Found problems: 85335
A function $f(a \tfrac bc)$ for a simplified mixed fraction $a \tfrac bc$ returns $\tfrac{a + b}{c}$. For instance, $f(2 \tfrac 57) = 1$ and $f(\tfrac45) = \tfrac45$. What is the sum of the three smallest positive rational $x$ where $f(x) = \tfrac 29$?
$\textbf{(A) }\dfrac52\qquad\textbf{(B) }\dfrac{68}{27}\qquad\textbf{(C) }\dfrac{23}{9}\qquad\textbf{(D) }\dfrac{74}{27}\qquad\textbf{(E) }\dfrac{13}4$
Determine all pairs $(x, y)$ of positive integers satisfying
$x + y + 1 | 2xy$ and $ x + y - 1 | x^2 + y^2 - 1$.
Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating $10$ pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have?
$\textbf{(A) } 10\qquad \textbf{(B) } 20\qquad \textbf{(C) } 30\qquad \textbf{(D) } 40\qquad \textbf{(E) } 50$
Prove the following inequality for any integer $ N\geq 4$.
\[ \sum_{p\equal{}4}^N \frac{p^2\plus{}2}{(p\minus{}2)^4}<5\]
Let $ABC$ be a triangle inscribed in the circle $(O)$. The bisector of $\angle BAC$ cuts the circle $(O)$ again at $D$. Let $DE$ be the diameter of $(O)$. Let $G$ be a point on arc $AB$ which does not contain $C$. The lines $GD$ and $BC$ intersect at $F$. Let $H$ be a point on the line $AG$ such that $FH \parallel AE$. Prove that the circumcircle of triangle $HAB$ passes through the orthocenter of triangle $HAC$.
Let $H$ be a convex, equilateral heptagon whose angles measure (in degrees) $168^\circ$, $108^\circ$, $108^\circ$, $168^\circ$, $x^\circ$, $y^\circ$, and $z^\circ$ in clockwise order. Compute the number $y$.
In a triangle $ABC$, $I$ is the incenter. $D$ is the reflection of $A$ to $I$. the incircle is tangent to $BC$ at point $E$. $DE$ cuts $IG$ at $P$ ($G$ is centroid). $M$ is the midpoint of $BC$. prove that
a) $AP||DM$.(15 points)
b) $AP=2DM$. (10 points)
Determine the set of all pairs (a,b) of positive integers for which the set of positive integers can be decomposed into 2 sets A and B so that $a\cdot A=b\cdot B$.
For $j=1,2,3$ let $x_{j},y_{j}$ be non-zero real numbers, and let $v_{j}=x_{j}+y_{j}$.Suppose that the following statements hold:
$x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}$
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2$
$v_{1},v_{2},v_{3}$ satisfy triangle inequality
$v_{1}^{2},v_{2}^{2},v_{3}^{2}$ also satisfy triangle inequality.
Prove that exactly one of $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$ is negative.
Grogg takes an a × b × c rectangular block (where a, b, c are positive integers), paints
the outside of it purple, and cuts it into abc small 1 × 1 × 1 cubes. He then places all the
small cubes into a bag, and Winnie reaches in and randomly picks one of the small cubes.
If the probability that Winnie picks a totally unpainted cube is 20%, determine all possible
values of the number of cubes in the bag.
$ABCD$ is a square. Describe the locus of points $P$, different from $A, B, C, D$, on that plane for which
\[\widehat{APB}+\widehat{CPD}=180^\circ\]
Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$.
Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$
Tarik wants to choose some distinct numbers from the set $S = \{2,...,111\}$ in such a way that each of the chosen numbers cannot be written as the product of two other distinct chosen numbers. What is the maximum number of numbers Tarik can choose ?
Determine all triples $(p, q, r)$ of prime numbers for which $p^q + p^r$ is a perfect square.
a. Determine all nonnegative integers $x$ and $y$ so that $3^x + 7^y$ is a perfect square and $y$ is even.
b. Determine all nonnegative integers $x$ and $y$ so that $3^x + 7^y$ is a perfect square and $y$ is odd
The diagram below shows a circle with center $F$. The angles are related with $\angle BFC = 2\angle AFB$, $\angle CFD = 3\angle AFB$, $\angle DFE = 4\angle AFB$, and $\angle EFA = 5\angle AFB$. Find the degree measure of $\angle BFC$.
[asy]
size(4cm);
pen dps = fontsize(10);
defaultpen(dps);
dotfactor=4;
draw(unitcircle);
pair A,B,C,D,E,F;
A=dir(90);
B=dir(66);
C=dir(18);
D=dir(282);
E=dir(210);
F=origin;
dot("$F$",F,NW);
dot("$A$",A,dir(90));
dot("$B$",B,dir(66));
dot("$C$",C,dir(18));
dot("$D$",D,dir(306));
dot("$E$",E,dir(210));
draw(F--E^^F--D^^F--C^^F--B^^F--A);
[/asy]
Let f be a function defined in the set $\{0, 1, 2,...\}$ of non-negative integers, satisfying $f(2x) = 2f(x), f(4x+1) =
4f(x) + 3$, and $f(4x-1) = 2f(2x - 1) -1$.
Show that $f $ is an injection, i.e. if $f(x) = f(y)$, then $x = y$.
If the six solutions of $x^6=-64$ are written in the form $a+bi$, where $a$ and $b$ are real, then the product of those solutions with $a>0$ is
$\text{(A)} \ -2 \qquad \text{(B)} \ 0 \qquad \text{(C)} \ 2i \qquad \text{(D)} \ 4 \qquad \text{(E)} \ 16$
Octavio writes an integer $n \geq 1$ on a blackboard and then he starts a process in which, at each step he erases the integer $k$ written on the blackboard and replaces it with one of the following numbers:
$$3k-1, \quad 2k+1, \quad \frac{k}{2}.$$
provided that the result is an integer.
Show that for any integer $n \geq 1$, Octavio can write on the blackboard the number $3^{2023}$ after a finite number of steps.
If two poles $ 20''$ and $ 80''$ high are $ 100''$ apart, then the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole is:
$ \textbf{(A)}\ 50'' \qquad\textbf{(B)}\ 40'' \qquad\textbf{(C)}\ 16'' \qquad\textbf{(D)}\ 60'' \qquad\textbf{(E)}\ \text{none of these}$
You have $14$ coins, dated $1901$ through $1914$. Seven of these coins are real and weigh $1.000$ ounce each. The other seven are counterfeit and weigh $0.999$ ounces each. You do not know which coins are real or counterfeit. You also cannot tell which coins are real by look or feel.
Fortunately for you, Zoltar the Fortune-Weighing Robot is capable of making very precise measurements. You may place any number of coins in each of Zoltar's two hands and Zoltar will do the following:
[list][*] If the weights in each hand are equal, Zoltar tells you so and returns all of the coins.
[*] If the weight in one hand is heavier than the weight in the other, then Zoltar takes one coin, at random, from the heavier hand as tribute. Then Zoltar tells you which hand was heavier, and returns the remaining coins to you.[/list]
Your objective is to identify a single real coin that Zoltar has not taken as tribute. Is there a strategy that guarantees this? If so, then describe the strategy and why it works. If not, then prove that no such strategy exists.
Let $ABCD$ be a cyclic quadrilateral which is not a trapezoid and whose diagonals meet at $E$. The midpoints of $AB$ and $CD$ are $F$ and $G$ respectively, and $\ell$ is the line through $G$ parallel to $AB$. The feet of the perpendiculars from E onto the lines $\ell$ and $CD$ are $H$ and $K$, respectively. Prove that the lines $EF$ and $HK$ are perpendicular.
Find the number of partitions of the set $\{1, 2, \cdots, n\}$ into three subsets $A_1,A_2,A_3$, some of which may be empty, such that the following conditions are satisfied:
$(i)$ After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity.
$(ii)$ If $A_1,A_2,A_3$ are all nonempty, then in exactly one of them the minimal number is even .
[i]Proposed by Poland.[/i]
Calculate the following indefinite integrals.
[1] $\int \sqrt{x}(\sqrt{x}+1)^2 dx$
[2] $\int (e^x+2e^{x+1}-3e^{x+2})dx$
[3] $\int (\sin ^2 x+\cos x)\sin x dx$
[4] $\int x\sqrt{2-x} dx$
[5] $\int x\ln x dx$
For $-1 < r < 1$, let $S(r)$ denote the sum of the geometric series \[12 + 12r + 12r^2 + 12r^3 + \ldots.\] Let $a$ between $-1$ and $1$ satisfy $S(a)S(-a)=2016$. Find $S(a) + S(-a)$.