle $ABC$ has circumcircle $\Omega$. Points $D$ and $E$ are chosen on minor arcs $AB$ and $AC$ of $\Omega$ respectively such that $BC=DE$. Given that triangle $ABE$ has area $3$ and triangle $ACD$ has area $4$, find the area of triangle $ABC$.

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

Two points $J$ and $H$ lie $26$ units apart on a given plane. Let $M$ be the locus of points $T$ on this plane such that $JT^2 + HT^2 = 2020$. Then, M encloses a region on the plane with area $a$ and perimeter $p$. If $q$ and $r$ are coprime positive integers and $\frac{a}{p} = \frac{q}{r}$ , then compute $q + r$.

Evaluate the sum \[1^2+2^2-3^2-4^2+5^2+6^2-7^2-8^2+\cdots-1000^2+1001^2\]

Prove that for any $a>0,b>0,c>0$ we have $8a^2 b^2 c^2 \geq (a^2 + ab + ac - bc)(b^2 + ba + bc - ac)(c^2 + ca + cb - ab)$.

Sarah pours four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then transfers half the coffee from the first cup to the second and, after stirring thoroughly, transfers half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream? $ \textbf{(A)}\ 1/4 \qquad \textbf{(B)}\ 1/3 \qquad \textbf{(C)}\ 3/8 \qquad \textbf{(D)}\ 2/5 \qquad \textbf{(E)}\ 1/2$

Let $n$ and $k$ be fixed positive integers , and $a$ be arbitrary nonnegative integer . Choose a random $k$-element subset $X$ of $\{1,2,...,k+a\}$ uniformly (i.e., all k-element subsets are chosen with the same probability) and, independently of $X$, choose random n-elements subset $Y$ of $\{1,2,..,k+a+n\}$ uniformly. Prove that the probability $P\left( \text{min}(Y)>\text{max}(X)\right)$ does not depend on $a$.

After the final exam, Mr. Liang asked each of his 17 students to guess the average final exam score. David, a very smart student, received a 100 and guessed the average would be 97. Each of the other 16 students guessed $30+\frac{n}{2}$ where $n$ was that student’s score. If the average of the final exam scores was the same as the average of the guesses, what was the average score on the final exam? [i]Proposed by Eric Wang[/i]

Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let $p$ be the probability that the object reaches $(2,2)$ in six or fewer steps. Given that $p$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

In a convex quadrilateral $ABCD$ rays $AB$ and $DC$ intersect at point $P$, and rays $BC$ and $AD$ at point $Q$. There is a point $T$ on the diagonal $AC$ such that the triangles $BTP$ and $DTQ$ are similar, in that order. Prove that $BD \Vert PQ$.

Define a sequence by $a_1=a_2=1, a_3=2,$ and $$a_n+a_{n-3}=a_{n-1}+a_{n-2}$$ for all $n>3.$ What is the value of $a_7?$

Consider the cube $ABCDA'B'C'D'$. The bisectors of the angles $\angle A' C'A$ and $\angle A' AC'$ intersect $AA'$ and $A'C$ in the points $P$, respectively $S$. The point $M$ is the foot of the perpendicular from $A'$ on $CP$ , and $N$ is the foot of the perpendicular from $A'$ to $AS$. Point $O$ is the center of the face $ABB'A'$ a) Prove that the planes $(MNO)$ and $(AC'B)$ are parallel. b) Calculate the distance between these planes, knowing that $AB = 1$.

Find the least real number $k$ with the following property: if the real numbers $x$, $y$, and $z$ are not all positive, then \[k(x^{2}-x+1)(y^{2}-y+1)(z^{2}-z+1)\geq (xyz)^{2}-xyz+1.\]

Find the smallest real number $M$ with the following property: Given nine nonnegative real numbers with sum $1$, it is possible to arrange them in the cells of a $3 \times 3$ square so that the product of each row or column is at most $M$. [i]Evan O' Dorney.[/i]

Let $AB$ and $AC$ be two distinct rays not lying on the same line, and let $\omega$ be a circle with center $O$ that is tangent to ray $AC$ at $E$ and ray $AB$ at $F$. Let $R$ be a point on segment $EF$. The line through $O$ parallel to $EF$ intersects line $AB$ at $P$. Let $N$ be the intersection of lines $PR$ and $AC$, and let $M$ be the intersection of line $AB$ and the line through $R$ parallel to $AC$. Prove that line $MN$ is tangent to $\omega$. [i]Warut Suksompong, Thailand[/i]

In the trapezoid $ABCD, AB = BC = CD, CH$ is the altitude. Prove that the perpendicular from $H$ on $AC$ passes through the midpoint of $BD$.

In triangle $ABC$, angles $B,C$ are acute. Point $D$ is on the side $BC$ such that $AD\perp{BC}$. Let the interior bisectors of $\angle B,\angle C$ meet $AD$ at $E,F$, respectively. If $BE=CF$, prove that $ABC$ is isosceles.