Found problems: 85335
Let $S$ be the sum of the interior angles of a polygon $P$ for which each interior angle is $7\frac{1}{2}$ times the exterior angle at the same vertex. Then
$ \textbf{(A)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may be regular}\qquad$
$\textbf{(B)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad$
$\textbf{(C)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is regular}\qquad$
$\textbf{(D)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad$
$\textbf{(E)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may or may not be regular} $
Let $r$ be a positive integer. Show that if a graph $G$ has no cycles of length at most $2r$, then it has at most $|V|^{2016}$ cycles of length exactly $2016r$, where $|V|$ denotes the number of vertices in the graph $G$.
Prove that, for all convex pentagons $P_1 P_2 P_3 P_4 P_5$ with area 1, there are indices $i$ and $j$ (assume $P_7 = P_2$ and $P_6 = P_1$) such that:
\[ \text{Area of} \ \triangle P_i P_{i+1} P_{i+2} \le \frac{5 - \sqrt 5}{10} \le \text{Area of} \ \triangle P_j P_{j+1} P_{j+2}\]
Every day you go to the music practice rooms at a random time between $12\text{AM}$ and $8\text{AM}$ and practice for $3$ hours, while your friend goes at a random time from $5\text{AM}$ to $11\text{AM}$ and practices for $1$ hour (the block of practice time need not be contained in he given time range for the arrival). What is the probability that you and your meet on at least $2$ days in a given span of $5$ days?
Show that, if the real numbers $a, b, c, A, B, C$ satisfy
$$aC -2bB + cA = 0 \ \ and \ \ ac - b^2 > 0,$$
then $$AC - B^2 \le 0.$$
Let $N = \{1, 2, \ldots, n\}$, $n \geq 3$. To each pair $i \neq j $ of elements of $N$ there is assigned a number $f_{ij} \in \{0, 1\}$ such that $f_{ij} + f_{ji} = 1$.
Let $r(i)=\sum_{i \neq j} f_{ij}$, and write $M = \max_{i\in N} r(i)$, $m = \min_{i\in N} r(i)$. Prove that for any $w \in N$ with $r(w) = m$ there exist $u, v \in N$ such that $r(u) = M$ and $f_{uv}f_{vw} = 1$.
Let $ n\ge12$ be an integer, and let $ P_1,P_2,...P_n, Q$ be distinct points in a plane. Prove that for some $ i$, at least $ \frac{n}{6}\minus{}1$ of the distances $ P_1P_i,P_2P_i,...P_{i\minus{}1}P_i,P_{i\plus{}1}P_i,...P_nP_i$ are less than $ P_iQ$.
a) Denote by $S(n)$ the sum of digits of a positive integer $n$. After the decimal point, we write one after the other the numbers $S(1),S(2),...$. Show that the number obtained is irrational.
b) Denote by $P(n)$ the product of digits of a positive integer $n$. After the decimal point, we write one after the other the numbers $P(1),P(2),...$. Show that the number obtained is irrational.
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$. Let $I$ be the center of $\omega$, and let $IA=12,$ $IB=16,$ $IC=14,$ and $ID=11$. Let $M$ be the midpoint of segment $AC$. Compute the ratio $\frac{IM}{IN}$, where $N$ is the midpoint of segment $BD$.
Susan had $\$50$ to spend at the carnival. She spent $\$12$ on food and twice as much on rides. How many dollars did she have left to spend?
$\textbf{(A)}\ 12 \qquad
\textbf{(B)}\ 14 \qquad
\textbf{(C)}\ 26 \qquad
\textbf{(D)}\ 38 \qquad
\textbf{(E)}\ 50 $
Let $a$, $b$ and $c$ be real numbers such that $a^2+b^2=c^2$, solve the system:
\[ z^2=x^2+y^2 \]
\[ (z+c)^2=(x+a)^2+(y+b)^2 \]
in real numbers $x, y$ and $z$.
Let $AB$ be a chord of the circle $(O)$. Denote M as the midpoint of the minor arc $AB$. A circle $(O')$ tangent to segment $AB$ and internally tangent to $(O)$. A line passes through $M$, perpendicular to $O'A$, $O'B$ and cuts $AB$ respectively at $C, D$. Prove that $AB = 2CD$.
Ben has $16$ balls labeled $1, 2, 3, \ldots, 16,$ as well as $4$ indistinguishable boxes. Two balls are [i]neighbors[/i] if their labels differ by $1.$ Compute the number of ways for him to put $4$ balls in each box such that each ball is in the same box as at least one of its neighbors. (The order in which the balls are placed does not matter.)
For each positive integer $n$ (written with no leading zeros), let $t(n)$ equal the number formed by reversing the digits of $n$. For example, $t(461) = 164$ and $t(560) = 65$. For how many three-digit positive integers $m$ is $m + t(t(m))$ odd?
[i]Proposed by David Altizio[/i]
Let $p\ge 3$ be a prime number. Show that there exist $p$ positive integers $a_1,a_2,\ldots ,a_p$ not exceeding $2p^2$ such that the $\frac{p(p-1)}{2}$ sums $a_i+a_j\ (i<j)$ are all distinct.
Circles $\Omega $ and $\omega $ are tangent at a point $P$ ($\omega $ lies inside $\Omega $). A chord $AB$ of $\Omega $ is tangent to $\omega $ at $C;$ the line $PC$ meets $\Omega $ again at $Q.$ Chords $QR$ and $QS$ of $ \Omega $ are tangent to $\omega .$ Let $I,X,$ and $Y$ be the incenters of the triangles $APB,$ $ARB,$ and $ASB,$ respectively. Prove that $\angle PXI+\angle PYI=90^{\circ }.$
Prove that $(1 - x)^n + (1 + x)^n < 2^n$ for an integer $n \ge 2$ and $|x| < 1$.
Let $A_1B_1C_1D_1$ be an arbitrary convex quadrilateral. $P$ is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point $P$ is acute. We recursively define points $A_k,B_k,C_k,D_k$ symmetric to $P$ with respect to lines $A_{k-1}B_{k-1}, B_{k-1}C_{k-1}, C_{k-1}D_{k-1},D_{k-1}A_{k-1}$ respectively for $k\ge 2$.
Consider the sequence of quadrilaterals $A_iB_iC_iD_i$.
i) Among the first 12 quadrilaterals, which are similar to the 1997th quadrilateral and which are not?
ii) Suppose the 1997th quadrilateral is cyclic. Among the first 12 quadrilaterals, which are cyclic and which are not?
On a circle we consider a set $M$ consisting of $n$ ($n \ge 3$) points, of which only one is colored red. Determine of which polygons inscribed in a circle having the vertices in the set $M$ are more: those that contain the red dot or those that do not contain those points? How many more are there than others?
Let $v(n)$ be the order of $2$ in $n!$. Prove that for any positive integers $a$ and $m$ there exists $n$ ($n>1$) such that $v(n) \equiv a (\mod m)$.
I have a book with Mongolian problems from this year, and this problem appeared in it. Perhaps I am terribly misinterpreting this problem, but it seems like it is wrong. Any ideas?
Every time these two wheels are spun, two numbers are selected by the pointers. What is the probability that the sum of the two selected numbers is even?
$\text{(A)}\ \frac{1}{6} \qquad \text{(B)}\ \frac{3}{7} \qquad \text{(C)}\ \frac{1}{2} \qquad \text{(D)}\ \frac{2}{3} \qquad \text{(E)}\ \frac{5}{7}$
[asy]
unitsize(36);
draw(circle((-3,0),1));
draw(circle((0,0),1));
draw((0,0)--dir(30)); draw((0,0)--(0,-1)); draw((0,0)--dir(150));
draw((-2.293,.707)--(-3.707,-.707)); draw((-2.293,-.707)--(-3.707,.707));
fill((-2.9,1)--(-2.65,1.25)--(-2.65,1.6)--(-3.35,1.6)--(-3.35,1.25)--(-3.1,1)--cycle,black);
fill((.1,1)--(.35,1.25)--(.35,1.6)--(-.35,1.6)--(-.35,1.25)--(-.1,1)--cycle,black);
label("$5$",(-3,.2),N); label("$3$",(-3.2,0),W); label("$4$",(-3,-.2),S); label("$8$",(-2.8,0),E);
label("$6$",(0,.2),N); label("$9$",(-.2,.1),SW); label("$7$",(.2,.1),SE);
[/asy]
Eight points are spaced around at intervals of one unit around a $2 \times 2$ square, as shown. Two of the $8$ points are chosen at random. What is the probability that the two points are one unit apart?
[asy]
size((50));
dot((5,0));
dot((5,5));
dot((0,5));
dot((-5,5));
dot((-5,0));
dot((-5,-5));
dot((0,-5));
dot((5,-5));
[/asy]
$ \textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{2}{7}\qquad\textbf{(C)}\ \frac{4}{11}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\textbf{(E)}\ \frac{4}{7} $
Find the value of $$\sum_{1\le a<b<c} \frac{1}{2^a3^b5^c}$$ (i.e. the sum of $\frac{1}{2^a3^b5^c}$ over all triples of positive integers $(a, b, c)$ satisfying $a<b<c$)
Let $S$ be a nonempty closed bounded convex set in the plane. Let $K$ be a line and $t$ a positive number. Let $L_1$ and $L_2$ be support lines for $S$ parallel to $K_1$, and let $ \overline {L} $ be the line parallel to $K$ and midway between $L_1$ and $L_2$. Let $B_S(K,t)$ be the band of points whose distance from $\overline{L}$ is at most $ \left( \frac {t}{2} \right) w $, where $w$ is the distance between $L_1$ and $L_2$. What is the smallest $t$ such that \[ S \cap \bigcap_K B_S (K, t) \ne \emptyset \]for all $S$? ($K$ runs over all lines in the plane.)
In a geometric progression whose terms are positive, any term is equal to the sum of the next two following terms. then the common ratio is:
$ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ \text{about }\frac{\sqrt{5}}{2} \qquad\textbf{(C)}\ \frac{\sqrt{5}\minus{}1}{2} \qquad\textbf{(D)}\ \frac{1\minus{}\sqrt{5}}{2} \qquad\textbf{(E)}\ \frac{2}{\sqrt{5}}$