Found problems: 85335
Jo claims that any two triangles, both having a perimeter of four, are congruent. Jann claims that two circles, both having a circumference of $4\pi$, are congruent. Julia claims that two squares, both having a perimeter of four, are congruent. Which of these students are correct?
$\textbf{(A) }\text{Jo}\qquad\textbf{(B) }\text{Jann}\qquad\textbf{(C) }\text{Julia}\qquad\textbf{(D) }\text{Jo, Julia}\qquad\textbf{(E) }\text{Jann, Julia}$
How many sequences of words (not necessarily grammatically correct) have the property that the first word has one letter, each word can be obtained by inserting a letter somewhere in the previous word, and the final word is CCAMT? Here are examples of possible sequences:
[center]
C,CA,CAM,CCAM,CCAMT.
[/center]
[center]
A,AT,CAT,CAMT,CCAMT.
[/center]
[i]2021 CCA Math Bonanza Team Round #1[/i]
In the mathematical talent show called “The $X^2$-factor” contestants are scored by a a panel of $8$ judges. Each judge awards a score of $0$ (‘fail’), $X$ (‘pass’), or $X^2$ (‘pass with distinction’). Three of the contestants were Ann, Barbara and David. Ann was awarded the same score as Barbara by exactly $4$ of the judges. David declares that he obtained different scores to Ann from at least $4$ of the judges, and also that he obtained different scores to Barbara from at least $4$ judges.
In how many ways could scores have been allocated to David, assuming he is telling the truth?
A convex 51-gon is given. For each of its vertices and each diagonal that does not contain this vertex, we mark in red a point symmetrical to the vertex relative to the middle of the diagonal. Prove that strictly inside the polygon there are no more than 20400 red dots.
[i]Proposed by P. Kozhevnikov[/i]
Let $a$ and $b$ be the solutions to $x^2-7x+17=0$. Compute $a^4+b^4$.
A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation
$$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$
Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.
Let $ABCD$ be a convex quadrilateral and let $P$ be the intersection of the diagonals $AC$ and $BD$. The radii of the circles inscribed in the triangles $\vartriangle ABP$, $\vartriangle BCP$, $\vartriangle CDP$ and $\vartriangle DAP$ are the same. Prove that $ABCD$ is a rhombus,
Determine the maximal area triangle such that all of its vertices satisfy $\frac{x^2}{9} + \frac{y^2}{16} = 1$.
Kostya drives a car from a village to a city, driving along three roads. Moreover, on each of these roads, he drives at a constant speed. Is it possible that a third of the distance traveled was completed earlier than a third of the time, half of the distance traveled later than half of the time, and two-thirds of the distance was earlier than two-thirds of the time?
Denote by $\mathbb{S}$ the set of all proper subsets of $\mathbb{Z}_{>0}$. Find all functions $f : \mathbb{S} \mapsto \mathbb{Z}_{>0}$ that satisfy the following:\\
[color=#FFFFFF]___[/color]1. For all sets $A, B \in \mathbb{S}$ we have \[f(A \cap B) = \text{min}(f(A), f(B)).\]
[color=#FFFFFF]___[/color]2. For all positive integers $n$ we have \[\sum \limits_{X \subseteq [1, n]}
f(X) = 2^{n+1}-1.\]
(Here, by a proper subset $X$ of $\mathbb{Z}_{>0}$ we mean $X \subset \mathbb{Z}_{>0}$ with $X \ne \mathbb{Z}_{>0}$. It is allowed for $X$ to have infinite size.) \\
[i]Proposed by MV Adhitya, Kanav Talwar, Siddharth Choppara, Archit Manas[/i]
A 3x3 magic square, with magic number $m$, is a $3\times 3$ matrix such that the entries on each row, each column and each diagonal sum to $m$. Show that if the square has positive integer entries, then $m$ is divisible by $3$, and each entry of the square is at most $2n-1$, where $m=3n$. An example of a magic square with $m=6$ is
\[\left( \begin{array}{ccccc}
2 & 1 & 3\\
3 & 2 & 1\\
1 & 3 & 2
\end{array} \right)\]
(1) Evaluate $ \int_1^{3\sqrt{3}} \left(\frac{1}{\sqrt[3]{x^2}}\minus{}\frac{1}{1\plus{}\sqrt[3]{x^2}}\right)\ dx.$
(2) Find the positive real numbers $ a,\ b$ such that for $ t>1,$ $ \lim_{t\rightarrow \infty} \left(\int_1^t \frac{1}{1\plus{}\sqrt[3]{x^2}}\ dx\minus{}at^b\right)$ converges.
each cells in a $9\times 9 $ grid is painted either blue or red.two cells are called [i]diagonal neighbors[/i] if their intersection is exactly a point.show that some cell has exactly two red neighbors,or exactly two blue neighbors, or both.
An occasionally unreliable professor has devoted his last book to a certain binary operation $*$. When this operation is applied to any two integers, the result is again an integer. The operation is known to satisfy the following axioms:
$\text{a})\ x*(x*y)=y$ for all $x,y\in\mathbb{Z}$;
$\text{b})\ (x*y)*y=x$ for all $x,y\in\mathbb{Z}$.
The professor claims in his book that
$1.$ The operation $*$ is commutative: $x*y=y*x$ for all $x,y\in\mathbb{Z}$.
$2.$ The operation $*$ is associative: $(x*y)*z=x*(y*z)$ for all $x,y,z\in\mathbb{Z}$.
Which of these claims follow from the stated axioms?
For triangle $ABC$, $P$ and $Q$ satisfy $\angle BPA + \angle AQC=90^{\circ}$. It is provided that the vertices of the triangle $BAP$ and $ACQ$ are ordered counterclockwise(or clockwise). Let the intersection of the circumcircles of the two triangles be $N$ ($A \neq N$, however if $A$ is the only intersection $A=N$), and the midpoint of segment $BC$ be $M$. Show that the length of $MN$ does not depend on $P$ and $Q$.
Let $W$ be a fixed positive integer. Let $S$ be the set of all pairs $(a, b)$ of positive integers such that $a \neq b$. For each $(a, b) \in S$, let $m(a,b)$ be the largest integer satisfying
\[
m(a, b) \leq \frac{1 + na}{1 + nb}
\]
for all integers $n \geq 1$.
(a) For each $(a, b) \in S$, prove that there exists a positive integer $k$ such that
\[
m(a,b) \leq \frac{1 + na}{W + nb}
\]
for all $n \geq k$.
(b) For each $(a, b) \in S$, let $k(a,b)$ be the smallest value of $k$ that satisfies the condition of part (a). Determine $\max \{k(a,b) \mid (a,b) \in S \}$ or prove that it does not exist.
Real numbers $X_1, X_2, \dots, X_{10}$ are chosen uniformly at random from the interval $[0,1]$. If the expected value of $\min(X_1,X_2,\dots, X_{10})^4$ can be expressed as a rational number $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, what is $m+n$?
[i]2016 CCA Math Bonanza Lightning #4.4[/i]
How many words with $n$ digits can be formed from the alphabet $\{0, 1, 2, 3, 4\}$, if neighboring digits must differ by exactly one?
[i]Proposed by Germany, FR.[/i]
Given a quadrilateral \(ABCD\), point \(M\) is the midpoint of the side \(CD\). It turns out that \(\angle BMA = 90^{\circ}\) and \(\angle MAB = \angle CBD\). Prove that \(AC = AB\).
[i]Proposed by Anton Trygub[/i]
Let $n$ be an integer. We consider $s (n)$, the sum of the $2001$ powers of $n$ with the exponents $0$ to $2000$. So $s (n) = \sum_{k=0}^{2000}n ^k$ . What is the unit digit of $s (n)$ in the decimal system?
Take $k\in \mathbb{Z}_{\ge 1}$ and the sets $A_1,A_2,\dots, A_k$ consisting of $x_1,x_2,\dots ,x_k$ positive integers, respectively. For any two sets $A$ and $B$, define $A+B=\{a+b~|~a\in A,~b\in B\}$.
Find the least and greatest number of elements the set $A_1+A_2+\dots +A_k$ may have.
[i] (Andrei Bâra)[/i]
Let us call "absolutely prime" the prime number, if having transposed its digits in an arbitrary order, we obtain prime number again. Prove that its notation cannot contain more than three different digits.
Big Bird has a polynomial $P$ with integer coefficients such that $n$ divides $P(2^n)$ for every positive integer $n$. Prove that Big Bird's polynomial must be the zero polynomial.
[i]Ashwin Sah[/i]
A point $C$ lies on a line segment $AB$ between $A$ and $B$ and circles are drawn having $AC$ and $CB$ as diameters. A common tangent to both circles touches the circle with $AC$ as diameter at $P \ne C$ and the circle with $CB$ as diameter at $Q \ne C$.
Prove that $AP, BQ$ and the common tangent to both circles at $C$ all meet at a single point which lies on the circumference of the circle with $AB$ as diameter.
Let $a,b$, and $c$ be odd positive integers such that $a$ is not a perfect square and
$$a^2+a+1 = 3(b^2+b+1)(c^2+c+1).$$
Prove that at least one of the numbers $b^2+b+1$ and $c^2+c+1$ is composite.