Found problems: 85335
Farmer Boso has a busy farm with lots of animals. He tends to $5b$ cows, $5a +7$ chickens, and $b^{a-5}$ insects. Note that each insect has $6$ legs. The number of cows is equal to the number of insects. The total number of legs present amongst his animals can be expressed as $\overline{LLL }+1$, where $L$ stands for a digit. Find $L$.
Find the graph of the function $y=x-|x+x^2|$
Define a number to be $boring$ if all the digits of the number are the same. How many positive integers less than $10000$ are both prime and boring?
Let $u_{jk}$ be a real number for each $j=1,2,3$ and each $k=1,2$ and let $N$ be an integer such that
\[\max_{1\le k \le 2} \sum_{j=1}^3 |u_{jk}| \leq N\]
Let $M$ and $l$ be positive integers such that $l^2 <(M+1)^3$. Prove that there exist integers $\xi_1,\xi_2,\xi_3$ not all zero, such that
\[\max_{1\le j \le 3}\xi_j \le M\ \ \ \ \text{and} \ \ \ \left|\sum_{j=1}^3 u_{jk}\xi_k\right| \le \frac{MN}{l} \ \ \ \ \text{for k=1,2}\]
A sequence $\left(a_n\right)$ with $a_1 = 1$ satisfies the following recursion: In the decimal expansion of $a_n$ (without trailing zeros) let $k$ be the smallest digest then $a_{n+1} = a_n + 2^k.$ How many digits does $a_{9 \cdot 10^{2010}}$ have in the decimal expansion?
For every $ n \ge 2$, $ a_n \equal{} \sqrt [3]{n^3 \plus{} n^2 \minus{} n \minus{} 1}/n$. What is the least value of positive integer $ k$ satisfying $ a_2a_3\cdots a_k > 3$ ?
$\textbf{(A)}\ 100 \qquad\textbf{(B)}\ 102 \qquad\textbf{(C)}\ 104 \qquad\textbf{(D)}\ 106 \qquad\textbf{(E)}\ \text{None}$
Let $p$ and $q$ be odd prime numbers. Assume that there exists a positive integer $n$ such that $pq-1= n^3$. Express $p+q$ in terms of $n$
Let $ABCD$ be a rhombus such that $m(\widehat{ABC}) = 40^\circ$. Let $E$ be the midpoint of $[BC]$ and $F$ be the foot of the perpendicular from $A$ to $DE$. What is $m(\widehat{DFC})$?
$
\textbf{a)}\ 100^\circ
\qquad\textbf{b)}\ 110^\circ
\qquad\textbf{c)}\ 115^\circ
\qquad\textbf{d)}\ 120^\circ
\qquad\textbf{e)}\ 135^\circ
$
Let $a_1, a_2,...,a_{2n}$ be positive real numbers such that $a_i + a_{n+i} = 1$, for all $i = 1,...,n$. Prove that there exist two different integers $1 \le j, k \le 2n$ for which $$\sqrt{a^2_j-a^2_k} < \frac{1}{\sqrt{n} +\sqrt{n - 1}}$$
A sequence $\{a_n\}$ is defined such that $a_i=i$ for $i=1,2,3\ldots,2020$ and for $i>2020$, $a_i$ is the average of the previous $2020$ terms. What is the largest integer less than or equal to $\displaystyle\lim_{n\to\infty}a_n$?
[i]2020 CCA Math Bonanza Lightning Round #4.4[/i]
Consider a circle of radius $4$ with center $O_1$, a circle of radius $2$ with center $O_2$ that lies on the circumference of circle $O_1$, and a circle of radius $1$ with center $O_3$ that lies on the circumference of circle $O_2$. The centers of the circle are collinear in the order $O_1$, $O_2$, $O_3$. Let $A$ be a point of intersection of circles $O_1$ and $O_2$ and $B$ be a point of intersection of circles $O_2$ and $O_3$ such that $A$ and $B$ lie on the same semicircle of $O_2$. Compute the length of $AB$.
Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get?
$\textbf{(A)}\ 200 \qquad \textbf{(B)}\ 202 \qquad \textbf{(C)}\ 220 \qquad \textbf{(D)}\ 380 \qquad \textbf{(E)}\ 398$
Let e>0. Prove that for a large enough natural n, there exist natural x,y,z st $n^2+x^2=y^2+z^2$ and $y,z\leq \frac{(1+e)n}{\sqrt{2}}$.
Let $ABC$ be a triangle such that $ABC$ isn't a isosceles triangle. $(I)$ is incircle of triangle touches $BC,CA,AB$ at $D,E,F$ respectively. The line through $E$ perpendicular to $BI$ cuts $(I)$ again at $K$. The line through $F$ perpendicular to $CI$ cuts $(I)$ again at $L$.$J$ is midpoint of $KL$.
[b]a)[/b] Prove that $D,I,J$ collinear.
[b]b)[/b] $B,C$ are fixed points,$A$ is moved point such that $\frac{AB}{AC}=k$ with $k$ is constant.$IE,IF$ cut $(I)$ again at $M,N$ respectively.$MN$ cuts $IB,IC$ at $P,Q$ respectively. Prove that bisector perpendicular of $PQ$ through a fixed point.
Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.
There is a pile with 2022 rocks. Ana y Beto play by turns to the following game, starting with Ana: in each turn, if there are $n$ rocks in the pile, the player can remove $S(n)$ rocks or $n-S(n)$ rocks, where $S(n)$ is the sum of the the digits of $n$. The person who removes the last rock wins. Determine which of the two players has a winning strategy and describe it.
On an escalator which is not moving, a person descends faster than he ascends. Is it faster for this person to descend and ascend once on an upward-moving escalator, or to descend and ascend once on a downward-moving escalator? (It is assumed that all the speeds mentioned here are constant, that the speed of the escalator is the same no matter if it is moving up or down and that the speed of the person relative to the escalator is always greater than the speed of the escalator.)
(Folklore)
One of the factors of $ x^4\plus{}4$ is:
$ \textbf{(A)}\ x^2\plus{}2 \qquad\textbf{(B)}\ x\plus{}1 \qquad\textbf{(C)}\ x^2\minus{}2x\plus{}2 \qquad\textbf{(D)}\ x^2\minus{}4\\
\textbf{(E)}\ \text{none of these}$
Find all integer solutions of \[ x^{4}\plus{}y^{2}\equal{}z^{4}\]
A triangle $\vartriangle ABC$ has $\angle B = 90^o$ . A circle is tangent to $AB$ at $B$ and also tangent to $AC$. Another circle is tangent to the first circle as well as the two sides $AB$ and $AC$. Suppose that $AB =\sqrt3$ and $BC = 3$. What is the radius of the second circle?
A $1 \times 5n$ rectangle is partitioned into tiles, each of the tile being either a separate $1 \times 1$ square or a broken domino consisting of two such squares separated by four squares (not belonging to the domino). Prove that the number of such partitions is a perfect fifth power.
[i](K. Kokhas)[/i]
Find the largest integer $n$ which equals the product of its leading digit and the sum of its digits.
Find all differentiable functions $ f:(0,\infty )\longrightarrow (-\infty ,\infty ) $ having the property that
$$ f'(\sqrt{x}) =\frac{1+x+x^2}{1+x} , $$
for any positive real numbers $ x. $
[i]Ovidiu Țâțan[/i]
A natural number $n$ is given. Determine all $(n - 1)$-tuples of nonnegative integers $a_1, a_2, ..., a_{n - 1}$ such that
$$\lfloor \frac{m}{2^n - 1}\rfloor + \lfloor \frac{2m + a_1}{2^n - 1}\rfloor + \lfloor \frac{2^2m + a_2}{2^n - 1}\rfloor + \lfloor \frac{2^3m + a_3}{2^n - 1}\rfloor + ... + \lfloor \frac{2^{n - 1}m + a_{n - 1}}{2^n - 1}\rfloor = m$$
holds for all $m \in \mathbb{Z}$.
$S$ is a $(k+1) \times (k+1)$ array of lattice points. How many squares have their vertices in $S$?