How many three-digit positive integers contain both even and odd digits?

Consider a plane $\epsilon$ and three non-collinear points $A,B,C$ on the same side of $\epsilon$; suppose the plane determined by these three points is not parallel to $\epsilon$. In plane $\epsilon$ take three arbitrary points $A',B',C'$. Let $L,M,N$ be the midpoints of segments $AA', BB', CC'$; Let $G$ be the centroid of the triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L,M,N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\epsilon$?

For a sequence with some ones and zeros, we count the number of continuous runs of equal digits in it. (For example the sequence $011001010$ has $7$ continuous runs: $0,11,00,1,0,1,0$.) Find the sum of the number of all continuous runs for all possible sequences with $2019$ ones and $2019$ zeros.

Point $P$ lies inside triangle of sides of length $3, 4, 5$. Show that if distances between $P$ and vertices of triangle are rational numbers then distances from $P$ to sides of triangle are rational numbers too.

Unit square $ABCD$ is drawn on a plane. Point $O$ is drawn outside of $ABCD$ such that lines $AO$ and $BO$ are perpendicular. Square $F ROG$ is drawn with $F$ on $AB$ such that $AF =\frac23$, $R$ is on $\overline{BO}$, and $G$ is on $\overline{AO}$. Extend segment $\overline{OF}$ past $\overline{AB}$ to intersect side $\overline{CD}$ at $E$. Compute $DE$.

For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in a row. What is the remainder when $S(2015)$ is divided by $12$? $\textbf{(A) }0\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }10$

Let $(a_n)_{n \geq 0}$ and $(b_n)_{n \geq 0}$ be sequences of real numbers such that $ a_0>\frac{1}{2}$ , $a_{n+1} \geq a_n$ and $b_{n+1}=a_n(b_n+b_{n+2})$ for all non-negative integers $n$ . Show that the sequence $(b_n)_{n \geq 0}$ is bounded .

Let $A$ and $B$ different points of a circle $k$ centered at $O$ in such a way such that $AB$ is not a diagonal of $k$. Furthermore, let $X$ be an arbitrary inner point of the segment $AB$. Let $k_1$ be the circle that passes through the points $A$ and $X$, and $A$ is the only common point of $k$ and $k_1$. Similarly, let $k_2$ be the circle that passes through the points $B$ and $X$, and $B$ is the only common point of $k$ and $k_2$. Let $M$ be the second intersection point of $k_1$ and $k_2$. Let $Q$ denote the center of circumscribed circle of the triangle $AOB$. Let $O_1$ and $O_2$ be the centers of $k_1$ and $k_2$. Show that the points $M,O,O_1,O_2,Q$ are on a circle.

Let $A_1A_2A_3A_4A_5A_6$ be a convex hexagon such that $A_iA_{i+2} \parallel A_{i+3}A_{i+5}$ for $i = 1, 2, 3$ (we take $A_{i+6} = A_i$ for each $i$). Segment $A_iA_{i+2}$ intersects segment $A_{i+1}A_{i+3}$ at $B_i$, for $1 \le i \le 6$, as shown. Furthermore, suppose that $\vartriangle A_1A_3A_5 \cong \vartriangle A_4A_6A_2$. Given that $[A_1B_5B_6] = 1$, $[A_2B_6B_1] = 4$, and $[A_3B_1B_2] = 9$ (by $[XY Z]$ we mean the area of $ \vartriangle XY Z$), determine the area of hexagon $B_1B_2B_3B_4B_5B_6$. [img]https://cdn.artofproblemsolving.com/attachments/d/0/1a8997c9eb7dea5223b6805dacd79c10a2cd33.png[/img]

Squares of an $8 \times 8$ chessboard are each allocated a number between $1$ and $32$ , with each number being used twice. Prove that it is possible to choose $32$ such squares, each allocated a different number, so that there is at least one such square on each row or column . (A . Andjans, Riga

If we add $1996$ to $1997$, we first add the unit digits $6$ and $7$. Obtaining $13$, we write down $3$ and “carry” $1$ to the next column. Thus we make a carry. Continuing, we see that we are to make three carries in total. Does there exist a positive integer $k$ such that adding $1996\cdot k$ to $1997\cdot k$ no carry arises during the whole calculation?

Given a regular $2013$-sided polygon, how many isosceles triangles are there whose vertices are vertices vertex of given polygon and haave an angle greater than $120^o$? Nguyen Tien Lam, High School for Natural Science,Hanoi National University.

The discriminant of the equation $ x^2\plus{}2x\sqrt{3}\plus{}3\equal{}0$ is zero. Hence, its roots are: $ \textbf{(A)}\ \text{real and equal} \qquad \textbf{(B)}\ \text{rational and equal} \qquad \textbf{(C)}\ \text{rational and unequal} \\ \textbf{(D)}\ \text{irrational and unequal} \qquad \textbf{(E)}\ \text{imaginary}$

For a finite non empty set of primes $P$, let $m(P)$ denote the largest possible number of consecutive positive integers, each of which is divisible by at least one member of $P$. (i) Show that $|P|\le m(P)$, with equality if and only if $\min(P)>|P|$. (ii) Show that $m(P)<(|P|+1)(2^{|P|}-1)$. (The number $|P|$ is the size of set $P$) [i]Dan Schwarz, Romania[/i]

$x,y,z,a,b,c$ are positive integers that satisfy $xy \equiv a \pmod z$, $yz \equiv b \pmod x$, $zx \equiv c \pmod y$. Prove that $\min{\{x,y,z\}} \le ab+bc+ca$.

Let two glasses, numbered $1$ and $2$, contain an equal quantity of liquid, milk in glass $1$ and coffee in glass $2$. One does the following: Take one spoon of mixture from glass $1$ and pour it into glass $2$, and then take the same spoon of the new mixture from glass $2$ and pour it back into the first glass. What happens after this operation is repeated $n$ times, and what as $n$ tends to infinity?

A positive integer is called [i]oneic[/i] if it consists of only $1$'s. For example, the smallest three oneic numbers are $1$, $11$, and $111$. Find the number of $1$'s in the smallest oneic number that is divisible by $63$.

Find all positive integers $n$ such that number $n^4-4n^3+22n^2-36n+18$ is perfect square of positive integer

Let $ABC$ be a triangle, let $B_{1}$ be the midpoint of the side $AB$ and $C_{1}$ the midpoint of the side $AC$. Let $P$ be the point of intersection, other than $A$, of the circumscribed circles around the triangles $ABC_{1}$ and $AB_{1}C$. Let $P_{1}$ be the point of intersection, other than $A$, of the line $AP$ with the circumscribed circle around the triangle $AB_{1}C_{1}$. Prove that $2AP=3AP_{1}$.

The tangents of the circumcircle $\Omega$ of the triangle $ABC$ at points $B$ and $C$ intersect at point $P$. The perpendiculars drawn from point $P$ to lines $AB$ and $AC$ intersect at points$ D$ and $E$ respectively. Prove that the altitudes of the triangle $ADE$ intersect at the midpoint of the segment $BC$.

There are positive integers that have these properties: * the sum of the squares of their digits is 50, and * each digit is larger than the one to its left. The product of the digits of the largest integer with both properties is $\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60$

A positive integer $n$ is called an untouchable number if there is no positive integer $m$ for which the sum of the factors of $m$ (including $m$ itself) is $n + m$. Find the sum of all of the untouchable numbers between $1$ and $10$ (inclusive)

Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took $ h$ hours and $ m$ minutes, with $ 0<m<60$, what is $ h\plus{}m$? $ \textbf{(A)}\ 46 \qquad \textbf{(B)}\ 47 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 53 \qquad \textbf{(E)}\ 54$

p1. Given the set $H = \{(x, y)|(x -y)^2 + x^2 - 15x + 50 = 0$ where x and y are natural numbers $\}$. Find the number of subsets of $H$. p2. A magician claims to be an expert at guessing minds with following show. One of the viewers was initially asked to hidden write a five-digit number, then subtract it with the sum of the digits that make up the number, then name four of the five digits that make up the resulting number (in order of any). Then the magician can guess the numbers hidden. For example, if the audience mentions four numbers result: $0, 1, 2, 3$, then the magician will know that the hidden number is $3$. a. Give an example of your own from the above process. b. Explain mathematically the general form of the process. p3. In a fruit basket there are $20$ apples, $18$ oranges, $16$ mangoes, $10$ pineapples and $6$ papayas. If someone wants to take $10$ pieces from the basket. After that, how many possible compositions of fruit are drawn? p4. Inside the Equator Park, a pyramid-shaped building will be made with base of an equilateral triangle made of translucent material with a side length of the base $8\sqrt3$ m long and $8$ m high. A globe will be placed in a pyramid the. Ignoring the thickness of the pyramidal material, determine the greatest possible length of the radius of the globe that can be made. p5. What is the remainder of $2012^{2012} + 2014^{2012}$ divided by $2013^2$?

Given a circle with center $O$, such that $A$ is not on the circumcircle. Let $B$ be the reflection of $A$ with respect to $O$. Now let $P$ be a point on the circumcircle. The line perpendicular to $AP$ through $P$ intersects the circle at $Q$. Prove that $AP \times BQ$ remains constant as $P$ varies.