Found problems: 85335
We call a function $g$ [i]special [/i] if $g(x)=a^{f(x)}$ (for all $x$) where $a$ is a positive integer and $f$ is polynomial with integer coefficients such that $f(n)>0$ for all positive integers $n$.
A function is called an [i]exponential polynomial[/i] if it is obtained from the product or sum of special functions. For instance, $2^{x}3^{x^{2}+x-1}+5^{2x}$ is an exponential polynomial.
Prove that there does not exist a non-zero exponential polynomial $f(x)$ and a non-constant polynomial $P(x)$ with integer coefficients such that
$$P(n)|f(n)$$
for all positive integers $n$.
For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid $3$ acorns in each of the holes it dug. The squirrel hid $4$ acorns in each of the holes it dug. They each hid the same
number of acorns, although the squirrel needed $4$ fewer holes. How many acorns did the chipmunk hide?
${{ \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48}\qquad\textbf{(E)}\ 54} $
Prove that there exists a constant $C > 0$ such that, for any integers $m, n$ with $n \geq m > 1$ and any real number $x > 1$, $$\sum_{k=m}^{n}\sqrt[k]{x} \leq C\bigg(\frac{m^2 \cdot \sqrt[m-1]{x}}{\log{x}} + n\bigg)$$
Cevians $AP$ and $AQ$ of a triangle $ABC$ are symmetric with respect to its bisector. Let $X$, $Y$ be the projections of $B$ to $AP$ and $AQ$ respectively, and $N$, $M$ be the projections of $C$ to $AP$ and $AQ$ respectively. Prove that $XM$ and $NY$ meet on $BC$.
[b]a)[/b] Find the group $ H $ that is isomorphic with the multiplicative group of positive real numbers, having an isomorphism $$ \iota :(0,\infty )\longrightarrow H,\quad\iota (x)=\frac{x-1}{x+1} . $$
[b]b)[/b] Calculate the $ 2012\text{-th} $ power of an arbitrary element of $ H. $
Let $f(x)=\frac{\sin x}{x}$, for $x>0$, and let $n$ be a positive integer. Prove that $|f^{(n)}(x)|<\frac{1}{n+1}$, where $f^{(n)}$ denotes the $n^{\mathrm{th}}$ derivative of $f$.
(Proposed by Alexander Bolbot, State University, Novosibirsk)
In triangle $ABC, O$ is the circumcenter, $H$ is the orthocenter. Construct the circumcircles of triangles $CHB, CHA$ and $AHB$, and let their centers be $A_1, B_1, C_1$, respectively. Prove that triangles $ABC$ and $A_1B_1C_1$ are congruent, and their nine-point circles coincide.
Let $ABC$ be a triangle and $D$ a point on the side $BC$. Point $E$ is the symmetric of $D$ with respect to $AB$. Point $F$ is the symmetric of $E$ with respect to $AC$. Point $P$ is the intersection of line $DF$ with line $AC$. Prove that the quadrilateral $AEDP$ is cyclic.
(Malik Talbi)
Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that
$ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$
(Proposed by Gerhard Woeginger, Austria)
A box of strawberries, containing $12$ strawberries total, costs $\$ 2$. A box of blueberries, containing $48$ blueberries total, costs $ \$ 3$. Suppose that for $\$ 12$, Sareen can either buy $m$ strawberries total or $n$ blueberries total. Find $n - m$.
[i]Proposed by Andrew Wu[/i]
[b]p1.[/b] Solve the equation $$\frac{\sqrt{x^2 - 2x + 1}}{x^2 - 1}+\frac{x^2 - 1}{\sqrt{x^2 - 2x + 1}}=\frac52.$$
[b]p2.[/b] Solve the inequality: $\frac{1 - 2\sqrt{1-x^2}}{x} \le 1$.
[b]p3.[/b] Let $ABCD$ be a convex quadrilateral such that the length of the segment connecting midpoints of the two opposite sides $AB$ and $CD$ equals $\frac{|AD| + |BC|}{2}$. Prove that $AD$ is parallel to $BC$.
[b]p4.[/b] Solve the equation: $\frac{1}{\cos x}+\frac{1}{\sin x}= 2\sqrt2$.
[b]p5.[/b] Long, long ago, far, far away there existed the Old Republic Galaxy with a large number of stars. It was known that for any four stars in the galaxy there existed a point in space such that the distance from that point to any of these four stars was less than or equal to $R$. Master Yoda asked Luke Skywalker the following question: Must there exist a point $P$ in the galaxy such that all stars in the galaxy are within a distance $R$ of the point $P$? Give a justified argument that will help Like answer Master Yoda’s question.
[b]p6.[/b] The Old Republic contained an odd number of inhabited planets. Some pairs of planets were connected to each other by space flights of the Trade Federation, and some pairs of planets were not connected. Every inhabited planet had at least one connections to some other inhabited planet. Luke knew that if two planets had a common connection (they are connected to the same planet), then they have a different number of total connections. Master Yoda asked Luke if there must exist a planet that has exactly two connections. Give a justified argument that will help Luke answer Master Yoda’s question.
PS. You should use hide for answers.
Consider two geometric sequences $16$, $a_1$, $a_2$, $ . . .$ and $56$, $b_1$, $b_2$, $. . . $ with the same common nonzero ratio. Given that $a_{2023} = b_{2020}$, compute $b_6 - a_6$.
Akash's birthday cake is in the form of a $4 \times 4 \times 4$ inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into $64$ smaller cubes, each measuring $1 \times 1 \times 1$ inch, as shown below. How many of the small pieces will have icing on exactly two sides?
[asy]
/*
Created by SirCalcsALot and sonone
Code modfied from https://artofproblemsolving.com/community/c3114h2152994_the_old__aops_logo_with_asymptote
*/
import three;
currentprojection=orthographic(1.75,7,2);
//++++ edit colors, names are self-explainatory ++++
//pen top=rgb(27/255, 135/255, 212/255);
//pen right=rgb(254/255,245/255,182/255);
//pen left=rgb(153/255,200/255,99/255);
pen top = rgb(170/255, 170/255, 170/255);
pen left = rgb(81/255, 81/255, 81/255);
pen right = rgb(165/255, 165/255, 165/255);
pen edges=black;
int max_side = 4;
//+++++++++++++++++++++++++++++++++++++++
path3 leftface=(1,0,0)--(1,1,0)--(1,1,1)--(1,0,1)--cycle;
path3 rightface=(0,1,0)--(1,1,0)--(1,1,1)--(0,1,1)--cycle;
path3 topface=(0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle;
for(int i=0; i<max_side; ++i){
for(int j=0; j<max_side; ++j){
draw(shift(i,j,-1)*surface(topface),top);
draw(shift(i,j,-1)*topface,edges);
draw(shift(i,-1,j)*surface(rightface),right);
draw(shift(i,-1,j)*rightface,edges);
draw(shift(-1,j,i)*surface(leftface),left);
draw(shift(-1,j,i)*leftface,edges);
}
}
picture CUBE;
draw(CUBE,surface(leftface),left,nolight);
draw(CUBE,surface(rightface),right,nolight);
draw(CUBE,surface(topface),top,nolight);
draw(CUBE,topface,edges);
draw(CUBE,leftface,edges);
draw(CUBE,rightface,edges);
// begin made by SirCalcsALot
int[][] heights = {{4,4,4,4},{4,4,4,4},{4,4,4,4},{4,4,4,4}};
for (int i = 0; i < max_side; ++i) {
for (int j = 0; j < max_side; ++j) {
for (int k = 0; k < min(heights[i][j], max_side); ++k) {
add(shift(i,j,k)*CUBE);
}
}
}
[/asy]
$\textbf{(A)}\ 12\qquad~~\textbf{(B)}\ 16\qquad~~\textbf{(C)}\ 18\qquad~~\textbf{(D)}\ 20\qquad~~\textbf{(E)}\ 24$
For every $n \in N $, let $d(n)$ denote the sum of digits of $n$. It is easy to see that the sequence $d(n), d(d(n))$, $d(d(d(n))), ... $ will eventually become a constant integer between $1$ and $9$ (both inclusive). This number is called the digital root of $n$ . Denote it by $b(n)$. Then for how many natural numbers $k<1000 , \lim_{n \to \infty} b(k^n)$ exists.
We wish to write $n$ distinct real numbers $(n\geq3)$ on the circumference of a circle in such a way that each number is equal to the product of its immediate neighbors to the left and right. Determine all of the values of $n$ such that this is possible.
Two circles of radius 1 are to be constructed as follows. The center of circle $ A$ is chosen uniformly and at random from the line segment joining $ (0,0)$ and $ (2,0)$. The center of circle $ B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $ (0,1)$ to $ (2,1)$. What is the probability that circles $ A$ and $ B$ intersect?
$ \textbf{(A)} \; \frac{2\plus{}\sqrt{2}}{4} \qquad \textbf{(B)} \; \frac{3\sqrt{3}\plus{}2}{8} \qquad \textbf{(C)} \; \frac{2 \sqrt{2} \minus{} 1}{2} \qquad \textbf{(D)} \; \frac{2\plus{}\sqrt{3}}{4} \qquad \textbf{(E)} \; \frac{4 \sqrt{3} \minus{} 3}{4}$
Given a set $ M$ of points $ (x,y)$ with integral coordinates satisfying $ x^2 + y^2\leq 10^{10}$. Two players play a game. One of them marks a point on his first move. After this, on each move the moving player marks a point, which is not yet marked and joins it with the previous marked point. Players are not allowed to mark a point symmetrical to the one just chosen. So, they draw a broken line. The requirement is that lengths of edges of this broken line must strictly increase. The player, which can not make a move, loses. Who have a winning strategy?
Let $ ABC$ be a triangle with circumradius $ R$, perimeter $ P$ and area $ K$. Determine the maximum value of: $ \frac{KP}{R^3}$.
each of the squares in a 2 x 2018 grid of squares is to be coloured black or white such that in any 2 x 2 block , at least one of the 4 squares is white. let P be the number of ways of colouring the grid. find the largest k so that $3^k$ divides P.
If two circles intersect at $A,B$ and common tangents of them intesrsect circles at $C,D$if $O_a$is circumcentre of $ACD$ and $O_b$ is circumcentre of $BCD$ prove $AB$ intersects $O_aO_b$ at its midpoint
You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensue that at least one of your guessed squares is covered by the rectangle?
$\textbf{(A)}~3\qquad\textbf{(B)}~5\qquad\textbf{(C)}~4\qquad\textbf{(D)}~8\qquad\textbf{(E)}~6$
Let $n$ be a positive integer. Prove that there are no positive integers $x$ and $y$ such as $\sqrt{n}+\sqrt{n+1} < \sqrt{x}+\sqrt{y} <\sqrt{4n+2} $
There are $n(n\ge 8)$ airports, some of which have one-way direct routes between them. For any two airports $a$ and $b$, there is at most one one-way direct route from $a$ to $b$ (there may be both one-way direct routes from $a$ to $b$ and from $b$ to $a$). For any set $A$ composed of airports $(1\le | A| \le n-1)$, there are at least $4\cdot \min \{|A|,n-|A| \}$ one-way direct routes from the airport in $A$ to the airport not in $A$.
Prove that: For any airport $x$, we can start from $x$ and return to the airport by no more than $\sqrt{2n}$ one-way direct routes.
Ahuiliztli is playing around with some coins (pennies, nickels, dimes, and quarters). She keeps grabbing $k$ coins and calculating the value of her handful. After a while, she begins to notice that if $k$ is even, she more often gets even sums, and if $k$ is odd, she more often gets odd sums. Help her prove this true! Given $k$ coins chosen uniformly and at random, prove that. the probability that the parity of $k$ is the same as the parity of the $k$ coins' value is greater than the probability that the parities are different.
In the triangle $ ABC$ we have $ AB \equal{} 5$ and $ AC \equal{} 6$. The area of the triangle when the $ \angle ACB$ is as large as possible is
$ \text{(A)}\ 15 \qquad \text{(B)}\ 5 \sqrt{7} \qquad \text{(C)}\ \frac{7}{2} \sqrt{7} \qquad \text{(D)}\ 3 \sqrt{11} \qquad \text{(E)}\ \frac{5}{2} \sqrt{11}$