Found problems: 85335
Elmer's new car gives $50\%$ percent better fuel efficiency, measured in kilometers per liter, than his old car. However, his new car uses diesel fuel, which is $20\%$ more expensive per liter than the gasoline his old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip?
$\textbf{(A) } 20\% \qquad \textbf{(B) } 26\tfrac23\% \qquad \textbf{(C) } 27\tfrac79\% \qquad \textbf{(D) } 33\tfrac13\% \qquad \textbf{(E) } 66\tfrac23\%$
Mr. Patrick teaches math to $15$ students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was $80$. After he graded Payton's test, the class average became $81$. What was Payton's score on the test?
$\textbf{(A) }81\qquad\textbf{(B) }85\qquad\textbf{(C) }91\qquad\textbf{(D) }94\qquad\textbf{(E) }95$
On a board, Ana and Bob start writing $0$s and $1$s alternatively until each of them has written $2021$ digits. Ana starts this procedure and each of them always adds a digit to the right of the already existing ones.
Ana wins the game if, after they stop writing, the resulting number (in binary) can be written as the sum of two squares. Otherwise, Bob wins. Determine who has a winning strategy.
Let $f(x) = 3(|x|+|x-1|-|x+1|)$ and let $x_{n+1}=f(x_n)$ $\forall n \ge 0$. How many real number $x_0$ are there, that satisfy $x_0=x_{2007}$ and $x_0,x_1,x_2,...,x_{2006}$ are distinct?
A [i]stick[/i] is defined as a $1 \times k$ or $k\times 1$ rectangle for any integer $k \ge 1$. We wish to partition the cells of a $2022 \times 2022$ chessboard into $m$ non-overlapping sticks, such that any two of these $m$ sticks share at most one unit of perimeter. Determine the smallest $m$ for which this is possible.
[i]Holden Mui[/i]
An automobile travels $a/6$ feet in $r$ seconds. If this rate is maintained for $3$ minutes, how many yards does it travel in $3$ minutes?
${{ \textbf{(A)}\ \frac{a}{1080r}\qquad\textbf{(B)}\ \frac{30r}{a}\qquad\textbf{(C)}\ \frac{30a}{r}\qquad\textbf{(D)}\ \frac{10r}{a} }\qquad\textbf{(E)}\ \frac{10a}{r} } $
Let \( AD \) and \( BE \) be altitudes of triangle \( \triangle ABC \) that meet at the orthocenter \( H \). The midpoints of segments \( AB \) and \( CH \) are \( X \) and \( Y \), respectively. Prove that the line \( XY \) is perpendicular to line \( DE \).
Let $a_1, a_2,...$ be an infinite sequence of positive integers such that for any $k,\ell\in \mathbb{Z_+}$, $a_{k+\ell}$ is divisible by $\gcd(a_k,a_\ell)$. Prove that for any integers $1\leqslant k\leqslant n$, $a_na_{n-1}\dots a_{n-k+1}$ is divisible by $a_ka_{k-1}\dots a_1$.
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.
Two disjoint circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ respectively lie on the same side of a line $p$ and touch the line at $A_1$ and $A_2$ respectively. The segment $O_1O_2$ intersects $k_1$ at $B_1$ and $k_2$ at $B_2$. Prove that $A_1B_1\perp A_2B_2$.
How many integer pairs $(x,y)$ satisfy $x^2+4y^2-2xy-2x-4y-8=0$?
In how many ways can $47$ be written as the sum of two primes?
$\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ \text{more than 3}$
Prove that the plane is not a union of the inner regions of finitely many parabolas. (The outer region of a parabola is the union of the lines not intersecting the parabola. The inner region of a parabola is the set of points of the plane that do not belong to the outer region of the parabola)
Let $ f(x)$ be a nonzero, bounded, real function on an Abelian group $ G$, $ g_1,...,g_k$ are given elements of $ G$ and $ \lambda_1,...,\lambda_k$ are real numbers. Prove that if \[ \sum_{i=1}^k \lambda_i f(g_ix) \geq 0\] holds for all $ x \in G$, then \[ \sum_{i=1}^k \lambda_i \geq 0.\]
[i]A. Mate[/i]
Let be a natural number $ n. $ Prove that there exists a number $ k\in\{ 0,1,2,\ldots n \} $ such that the floor of $ 2^{n+k}\sqrt 2 $ is even.
Let $ A$, $ B$, $ C$, $ A^{\prime}$, $ B^{\prime}$, $ C^{\prime}$, $ X$, $ Y$, $ Z$, $ X^{\prime}$, $ Y^{\prime}$, $ Z^{\prime}$ and $ P$ be pairwise distinct points in space such that
$ A^{\prime} \in BC;\ B^{\prime}\in CA;\ C^{\prime}\in AB;\ X^{\prime}\in YZ;\ Y^{\prime}\in ZX;\ Z^{\prime}\in XY;$
$ P \in AX;\ P\in BY;\ P\in CZ;\ P\in A^{\prime}X^{\prime};\ P\in B^{\prime}Y^{\prime};\ P\in C^{\prime}Z^{\prime}$.
Prove that
$ \frac {BA^{\prime}}{A^{\prime}C}\cdot\frac {CB^{\prime}}{B^{\prime}A}\cdot\frac {AC^{\prime}}{C^{\prime}B} \equal{} \frac {YX^{\prime}}{X^{\prime}Z}\cdot\frac {ZY^{\prime}}{Y^{\prime}X}\cdot\frac {XZ^{\prime}}{Z^{\prime}Y}$.
Let $ABCDEF$ be a regular hexagon with sidelength s. The points $P$ and $Q$ are on the diagonals $BD$ and $DF$, respectively, such that $BP = DQ = s$. Prove that the three points $C$, $P$ and $Q$ are on a line.
[i](Walther Janous)[/i]
Find the smallest positive real $k$ satisfying the following condition: for any given four DIFFERENT real numbers $a,b,c,d$, which are not less than $k$, there exists a permutation $(p,q,r,s)$ of $(a,b,c,d)$, such that the equation $(x^{2}+px+q)(x^{2}+rx+s)=0$ has four different real roots.
A square piece of paper has sides of length $ 100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at distance $ \sqrt {17}$ from the corner, and they meet on the diagonal at an angle of $ 60^\circ$ (see the figure below). The paper is then folded up along the lines joining the vertices of adjacent cuts. When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of the tray, that is, the perpendicular distance between the plane of the base and the plane formed by the upper edges, can be written in the form $ \sqrt [n]{m}$, where $ m$ and $ n$ are positive integers, $ m < 1000$, and $ m$ is not divisible by the $ n$th power of any prime. Find $ m \plus{} n$.
[asy]import math;
unitsize(5mm);
defaultpen(fontsize(9pt)+Helvetica()+linewidth(0.7));
pair O=(0,0);
pair A=(0,sqrt(17));
pair B=(sqrt(17),0);
pair C=shift(sqrt(17),0)*(sqrt(34)*dir(75));
pair D=(xpart(C),8);
pair E=(8,ypart(C));
draw(O--(0,8));
draw(O--(8,0));
draw(O--C);
draw(A--C--B);
draw(D--C--E);
label("$\sqrt{17}$",(0,2),W);
label("$\sqrt{17}$",(2,0),S);
label("cut",midpoint(A--C),NNW);
label("cut",midpoint(B--C),ESE);
label("fold",midpoint(C--D),W);
label("fold",midpoint(C--E),S);
label("$30^\circ$",shift(-0.6,-0.6)*C,WSW);
label("$30^\circ$",shift(-1.2,-1.2)*C,SSE);[/asy]
RyNo, a little rhinoceros, has $17$ scratch marks on its body. Some are horizontal and the rest are vertical. Some are on the left side and the rest are on the right side. If RyNo rubs one side of its body against a tree, two scratch marks, either both horizontal or both vertical, will disappear from that side. However, at the same time, two new scratch marks, one horizontal and one vertical, will appear on the other side. If there are less than two horizontal and less than two vertical scratch marks on the side being rubbed, then nothing happens. If RyNo continues to rub its body against trees, is it possible that at some point in time, the numbers of horizontal and vertical scratch marks have interchanged on each side of its body?
Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz start on the ground, make a sequence of $6$ hops, and end up back on the ground? (For example, one sequence of hops is up-up-down-down-up-down.)
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }12$
Find all regular $n$-gons with the following properties:
(a) a diagonal is equal to the sum of two other diagonals
(b) a diagonal is equal to the sum of a side and another diagonal
Solve the system
$$\begin{cases} x+\log\left(x+\sqrt{x^2+1}\right)=y \\
y+\log\left(y+\sqrt{y^2+1}\right)=z \\
z+\log\left(z+\sqrt{z^2+1}\right)=x \end{cases}$$
Fill in the grid below with the numbers $1$ through $25$, with each number used exactly once, subject to the following constraints:
[list=1]
[*] Each shaded square contains an even number, and each unshaded square contains an odd number. [/*]
[*] For any pair of squares that share a side, if $x$ and $y$ are the two numbers in those squares, then either $x\geq2y$ or $y\geq2x$. [/*]
[/list]
Four numbers have been filled in already.
[asy]
size(10cm);
for(int i=1; i<5; ++i){
draw((-2i+1,-2i+9)--(2i-1,-2i+9));
draw((-2i+1,2i-9)--(2i-1,2i-9));
draw((-2i+9,-2i+1)--(-2i+9,2i-1));
draw((2i-9,-2i+1)--(2i-9,2i-1));
}
for(int i=1; i<3; ++i){
filldraw((-1,2i+1)--(-1,2i-1)--(-3,2i-1)--(-3,2i+1)--cycle,lightgray);
}
for(int i=2; i<4; ++i){
filldraw((1,2i+1)--(1,2i-1)--(-1,2i-1)--(-1,2i+1)--cycle,lightgray);
}
for(int i=1; i<5; ++i){
filldraw((3,2i-5)--(3,2i-7)--(1,2i-7)--(1,2i-5)--cycle,lightgray);
}
filldraw((-5,1)--(-5,-1)--(-7,-1)--(-7,1)--cycle,lightgray);
filldraw((-1,-1)--(-1,-3)--(-3,-3)--(-3,-1)--cycle,lightgray);
filldraw((1,-5)--(1,-7)--(-1,-7)--(-1,-5)--cycle,lightgray);
filldraw((5,3)--(5,1)--(3,1)--(3,3)--cycle,lightgray);
label("\Huge{25}",(-4,2));
label("\Huge{13}",(0,0));
label("\Huge{16}",(0,6));
label("\Huge{21}",(4,-2));
[/asy]
The sequence $a_n$ is defined by the following conditions: $a_1=1$, and for any $n\in \mathbb N$, the number $a_{n+1}$ is obtained from $a_n$ by adding three if $n$ is a member of this sequence, and two if it is not. Prove that $a_n<(1+\sqrt 2)n$ for all $n$.
[i]Proposed by Mikhail Ivanov[/i]