Found problems: 85335
1 - Prove that $55 < (1+\sqrt{3})^4 < 56$ .
2 - Find the largest power of $2$ that divides $\lceil(1+\sqrt{3})^{2n}\rceil$ for the positive integer $n$
$21$ Savage has a $12$ car garage, with a row of spaces numbered $1,2,3,\ldots,12$. How many ways can he choose $6$ of them to park his $6$ identical cars in, if no $3$ spaces with consecutive numbers may be all occupied?
[i]2018 CCA Math Bonanza Team Round #9[/i]
[u]Round 1[/u]
[b]p1.[/b] At a fortune-telling exam, $13$ witches are sitting in a circle. To pass the exam, a witch must correctly predict, for everybody except herself and her two neighbors, whether they will pass or fail. Each witch predicts that each of the $10$ witches she is asked about will fail. How many witches could pass?
[b]p2.[/b] Out of $152$ coins, $7$ are counterfeit. All counterfeit coins have the same weight, and all real coins have the same weight, but counterfeit coins are lighter than real coins. How can you find $19$ real coins if you are allowed to use a balance scale three times?
[b]p3.[/b] The digits of a number $N$ increase from left to right. What could the sum of the digits of $9 \times N$ be?
[b]p4.[/b] The sides and diagonals of a pentagon are colored either blue or red. You can choose three vertices and flip the colors of all three lines that join them. Can every possible coloring be turned all blue by a sequence of such moves?
[img]https://cdn.artofproblemsolving.com/attachments/5/a/644aa7dd995681fc1c813b41269f904283997b.png[/img]
[b]p5.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order. Count the number of blueberries in the top pancake and call that number $N$. Pick up the stack of the top $N$ pancakes and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack.
[u]Round 2[/u]
[b]p6.[/b] A circus owner will arrange $100$ fleas on a long string of beads, each flea on her own bead. Once arranged, the fleas start jumping using the following rules. Every second, each flea chooses the closest bead occupied by one or more of the other fleas, and then all fleas jump simultaneously to their chosen beads. If there are two places where a flea could jump, she jumps to the right. At the start, the circus owner arranged the fleas so that, after some time, they all gather on just two beads. What is the shortest amount of time it could take for this to happen?
[b]p7.[/b] The faraway land of Noetheria has $2016$ cities. There is a nonstop flight between every pair of cities. The price of a nonstop ticket is the same in both directions, but flights between different pairs of cities have different prices. Prove that you can plan a route of $2015$ consecutive flights so that each flight is cheaper than the previous one. It is permissible to visit the same city several times along the way.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Consider the set $M=\{1,2,...,n\},n\in\mathbb N$. Find the smallest positive integer $k$ with the following property: In every $k$-element subset $S$ of $M$ there exist two elements, one of which divides the other one.
Maggie Waggie organizes a pile of 127 calculus tests in alphabetical order, with Joccy Woccy's test being 64th in the pile. While Maggie isn't looking, Joccy walks over and randomly scrambles the entire pile of tests. When Maggie returns, she is oblivious to the fact that Joccy has tampered with the list. She uses a binary search algorithm to find Joccy's test, where she looks at the test in the middle of the pile. If the test is not Joccy's, she binary searches the top half of the list if the test appears after Joccy's name when arranged alphabetically, or the bottom half of the list otherwise. The probability that Maggie finds Joccy's test can be expressed as $\frac{p}{q}$. Compute $p+q$.
[i]2022 CCA Math Bonanza Team Round #5[/i]
Let be a nontrivial finite ring having the property that any element of it has an even power that is equal to itself. Prove that
[b]a)[/b] the order of the ring is a power of $ 2. $
[b]b)[/b] the sum of all elements of the ring is $ 0. $
Let $ABCD$ be a cyclic quadrilateral and triangles $ACD, BCD$ are acute. Suppose that the lines $AB$ and $CD$ meet at $S$. Denote by $E$ the intersection of $AC, BD$. The circles $(ADE)$ and $(BC E)$ meet again at $F$.
a) Prove that $SF \perp EF.$
b) The point $G$ is taken out side of the quadrilateral $ABCD$ such that triangle $GAB$ and $FDC$ are similar. Prove that $GA+ FB = GB + FA$
The operation $*$ is defined by the following: $a*b=a!-ab-b.$ Compute the value of $5*8.$
[i]2015 CCA Math Bonanza Individual Round #2[/i]
The figure shows a $2\times 2$ grid that has been filled with the numbers $a, b, c$, and $d$.
We say that this grid is [i]ordered[/i] if it is true that $a > b > c > d$ or that $a > d > c > b$.
$\begin{tabular}{|l|l|}
\hline
a & b \\ \hline
d & c \\ \hline
\end{tabular}$
In how many ways can the numbers from $1$ to $1000$ be arranged in the cells of a $2 \times 500$ grid ($2$ rows and $500$ columns) so that each $2 \times 2$ sub-grid is ordered?
In triangle $ABC$, $M,N$ and $P$ are midpoints of sides $BC,CA$ and $AB$. Point $K$ lies on segment $NP$ so that $AK$ bisects $\angle BKC$. Lines $MN,BK$ intersects at $E$ and lines $MP,CK$ intersects at $F$. Suppose that $H$ be the foot of perpendicular line from $A$ to $BC$ and $L$ the second intersection of circumcircle of triangles $AKH, HEF$. Prove that $MK,EF$ and $HL$ are concurrent.
[i]Proposed by Alireza Dadgarnia[/i]
[b]p1.[/b] Find all integer solutions of the equation $a^2 - b^2 = 2002$.
[b]p2.[/b] Prove that the disks drawn on the sides of a convex quadrilateral as on diameters cover this quadrilateral.
[b]p3.[/b] $30$ students from one school came to Mathematical Olympiad. In how many different ways is it possible to place them in four rooms?
[b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let
$$A = \{2, 4, \ldots, 1000\},$$
$$B = \{3, 6, \ldots, 999\},$$
$$C = \{5, 10, \ldots, 1000\},$$
$$D = \{7, 14, \ldots, 994\},$$
$$E = \{11, 22, \ldots, 990\},$$
$$\textrm{and } F = \{13, 26, \ldots, 988\}.$$
Find the number of elements in the set $(((((A\cup B)\cap C)\cup D)\cap E)\cup F)$.
[i]2022 CCA Math Bonanza Individual Round #7[/i]
Point $ P$ is inside equilateral $ \triangle ABC$. Points $ Q, R$ and $ S$ are the feet of the perpendiculars from $ P$ to $ \overline{AB}, \overline{BC}$, and $ \overline{CA}$, respectively. Given that $ PQ \equal{} 1, PR \equal{} 2$, and $ PS \equal{} 3$, what is $ AB$?
$ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 3 \sqrt {3}\qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 4 \sqrt {3}\qquad \textbf{(E)}\ 9$
A regular hexagon and an equilateral triangle have equal perimeter.
If the area of the triangle is $4\sqrt3$ square units, the area of the hexagon is
(A): $5\sqrt3$, (B): $6\sqrt3$, (C): $7\sqrt3$, (D): $8\sqrt3$, (E): None of the above.
The set $\{1, 2, 3, 4\}$ can be partitioned into two subsets $A = \{1, 4\}$ and $B = \{3, 2\}$ with no common elements and such that the sum of the elements of $A$ is equal to the sum of the elements of B. Such a partition is impossible for the set $\{1, 2, 3, 4, 5\}$ and also for the set $\{1, 2, 3, 4, 5, 6\}$.
Determine all values of $n$ for which the set of the first $n$ natural numbers can be partitioned into two subsets with no common elements such that the sum of the elements of each subset is the same.
What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with \[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\]
In the hexagon $ABCDEF$, all angles are equally large. The side lengths satisfy $AB = CD = EF = 3$ and $BC = DE = F A = 2$. The diagonals $AD$ and $CF$ intersect each other in the point $G$. The point $H$ lies on the side $CD$ so that $DH = 1$. Prove that triangle $EGH$ is equilateral.
Find the smallest positive integer $k$ such that $ \underbrace{11\cdots 11}_{k\text{ 1's}}$ is divisible by $9999$.
Let $T = TNYWR$. Circles $\omega_1$ and $\omega_2$ intersect at $P$ and $Q$. The common external tangent $\ell$ to the two circles closer to $Q$ touches $\omega_1$ and $\omega_2$ at $A$ and $B$ respectively. Line $AQ$ intersects $\omega_2$ at $X$ while $BQ$ intersects $\omega_1$ again at $Y$. Let $M$ and $N$ denote the midpoints of $\overline{AY}$ and $\overline{BX}$, also respectively. If $AQ=\sqrt{T}$, $BQ=7$, and $AB=8$, then find the length of $MN$.
Let $n$ be a positive integer. An equilateral triangle with side $n$ will be denoted by $T_n$ and is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two).
Let also $m$ be a positive integer with $m<n$ and suppose that $T_n$ and $T_m$ can be tiled with "trapezoids".
Prove that, if from $T_n$ we remove a $T_m$ with the same orientation, then the rest can be tiled with "trapezoids".
The four circumcircles of the four faces of a tetrahedron have equal radii. Prove that the four faces of the tetrahedron are congruent triangles.
Let $\omega$ be the circumcircle of isosceles triangle $ABC$ ($AB=AC$). Points $P$ and $Q$ lie on $\omega$ and $BC$ respectively such that $AP=AQ$ .$AP$ and $BC$ intersect at $R$. Prove that the tangents from $B$ and $C$ to the incircle of $\triangle AQR$ (different from $BC$) are concurrent on $\omega$.
[i]Proposed by Ali Zamani, Hooman Fattahi[/i]
Let $m,\ n$ be positive integer such that $2\leq m<n$.
(1) Prove the inequality as follows.
\[\frac{n+1-m}{m(n+1)}<\frac{1}{m^2}+\frac{1}{(m+1)^2}+\cdots +\frac{1}{(n-1)^2}+\frac{1}{n^2}<\frac{n+1-m}{n(m-1)}\]
(2) Prove the inequality as follows.
\[\frac 32\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq 2\]
(3) Prove the inequality which is made precisely in comparison with the inequality in (2) as follows.
\[\frac {29}{18}\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq \frac{61}{36}\]
Ten different odd primes are given. Is it possible that for any two of them, the difference of their sixteenth powers to be divisible by all the remaining ones ?
Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.
$x_1,x_2,...,x_n>1$ are natural numbers and $n \geq 3$
Prove that : $(x_1x_2...x_n)^2 \ne x_1^3 + x_2^3 +...+x_n^3$