Found problems: 85335
Mr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves one problem every $3$ minutes. When he listens to rap music, however, he only solves one problem every $5$ minutes. Mr. DoBa listens to a playlist comprised of $60\%$ classical music and $40\%$ rap music. Each song is exactly $4$ minutes long. Suppose that the expected number of problems he solves in an hour does not depend on whether or not Mr. DoBa is listening to music at any given moment, and let $m$ the average number of problems Mr. DoBa solves per minute when he is not listening to music. Determine the value of $1000m$.
Prove that there exist many natural numbers n so that both roots of the quadratic equation $x^2+(2-3n^2)x+(n^2-1)^2=0$ are perfect squares.
S. Kuzmich
[b]p1.[/b] $17$ rooks are placed on an $8\times 8$ chess board. Prove that there must be at least one rook that is attacking at least $2$ other rooks.
[b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $99$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a $6$ cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $500$ coins?
[b]p3.[/b] Find all solutions $a, b, c, d, e, f, g, h, i$ if these letters represent distinct digits and the following multiplication is correct:
$\begin{tabular}{ccccc}
& & a & b & c \\
x & & & d & e \\
\hline
& f & a & c & c \\
+ & g & h & i & \\
\hline
f & f & f & c & c \\
\end{tabular}$
[b]p4.[/b] Pinocchio rode a bicycle for $3.5$ hours. During every $1$-hour period he went exactly $5$ km. Is it true that his average speed for the trip was $5$ km/h? Explain your reasoning.
[b]p5.[/b] Let $a, b, c$ be odd integers. Prove that the equation $ax^2 + bx + c = 0$ cannot have a rational solution.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
A positive integer $n$ is said to be a [i]perfect power[/i] if $n=a^b$ for some integers $a,b$ with $b>1$.
$(\text{a})$ Find $2004$ perfect powers in arithmetic progression.
$(\text{b})$ Prove that perfect powers cannot form an infinite arithmetic progression.
A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying
$\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.
Prove that there exists a positive integer $k>100$, such that for any set $A$ of $k$ positive reals, there exists a subset $B$ of $100$ numbers, so that none of the sums of at least two numbers in $B$ is in the set $A$.
The parabola with equation $ y \equal{} ax^2 \plus{} bx \plus{} c$ and vertex $ (h,k)$ is reflected about the line $ y \equal{} k$. This results in the parabola with equation $ y \equal{} dx^2 \plus{} ex \plus{} f$. Which of the following equals $ a \plus{} b \plus{} c \plus{} d \plus{} e \plus{} f$?
$ \textbf{(A)} \ 2b \qquad \textbf{(B)} \ 2c \qquad \textbf{(C)} \ 2a \plus{} 2b \qquad \textbf{(D)} \ 2h \qquad \textbf{(E)} \ 2k$
Find the number of sets $A$ containing $9$ positive integers with the following property: for any positive integer $n\le 500$, there exists a subset $B\subset A$ such that $\sum_{b\in B}{b}=n$.
[i]Bogdan Enescu & Dan Ismailescu[/i]
A rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has $20$ regular triangular faces, $30$ square faces, and $12$ regular pentagonal faces, as shown below. How many rotational symmetries does a rhombicosidodecahedron have?
[i]2022 CCA Math Bonanza Lightning Round 4.2[/i]
Suppose $ \,G\,$ is a connected graph with $ \,k\,$ edges. Prove that it is possible to label the edges $ 1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.
[b]Note: Graph-Definition[/b]. A [b]graph[/b] consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices $ \,u,v\,$ belongs to at most one edge. The graph $ G$ is connected if for each pair of distinct vertices $ \,x,y\,$ there is some sequence of vertices $ \,x \equal{} v_{0},v_{1},v_{2},\cdots ,v_{m} \equal{} y\,$ such that each pair $ \,v_{i},v_{i \plus{} 1}\;(0\leq i < m)\,$ is joined by an edge of $ \,G$.
Two men set out at the same time to walk towards each other from $ M$ and $ N$, $ 72$ miles apart. The first man walks at the rate of $ 4$ mph. The second man walks $ 2$ miles the first hour, $ 2\frac {1}{2}$ miles the second hour, $ 3$ miles the third hour, and so on in arithmetic progression. Then the men will meet:
$ \textbf{(A)}\ \text{in 7 hours} \qquad \textbf{(B)}\ \text{in }{8\frac {1}{4}}\text{ hours}\qquad \textbf{(C)}\ \text{nearer }{M}\text{ than }{N}\qquad \\
\textbf{(D)}\ \text{nearer }{N}\text{ than }{M}\qquad \textbf{(E)}\ \text{midway between }{M}\text{ and }{N}$
There are $33!$ empty boxes labeled from $1$ to $33!$. In every move, we find the empty box with the smallest label, then we transfer $1$ ball from every box with a smaller label and we add an additional $1$ ball to that box. Find the smallest labeled non-empty box and the number of the balls in it after $33!$ moves.
We want to draw a number of straight lines such that for each square of a chessboard, at least one of the lines passes through an interior point of the square. What is the smallest number of lines needed for a
(a) $3\times 3$;
(b) $4\times 4$
chessboard? Use a picture to show that this many lines are enough, and prove that no smaller number would do.
(M Vyalyi)
Let $ABC$ be a triangle such that $|AB|=13 , |BC|=12$ and $|CA|=5$. Let the angle bisectors of $A$ and $B$ intersect at $I$ and meet the opposing sides at $D$ and $E$, respectively. The line passing through $I$ and the midpoint of $[DE]$ meets $[AB]$ at $F$. What is $|AF|$?
$
\textbf{(A)}\ \dfrac{3}{2}
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ \dfrac{5}{2}
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \dfrac{7}{2}
$
Find all values of the parameter $a$ for which the sum of all solutions (meaning real solutions) of the equation $x^4 - 5x + a = 0$ is equal to $a$
There are $2015$ distinct circles in a plane, with radius $1$.
Prove that you can select $27$ circles, which form a set $C$, which satisfy the following.
For two arbitrary circles in $C$, they intersect with each other or
For two arbitrary circles in $C$, they don't intersect with each other.
Let $ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$. Let the height from $A$ cut its side
$BC$ at $D$. Let $I, I_B, I_C$ be the incenters of triangles $ABC, ABD, ACD$ respectively. Let also
$EB, EC$ be the excenters of $ABC$ with respect to vertices $B$ and $C$ respectively. If $K$ is the
point of intersection of the circumcircles of $E_CIB_I$ and $E_BIC_I$, show that $KI$ passes through
the midpoint $M$ of side $BC$.
Two sequences of positive reals, $ \left(x_n\right)$ and $ \left(y_n\right)$, satisfy the relations $ x_{n \plus{} 2} \equal{} x_n \plus{} x_{n \plus{} 1}^2$ and $ y_{n \plus{} 2} \equal{} y_n^2 \plus{} y_{n \plus{} 1}$ for all natural numbers $ n$. Prove that, if the numbers $ x_1$, $ x_2$, $ y_1$, $ y_2$ are all greater than $ 1$, then there exists a natural number $ k$ such that $ x_k > y_k$.
Consider the equation $ x^4 \minus{} 4x^3 \plus{} 4x^2 \plus{} ax \plus{} b \equal{} 0$, where $ a,b\in\mathbb{R}$. Determine the largest value $ a \plus{} b$ can take, so that the given equation has two distinct positive roots $ x_1,x_2$ so that $ x_1 \plus{} x_2 \equal{} 2x_1x_2$.
A circle inscribed in an angle with vertex $O$ touches its sides at points $A$ and $B$, $K$ is an arbitrary point on the smaller of the two arcs $AB$ of this circle. On the line $OB$ a point $L$ is taken such that the lines $OA$ and $KL$ are parallel. Let $M$ be the intersection point of the circle $\omega$ circumscribed around triangle $KLB$, with line $AK$, with $M$ different from $K$. Prove that line $OM$ touches circle $\omega$.
Find all functions $ f,g: \mathbb Q \to \mathbb Q$ such that for all rational numbers $ x,y$ we have
\[ f(f(x) \plus{} g(y) ) \equal{} g(f(x)) \plus{} y .
\]
Let $(a_m)$ be a sequence satisfying $a_n \geq 0$, $n=0,1,2,\ldots$ Suppose there exists $A >0$, $a_m - a_{m+1}$ $\geq A a_m ^2$ for all $m \geq 0$. Prove that there exists $B>0$ such that
\begin{align*} a_n \le \frac{B}{n} \qquad \qquad \text{for }1 \le n \end{align*}
For a given triangle $A_1A_2A_3$ and a point $X{}$ inside of it we denote by $X_i$ the intersection of line $A_iX$ with the side opposite to $A_i$ for all $1\leqslant i \leqslant 3.$ Let $P{}$ and $Q{}$ be distinct points inside the triangle. We then say that the two points are isotomic (i.e. they form an isotomic pair) if for all $i{}$ the points $P_i$ and $Q_i$ are symmetric with respect to the midpoint of the side opposite to $A_i.$
Augustus wants to construct isotomic pairs with his favourite app, [i]GeoZebra[/i]. He already constructed the vertices and sidelines of a non-isosceles acute triangle when suddenly his computer got infected with a virus. Most tools became unavailable, only a few are usable, some of which even require a fee:
[list]
[*][b]Point:[/b] select an arbitrary point (with respect to the position of the mouse) on the plane or on a figure (circle or line) [b]- free[/b]
[*][b]Intersection:[/b] intersection points of two figures (where each figure is a circle or a line) [b]- free[/b]
[*][b]Line:[/b] line through two points [b]- \$5 per use[/b]
[*][b]Perpendicular:[/b] perpendicular from a point to an already constructed line [b]- \$50 per use[/b]
[*][b]Circumcircle:[/b] circle through three points [b]- \$10 per use[/b]
[/list]
[list=a]
[*]Agatha selected a point $P{}$ inside the triangle, which is not the centroid of the triangle. Show that Augustus can construct a point $Q{}$ at a cost of at most 1000 dollars such that $P{}$ and $Q{}$ are isotomic.
[*]Prove that for any $n\geqslant 1$ Augustus can construct $n{}$ different isotomic pairs at a cost of at most $200 + 10n$ dollars.
[/list]
[i]Note: The parts are unrelated, that is Augustus can’t use his constructions from part a) in part b).[/i]
Let $v_1, v_2, \ldots, v_m$ be vectors in $\mathbb{R}^n$, such that each has a strictly positive first coordinate. Consider the following process. Start with the zero vector $w = (0, 0, \ldots, 0) \in \mathbb{R}^n$. Every round, choose an $i$ such that $1 \le i \le m$ and $w \cdot v_i \le 0$, and then replace $w$ with $w + v_i$.
Show that there exists a constant $C$ such that regardless of your choice of $i$ at each step, the process is guaranteed to terminate in (at most) $C$ rounds. The constant $C$ may depend on the vectors $v_1, \ldots, v_m$.
The President of the Bank "Glavny Central" Gerasim Shchenkov announced that from January $2$, $2001$ until January $31$ of the same year, the dollar exchange rate would not go beyond the boundaries of the corridor of $27$ rubles $50$ kopecks. and $28$ rubles $30$ kopecks for the dollar. On January $2$, the rate will be a multiple of $5$ kopecks, and starting from January 3, each day will differ from the rate of the previous day by exactly $5$ kopecks. Mr. Shchenkov suggested that citizens try to guess what the dollar exchange rate will be during the specified period. Anyone who can give an accurate forecast for at least one day, he promised to give a cash prize. One interesting person lives in our house, a tireless arguer. For his passion for arguments and constant winnings, he was even nicknamed Zhora Sporos. Zhora claims that he can give such a forecast of the dollar exchange rate for every day from January 424 to January 4314, which he will surely guess at least once, if, of course, the banker strictly acts in accordance with the announced rules. Is Zhora right?
Note: 1 ruble =100 kopecks
[hide=original wording]10-I-7. Президент банка "Главный централ" Герасим Щенков объявил, что со 2-го января 2001 года и до 31-го января этого же года курс доллара не будет выходить за границы коридора 27 руб. 50 коп. и 28 руб. 30 коп. за доллар. 2-го января курс будет кратен 5 копейкам, а, начиная с 3-го января, каждый день будет отличаться от курса предыдущего дня ровно на 5 копеек. Господин Щенков предложил гражданам попробовать угадать, каким будет курс доллара в течение указанного периода. Тому, кто сумеет дать точный прогноз хотя бы на один день, он обещал выдать денежный приз. В нашем доме живет один интересный человек, неутомимый спорщик. За страсть к спорам и постоянные выигрыши его даже прозвали Жора Спорос. Жора утверждает, что может дать такой прогноз курса доллара на каждый день со 2-го по 31-е января, что обязательно хотя бы один раз угадает, если, конечно, банкир будет строго действовать в соответствии с объявленными правилами. Прав ли Жора?
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