Found problems: 85335
Moist air is less dense than dry air at the same temperature and barometric pressure. Which is the best explanation for this observation?
$ \textbf{(A)}\hspace{.05in}\ce{H2O} \text{ is a polar molecule but } \ce{N2} \text{ and } \ce{O2} \text{ are not} \qquad$
$\textbf{(B)}\hspace{.05in} \ce{H2O} \text{has a higher boiling point than } \ce{N2} \text{or} \ce{O2}\qquad$
$\textbf{(C)}\hspace{.05in}\ce{H2O} \text{has a lower molar mass than} \ce{N2} \text{or} \ce{O2}\qquad$
$\textbf{(D)}\hspace{.05in}\ce{H2O} \text{has a higher heat capacity than} \ce{N2} \text{or} \ce{O2}\qquad$
Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?
Alice and Bob play the following game. They alternate selecting distinct nonzero digits (from $1$ to $9$) until they have chosen seven such digits, and then consider the resulting seven-digit number by concatenating the digits in the order selected, with the seventh digit appearing last (i.e. $\overline{A_1B_2A_3B_4A_6B_6A_7}$). Alice wins if and only if the resulting number is the last seven decimal digits of some perfect seventh power. Please determine which player has the winning strategy.
$300$ couples (one man, one woman) are invited to a party. Everyone at the party either always tells the truth or always lies. Exactly $2/3$ of the men say their partner always tells the truth and the remaining $1/3$ say their partner always lies. Exactly $2/3$ of the women say their partner is the same type as themselves and the remaining $1/3$ say their partner is different. Find $a$, the maximum possible number of people who tell the truth, and $b$, the minimum possible number of people who tell the truth. Express your answer as $(a,b)$.
Inside a right circular cone with base radius $5$ and height $12$ are three congruent spheres each with radius $r$. Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. What is $r?$
$\textbf{(A) }\dfrac32\qquad\textbf{(B) }\dfrac{90 - 40\sqrt3}{11}\qquad\textbf{(C) }2\qquad\textbf{(D) }\dfrac{144 - 25\sqrt3}{44}\qquad\textbf{(E) }\dfrac52$
Let $ABC$ be an arbitrary triangle and $O$ is the circumcenter of $\triangle {ABC}$.Points $X,Y$ lie on $AB,AC$,respectively such that the reflection of $BC$ WRT $XY$ is tangent to circumcircle of $\triangle {AXY}$.Prove that the circumcircle of triangle $AXY$ is tangent to circumcircle of triangle $BOC$.
Let be two real numbers $ x,y, $ and a natural number $ n_0 $ such that $ \{ n_0x \} = \{ n_0y \} $ and $ \{ (n_0+1)x \} = \{ (n_0+1)y \} ,$ where $ \{\} $ denotes the fractional part. Show that $ \{ nx \} =\{ ny \} , $ for any natural number $ n. $
[i]Ovidiu Pop[/i]
Find two positive integers $a$ and $b$, when their sum and their least common multiple is given. Find the numbers when the sum is $3972$ and the least common multiple is $985928$.
Let $x,y$ be real numbers such that $(x+1)(y+2)=8.$ Prove that $$(xy-10)^2\ge 64.$$
Let \( ABC \) be an acute scalene triangle with orthocenter \( H \), and consider \( M \) to be the midpoint of side \( BC \). Define \( P \neq A \) as the intersection point of the circle with diameter \( AH \) and the circumcircle of triangle \( ABC \), and let \( Q \) be the intersection of \( AP \) with \( BC \). Let \( G \neq M \) be the intersection of the circumcircle of triangle \( MPQ \) with the circumcircle of triangle \( AHM \). Show that \( G \) lies on the circle that passes through the feet of the altitudes of triangle \( ABC \).
Some of $A,B,C,D,$ and $E$ are truth tellers, and the others are liars. Truth tellers always tell the truth. Liars always lie. We know $A$ is a truth teller. According to below conversation,
$B: $ I'm a truth teller.
$C: $ $D$ is a truth teller.
$D: $ $B$ and $E$ are not both truth tellers.
$E: $ $A$ and $B$ are truth tellers.
How many truth tellers are there?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ \text{More information is needed}
$
Given positive integers $a_1,a_2,\ldots, a_n$ with $a_1<a_2<\cdots<a_n)$, and a positive real $k$ with $k\geq 1$. Prove that
\[\sum_{i=1}^{n}a_i^{2k+1}\geq \left(\sum_{i=1}^{n}a_i^k\right)^2.\]
The roots of the equation $x^2+5x-7=0$ are $x_1$ and $x_2$. What is the value of $x_1^3+5x_1^2-4x_1+x_1^2x_2-4x_2$ ?
$\textbf{(A)}\ -15 \qquad\textbf{(B)}\ 175+25\sqrt{53} \qquad\textbf{(C)}\ -50 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ \text{None}$
Let $n$ be a positive integer. There are $n$ islands with $n-1$ bridges connecting them such that one can travel from any island to another. One afternoon, a fire breaks out in one of the islands. Every morning, it spreads to all neighbouring islands. (Two islands are neighbours if they are connected by a bridge.) To control the spread, one bridge is destroyed every night until the fire has nowhere to spread the next day. Let $X$ be the minimum possible number of bridges one has to destroy before the fire stops spreading. Find the maximum possible value of $X$ over all possible configurations of bridges and island where the fire starts at.
Evaluate $\int_0^1 \frac{x^2+x+1}{x^4+x^3+x^2+x+1}\ dx.$
What is the minimum number of cells that can be colored black in white square $ 300 \times 300 $ so that no three black cells formed a corner, and after painting any white cell this condition violated?
Find the smallest exact square with last digit not $0$, such that after deleting its last two digits we shall obtain another exact square.
In a school with $101$ students, each student has at least one friend among the other students. Show that for every integer $1<n<101$, a group of $n$ students can be selected from this school in such a way that each selected student has at least one friend among the other selected students.
Find all real solutions of the system of equations
$$x^2-y^2=2(xz+yz+x+y),$$$$y^2-z^2=2(yx+zx+y+z),$$$$z^2-x^2=2(zy+xy+z+x).$$
Find all $n$ natural numbers such that for each of them there exist $p , q$ primes such that these terms satisfy.
$1.$ $p+2=q$
$2.$ $2^n+p$ and $2^n+q$ are primes.
For each positive integer $n$, let $s(n)$ be the sum of the squares of the digits of $n$. For example, $s(15)=1^2+5^2=26$. Determine all integers $n\geq 1$ such that $s(n)=n$.
How many real solutions are there to the equation $x = 1964 \sin x - 189$ ?
On each horizontal line in the figure below, the five large dots indicate the populations of cities $A$, $B$, $C$, $D$ and $E$ in the year indicated. Which city had the greatest percentage increase in population from 1970 to 1980?
[asy]
size(300);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=(5,0), B=(7,0), C=(10,0), D=(13,0), E=(16,0);
pair F=(4,3), G=(5,3), H=(7,3), I=(10,3), J=(12,3);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(H);
dot(I);
dot(J);
draw((0,0)--(18,0)^^(0,3)--(18,3));
draw((0,0)--(0,.5)^^(5,0)--(5,.5)^^(10,0)--(10,.5)^^(15,0)--(15,.5));
draw((0,3)--(0,2.5)^^(5,3)--(5,2.5)^^(10,3)--(10,2.5)^^(15,3)--(15,2.5));
draw((1,0)--(1,.2)^^(2,0)--(2,.2)^^(3,0)--(3,.2)^^(4,0)--(4,.2)^^(6,0)--(6,.2)^^(7,0)--(7,.2)^^(8,0)--(8,.2)^^(9,0)--(9,.2)^^(10,0)--(10,.2)^^(11,0)--(11,.2)^^(12,0)--(12,.2)^^(13,0)--(13,.2)^^(14,0)--(14,.2)^^(16,0)--(16,.2)^^(17,0)--(17,.2)^^(18,0)--(18,.2));
draw((1,3)--(1,2.8)^^(2,3)--(2,2.8)^^(3,3)--(3,2.8)^^(4,3)--(4,2.8)^^(6,3)--(6,2.8)^^(7,3)--(7,2.8)^^(8,3)--(8,2.8)^^(9,3)--(9,2.8)^^(10,3)--(10,2.8)^^(11,3)--(11,2.8)^^(12,3)--(12,2.8)^^(13,3)--(13,2.8)^^(14,3)--(14,2.8)^^(16,3)--(16,2.8)^^(17,3)--(17,2.8)^^(18,3)--(18,2.8));
label("A", A, S);
label("B", B, S);
label("C", C, S);
label("D", D, S);
label("E", E, S);
label("A", F, N);
label("B", G, N);
label("C", H, N);
label("D", I, N);
label("E", J, N);
label("1970", (0,3), W);
label("1980", (0,0), W);
label("0", (0,1.5));
label("50", (5,1.5));
label("100", (10,1.5));
label("150", (15,1.5));
label("Population", (21,2));
label("in thousands", (21.4,1));[/asy]
$ \textbf{(A)}\ A\qquad\textbf{(B)}\ B\qquad\textbf{(C)}\ C\qquad\textbf{(D)}\ D\qquad\textbf{(E)}\ E $
Given the real number $k$, find all differentiable real-valued functions $f(x)$ defined on the reals such that $f(x+y) = f(x) + f(y) + f(kxy)$ for all $x, y$.
Let $a_1$, $a_2$, $a_3$, $a_4$, and $a_5$ be random integers chosen independently and uniformly from the set $\{ 0, 1, 2, \dots, 23 \}$. (Note that the integers are not necessarily distinct.) Find the probability that
\[
\sum_{k=1}^{5} \operatorname{cis} \Bigl( \frac{a_k \pi}{12} \Bigr) = 0.
\]
(Here $\operatorname{cis} \theta$ means $\cos \theta + i \sin \theta$.)