Found problems: 85335
Professor Shian Bray is buying CCA Math Bananas$^{\text{TM}}$. He starts with $\$500$. The first CCA Math Bananas$^{\text{TM}}$ he buys costs $\$1$. Each time after he buys a CCA Math Banana$^{\text{TM}}$, the cost of a CCA Math Bananas$^{\text{TM}}$ doubles with probability $\frac{1}{2}$ (otherwise staying the same). Professor Bray buys CCA Math Bananas$^{\text{TM}}$ until he cannot afford any more, ending with $n$ CCA Math Bananas$^{\text{TM}}$. Estimate the expected value of $n$. An estimate of $E$ earns $2^{1-0.25|E-A|}$ points, where $A$ is the actual answer.
[i]2020 CCA Math Bonanza Lightning Round #5.1[/i]
It is given set $A=\{1,2,3,...,2n-1\}$. From set $A$, at least $n-1$ numbers are expelled such that:
$a)$ if number $a \in A$ is expelled, and if $2a \in A$ then $2a$ must be expelled
$b)$ if $a,b \in A$ are expelled, and $a+b \in A$ then $a+b$ must be also expelled
Which numbers must be expelled such that sum of numbers remaining in set stays minimal
Prove that
\[ \int^1_0 \int^1_0 \frac { dx \ dy }{ \frac 1x + |\log y| -1 } \leq 1 . \]
Cows Alpha and Beta are tied by eight-meter ropes, on the midpoints of adjacent sides of a rectangular fence. Both cows are outside the fence; Alpha can wander in a region with an area of $34\pi$ square meters and Beta can wander in a region with an area of $40\pi$ square meters. What is the area enclosed by the rectangular fence?
$\textbf{(A) }45\qquad\textbf{(B) }48\qquad\textbf{(C) }96\qquad\textbf{(D) }120\qquad\textbf{(E) }144$
Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove:
\[ a^{2} \plus{} b^{2} \plus{} c^{2}\geq 4S \sqrt {3}
\]
In what case does equality hold?
The maximum value of function $f(x)=\sqrt{x^4-3x^2-6x+13}-\sqrt{x^4-x^2+1}$ is________.
There are written $n$ distinct positive integers. An operation is defined as follows: we chose two numers $a$ and $b$ written on the table; we erase them; we write at their places $a+1$ and $b-1$. Find the smallest value of the difference the biggest and the smallest written numbers after some operations.
Let $A,B,P$ be points on a line $\ell$, with $P$ outside the segment $AB$. Lines $a$ and $b$ pass through $A$ and $B$ and are perpendicular to $\ell$. A line $m$ through $P$, which is neither parallel nor perpendicular to $\ell$, intersects $a$ and $b$ at $Q$ and $R$, respectively. The perpendicular from $B$ to $AR$ meets $a$ and $AR$ at $S$ and $U$, and the perpendicular from $A$ to $BQ$ meets $b$ and $BQ$ at $T$ and $V$, respectively.
(a) Prove that $P,S,T$ are collinear.
(b) Prove that $P,U,V$ are collinear.
All of David's telephone numbers have the form $ 555\minus{}abc\minus{}defg$, where $ a$, $ b$, $ c$, $ d$, $ e$, $ f$, and $ g$ are distinct digits and in increasing order, and none is either $ 0$ or $ 1$. How many different telephone numbers can David have?
$ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 7\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$
Let $(O)$ be the circumcircle of triangle $ABC$, and $P$ a point on the arc $BC$ not containing $A$. $(Q)$ is the $A$-mixtilinear circle of triangle $ABC$, and $(K), (L)$ are the $P$-mixtilinear circles of triangle $PAB, PAC$ respectively. Prove that there is a line tangent to all the three circles $(Q), (K)$ and $(L)$.
Nguyen Van Linh, a student at Hanoi Foreign Trade University Cabinet
Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number
$x^p + y^p + z^p - x - y - z$
is a product of exactly three distinct prime numbers.
[b]p1.[/b] Find the largest integer which is a factor of all numbers of the form $n(n +1)(n + 2)$ where $n$ is any positive integer with unit digit $4$. Prove your claims.
[b]p2.[/b] Each pair of the towns $A, B, C, D$ is joined by a single one way road. See example. Show that for any such arrangement, a salesman can plan a route starting at an appropriate town that: enables him to call on a customer in each of the towns.
Note that it is not required that he return to his starting point.
[img]https://cdn.artofproblemsolving.com/attachments/6/5/8c2cda79d2c1b1c859825f3df0163e65da761b.png[/img]
[b]p3.[/b] $A$ and $B$ are two points on a circular race track . One runner starts at $A$ running counter clockwise, and, at the same time, a second runner starts from $B$ running clockwise. They meet first $100$ yds from A, measured along the track. They meet a second time at $B$ and the third time at $A$. Assuming constant speeds, now long is the track?
[b]p4.[/b] $A$ and $B$ are points on the positive $x$ and positive $y$ axis, respectively, and $C$ is the point $(3,4)$. Prove that the perimeter of $\vartriangle ABC$ is greater than $10$.
Suggestion: Reflect!!
[b]p5.[/b] Let $A_1,A_2,...,A_8$ be a permutation of the integers $1,2,...,8$ so chosen that the eight sums $9 + A_1$, $10 + A_2$, $...$, $16 + A_8$ and the eight differences $9 -A_1$ , $10 - A_2$, $...$, $16 - A_8$ together comprise $16$ different numbers.
Show that the same property holds for the eight numbers in reverse order. That is, show that the $16$ numbers $9 + A_8$, $10 + A_7$, $...$, $16 + A_1$ and $9 -A_8$ , $10 - A_7$, $...$, $16 - A_1$ are also pairwise different.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
I have 6 friends and during a vacation I met them during several dinners. I found that I dined with all the 6 exactly on 1 day; with every 5 of them on 2 days; with every 4 of them on 3 days; with every 3 of them on 4 days; with every 2 of them on 5 days. Further every friend was present at 7 dinners and every friend was absent at 7 dinners. How many dinners did I have alone?
Find all positive integers $N$ such that the following holds: There exist pairwise coprime positive integers $a,b,c$ with
$$\frac1a+\frac1b+\frac1c=\frac N{a+b+c}.$$
Let $\{a_n\}_{n=0}^{\infty}$ be the sequence defined by the recurrence relation $a_{n+3}=2a_{n+2} - 23a_{n+1}+3a_n$ for all $n \ge 0,$ with initial conditions $a_0=20, a_1=0,$ and $a_2=23.$ Let $b_n=a_n^3$ for all $n \ge 0.$ There exists a unique positive integer $k$ and constants $c_0, \ldots, c_{k-1}$ with $c_0 \neq 0$ and $c_{k-1} \neq 0$ such that for all sufficiently large $n,$ we have the recurrence relation $b_{n+k} = \sum_{t=0}^{k-1} c_t b_{n+t}.$ Find $k+\sqrt{|c_{k-1}|}+\sqrt{|c_0|}.$
Two circles $\Gamma_a$ and $\Gamma_b$ with their centres lying on the legs $BC$ and $CA$ of a right triangle, both touching the hypotenuse $AB$, and both passing through the vertex $C$ are given. Let the radii of these circles be denoted by $\gamma_a$ and $\gamma_b$.
Find the greatest real number $p$ such that the inequality $\frac{1}{\gamma_a}+\frac{1}{\gamma_b}\ge p \left(\frac{1}{a}+\frac{1}{b}\right)$
($BC = a,CA = b$) holds for all right triangles $ABC$.
[b]p1. [/b](a) Show that for every positive integer $m > 1$, there are positive integers $x$ and $y$ such that $x^2 - y^2 = m^3$.
(b) Find all pairs of positive integers $(x, y)$ such that $x^6 = y^2 + 127$.
[b]p2.[/b] (a) Let $P(x)$ be a polynomial with integer coefficients. Suppose that $P(0)$ is an odd integer and that $P(1)$ is also an odd integer. Show that if $c$ is an integer then $P(c)$ is not equal to $0$.
(b) Let P(x) be a polynomial with integer coefficients. Suppose that $P(1,000) = 1,000$ and $P(2,000) = 2,000.$ Explain why $P(3,000)$ cannot be equal to $1,000$.
[b]p3.[/b] Triangle $\vartriangle ABC$ is created from points $A(0, 0)$, $B(1, 0)$ and $C(1/2, 2)$. Let $q, r$, and $s$ be numbers such that $0 < q < 1/2 < s < 1$, and $q < r < s$. Let D be the point on $AC$ which has $x$-coordinate $q$, $E$ be the point on AB which has $x$-coordinate $r$, and $F$ be the point on $BC$ that has $x$-coordinate $s$.
(a) Find the area of triangle $\vartriangle DEF$ in terms of $q, r$, and $s$.
(b) If $r = 1/2$, prove that at least one of the triangles $\vartriangle ADE$, $\vartriangle CDF$, or $\vartriangle BEF$ has an area of at least $1/4$.
[b]p4.[/b] In the Gregorian calendar:
(i) years not divisible by $4$ are common years,
(ii) years divisible by $4$ but not by $100$ are leap years,
(iii) years divisible by $100$ but not by $400$ are common years,
(iv) years divisible by $400$ are leap years,
(v) a leap year contains $366$ days, a common year $365$ days.
From the information above:
(a) Find the number of common years and leap years in $400$ consecutive Gregorian years. Show that $400$ consecutive Gregorian years consists of an integral number of weeks.
(b) Prove that the probability that Christmas falls on a Wednesday is not equal to $1/7$.
[b]p5.[/b] Each of the first $13$ letters of the alphabet is written on the back of a card and the $13$ cards are placed in a row in the order $$A,B,C,D,E, F, G,H, I, J,K, L,M$$
The cards are then turned over so that the letters are face down. The cards are rearranged and again placed in a row, but of course they may be in a different order. They are rearranged and placed in a row a second time and both rearrangements were performed exactly the same way. When the cards are turned over the letters are in the order $$B,M, A,H, G,C, F,E,D, L, I,K, J$$ What was the order of the letters after the cards were rearranged the first time?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
In the triangle $ABC$, we have that $M$ and $N$ are points on $AB$ and $AC$, respectively, such that $BC$ is parallel to $MN$. A point $D$ is chosen inside the triangle $AMN$. Let $E$ and $F$ be the points of intersection of $MN$ with $BD$ and $CD$, respectively. Show that the line joining the centers of the circles circumscribed to the triangles $DEN$ and $DFM$ is perpendicular to $AD$.
An infinite sequence $p_1, p_2, p_3, \ldots$ of natural numbers in the decimal system has the following property: For every $i \in \mathbb{N}$ the last digit of $p_{i+1}$ is different from $9$, and by omitting this digit one obtains number $p_i$. Prove that this sequence contains infinitely many composite numbers.
In square $ABCD$, points $E$ and $H$ lie on $\overline{AB}$ and $\overline{DA}$, respectively, so that $AE=AH.$ Points $F$ and $G$ lie on $\overline{BC}$ and $\overline{CD}$, respectively, and points $I$ and $J$ lie on $\overline{EH}$ so that $\overline{FI} \perp \overline{EH}$ and $\overline{GJ} \perp \overline{EH}$. See the figure below. Triangle $AEH$, quadrilateral $BFIE$, quadrilateral $DHJG$, and pentagon $FCGJI$ each has area $1.$ What is $FI^2$?
[asy]
real x=2sqrt(2);
real y=2sqrt(16-8sqrt(2))-4+2sqrt(2);
real z=2sqrt(8-4sqrt(2));
pair A, B, C, D, E, F, G, H, I, J;
A = (0,0);
B = (4,0);
C = (4,4);
D = (0,4);
E = (x,0);
F = (4,y);
G = (y,4);
H = (0,x);
I = F + z * dir(225);
J = G + z * dir(225);
draw(A--B--C--D--A);
draw(H--E);
draw(J--G^^F--I);
draw(rightanglemark(G, J, I), linewidth(.5));
draw(rightanglemark(F, I, E), linewidth(.5));
dot("$A$", A, S);
dot("$B$", B, S);
dot("$C$", C, dir(90));
dot("$D$", D, dir(90));
dot("$E$", E, S);
dot("$F$", F, dir(0));
dot("$G$", G, N);
dot("$H$", H, W);
dot("$I$", I, SW);
dot("$J$", J, SW);
[/asy]
$\textbf{(A) } \frac{7}{3} \qquad \textbf{(B) } 8-4\sqrt2 \qquad \textbf{(C) } 1+\sqrt2 \qquad \textbf{(D) } \frac{7}{4}\sqrt2 \qquad \textbf{(E) } 2\sqrt2$
In plane there are two circles with radiuses $r_1$ and $r_2$, one outside the other. There are two external common tangents on those circles and one internal common tangent. The internal one intersects external ones in points $A$ and $B$ and touches one of the circles in point $C$. Prove that
$AC \cdot BC=r_1\cdot r_2$
Let $C$ be the set of non-negative integers which can be expressed as $1999s+2000t$, where $s$ and $t$ are also non-negative integers.
(a) Show that $3,994,001$ is not in $C$.
(b) Show that if $0\leq n \leq 3,994,001$ and $n$ is an integer not in $C$, then $3,994,001-n$ is in $C$.
Let $S$ be the set of integers which are both a multiple of $70$ and a factor of $630{,}000$. A random element $c$ of $S$ is selected. If the probability that there exists an integer $d$ with $\gcd (c,d) = 70$ and $\operatorname{lcm} (c,d) = 630{,}000$ is $\frac mn$ for some relatively prime integers $m$ and $n$, compute $100m+n$.
[i]Proposed by Eugene Chen[/i]
Find all functions $f : Z \to Z$ satisfying the following two conditions:
(i) for all integers $x$ we have $f(f(x)) = x$,
(ii) for all integers $x$ and y such that $x + y$ is odd, we have $f(x) + f(y) \ge x + y$.
For a positive integer $n$, define $n?=1^n\cdot2^{n-1}\cdot3^{n-2}\cdots\left(n-1\right)^2\cdot n^1$. Find the positive integer $k$ for which $7?9?=5?k?$.
[i]Proposed by Tristan Shin[/i]