Found problems: 85335
Danica drove her new car on a trip for a whole number of hours, averaging $55$ miles per hour. At the beginning of the trip, $abc$ miles were displayed on the odometer, where $abc$ is a 3-digit number with $a \ge 1$ and $a+b+c \le 7$. At the end of the trip, where the odometer showed $cba$ miles. What is $a^2+b^2+c^2$?
$ \textbf{(A) } 26 \qquad\textbf{(B) }27\qquad\textbf{(C) }36\qquad\textbf{(D) }37\qquad\textbf{(E) }41\qquad $
Prove that for $ x,y\in\mathbb{R^ \plus{} }$,
$ \frac {1}{(1 \plus{} \sqrt {x})^{2}} \plus{} \frac {1}{(1 \plus{} \sqrt {y})^{2}} \ge \frac {2}{x \plus{} y \plus{} 2}$
Let $U$ be the set of all complex numbers $m$ such that the $4$ roots of $(x^2+2x+5)(x^2-2mx+25)=0$ are concyclic in the complex plane. One can show that when the points of $U$ are plotted on the complex plane, it is visualized as the finite union of some curves. Find the sum of the lengths of those curves (i.e. the perimeter of $U$).
Let $a_1$ be a natural number not divisible by $5$. The sequence $a_1,a_2,a_3, . . .$ is defined by $a_{n+1} =a_n+b_n$, where $b_n$ is the last digit of $a_n$. Prove that the sequence contains infinitely many powers of two.
(N. Agakhanov)
The equation $166\times 56 = 8590$ is valid in some base $b \ge 10$ (that is, $1, 6, 5, 8, 9, 0$ are digits in base $b$ in the above equation). Find the sum of all possible values of $b \ge 10$ satisfying the equation.
Let $f(x)$ be a function defined in $0<x<\frac{\pi}{2}$ satisfying:
(i) $f\left(\frac{\pi}{6}\right)=0$
(ii) $f'(x)\tan x=\int_{\frac{\pi}{6}}^x \frac{2\cos t}{\sin t}dt$.
Find $f(x)$.
[i]1987 Sapporo Medical University entrance exam[/i]
For all $ |x| \ge n$, the inequality $ |x^3 \plus{} 3x^2 \minus{} 33x \minus{} 3| \ge 2x^2$ holds. Integer $ n$ can be at least ?
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 5$
Each day, Jenny ate $ 20\%$ of the jellybeans that were in her jar at the beginning of the day. At the end of the second day, $ 32$ remained. How many jellybeans were in the jar originally?
$ \textbf{(A)}\ 40\qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 55 \qquad \textbf{(D)}\ 60\qquad \textbf{(E)}\ 75$
Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
$ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 180\qquad \textbf{(E)}\ 231$
Find all pairs $(a, b)$ of natural numbers such that $$\frac{a^3 + 1}{2ab^2 + 1}$$ is an integer.
We're given two congruent, equilateral triangles $ABC$ and $PQR$ with parallel sides, but one has one vertex pointing up and the other one has the vertex pointing down. One is placed above the other so that the area of intersection is a hexagon $A_1A_2A_3A_4A_5A_6$ (labelled counterclockwise). Prove that $A_1A_4$, $A_2A_5$ and $A_3A_6$ are concurrent.
Let $n$ be a positive integer. Compute the number of words $w$ that satisfy the following three properties.
1. $w$ consists of $n$ letters from the alphabet $\{a,b,c,d\}.$
2. $w$ contains an even number of $a$'s
3. $w$ contains an even number of $b$'s.
For example, for $n=2$ there are $6$ such words: $aa, bb, cc, dd, cd, dc.$
Determine the smallest $n$ such that $n \equiv (a - 1)$ mod $a$ for all $a \in \{2,3,..., 10\}$.
Let $E$ and $F$ be the intersections of opposite sides of a convex quadrilateral $ABCD$. The two diagonals meet at $P$. Let $O$ be the foot of the perpendicular from $P$ to $EF$. Show that $\angle BOC=\angle AOD$.
There are 100 persons in a hall, everyone knowing at least 66 of the others. Prove that there is a case in which among any four some two don’t know each other.
Show that the cube roots of three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression.
Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.
Find all prime numbers $ p,q < 2005$ such that $ q | p^{2} \plus{} 8$ and $ p|q^{2} \plus{} 8.$
Two mirror walls are placed to form an angle of measure $\alpha$. There is a candle inside the angle. How many reflections of the candle can an observer see?
Let $S = \{2,5,8,11,14,17,20,\dots\}$. Given that one can choose $n$ different numbers from $S$, $\{A_1,A_2,\dots,A_n\}$ s.t. $\sum_{i=1}^n \frac{1}{A_i} = 1$, find the minimum possible value of $n$.
Let $a\in(0,1)$ ,$f(x)=ax^3+(1-4a)x^2+(5a-1)x-5a+3 $ , $g(x)=(1-a)x^3-x^2+(2-a)x-3a-1 $.
Prove that:For any real number $x$ ,at least one of $|f(x)|$ and $|g(x)|$ not less than $a+1$.
Let $f$ be a function defined on positive integers such that $f(1)=4$, $f(2n)=f(n)$ and $f(2n+1)=f(n)+2$ for every positive integer $n$. For how many positive integers $k$ less than $2014$, it is $f(k)=8$?
$
\textbf{(A)}\ 45
\qquad\textbf{(B)}\ 120
\qquad\textbf{(C)}\ 165
\qquad\textbf{(D)}\ 180
\qquad\textbf{(E)}\ 215
$
Consider a finite binary string $b$ with at least $2017$ ones. Show that one can insert some plus signs in between pairs of digits such that the resulting sum, when performed in base $2$, is equal to a power of two.
[i]Proposed by David Stoner
Let $p{}$ be a prime number. There are $p{}$ integers $a_0,\ldots,a_{p-1}$ around a circle. In one move, it is allowed to select some integer $k{}$ and replace the existing numbers via the operation $a_i\mapsto a_i-a_{i+k}$ where indices are taken modulo $p{}.$ Find all pairs of natural numbers $(m, n)$ with $n>1$ such that for any initial set of $p{}$ numbers, after performing any $m{}$ moves, the resulting $p{}$ numbers will all be divisible by $n{}.$
[i]Proposed by P. Kozhevnikov[/i]
Let $a, b, c$ be integers and let $p$ be an odd prime with \[p \not\vert a \;\; \text{and}\;\; p \not\vert b^{2}-4ac.\] Show that \[\sum_{k=1}^{p}\left( \frac{ak^{2}+bk+c}{p}\right) =-\left( \frac{a}{p}\right).\]