Found problems: 85335
Denote by $d(n)$ the number of positive divisors of a positive integer $n$. Find the smallest constant $c$ for which $d(n)\le c\sqrt n$ holds for all positive integers $n$.
$x,y,z,a,b,c$ are positive integers that satisfy $xy \equiv a \pmod z$, $yz \equiv b \pmod x$, $zx \equiv c \pmod y$. Prove that
$\min{\{x,y,z\}} \le ab+bc+ca$.
Let $ABC$ be a triangle with $AB=5$, $BC=7$, $CA=8$, and let $M$ be the midpoint of $BC$. Points $P$ and $Q$ are chosen on the circumcircle of $ABC$ such that $MPQ$ and $ABC$ are similar (with vertices in that order). The product of all different possible areas of $MPQ$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
[i]2021 CCA Math Bonanza Lightning Round #4.4[/i]
Real numbers $ a$ and $ b$ satisfy the equations $ 3^{a} \equal{} 81^{b \plus{} 2}$ and $ 125^{b} \equal{} 5^{a \minus{} 3}$. What is $ ab$?
$ \textbf{(A)} \minus{} \!60 \qquad \textbf{(B)} \minus{} \!17 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 60$
Let $n\geq 2$ be a given integer. Initially, we write $n$ sets on the blackboard and do a sequence of moves as follows: choose two sets $A$ and $B$ on the blackboard such that none of them is a subset of the other, and replace $A$ and $B$ by $A\cap B$ and $A\cup B$. This is called a $\textit{move}$.
Find the maximum number of moves in a sequence for all possible initial sets.
Find all positive integers $n$ such that $n^3$ is the product of all divisors of $n$.
In an isosceles triangle, we drew one of the angle bisectors. At least one of the resulting two smaller ones triangles is similar to the original. What can be the leg of the original triangle if the length of its base is $1$ unit?
Assume every side length of a triangle $ABC$ is more than $2$ and two of its angles are given by $\angle ABC = 57^\circ$ and $ACB = 63^\circ$. Point $P$ is chosen on side $BC$ with $BP:PC = 2:1$. Points $M,N$ are chosen on sides $AB$ and $AC$, respectively so that $BM = 2$ and $CN = 1$. Let $Q$ be the point on segment $MN$ for which $MQ:QN = 2:1$. Find the value of $PQ$. Your answer must be in simplest form.
If a triangle has three altitudes of lengths $6, 6, \text{and} 6,$ what is its perimeter?
Kevin has a square piece of paper with creases drawn to split the paper in half in both directions, and then each of the four small formed squares diagonal creases drawn, as shown below.
[img]https://cdn.artofproblemsolving.com/attachments/2/2/70d6c54e86856af3a977265a8054fd9b0444b0.png[/img]
Find the sum of the corresponding numerical values of figures below that Kevin can create by folding the above piece
of paper along the creases. (The figures are to scale.) Kevin cannot cut the paper or rip it in any way.
[img]https://cdn.artofproblemsolving.com/attachments/a/c/e0e62a743c00d35b9e6e2f702106016b9e7872.png[/img]
$A,B,C,D,E$ are points on a circle $O$ with radius equal to $r$. Chords $AB$ and $DE$ are parallel to each other and have length equal to $x$. Diagonals $AC,AD,BE, CE$ are drawn. If segment $XY$ on $O$ meets $AC$ at $X$ and $EC$ at $Y$ , prove that lines $BX$ and $DY$ meet at $Z$ on the circle.
Real numbers are placed at the vertices of an $n{}$-gon. On each side, we write the sum of the numbers on its endpoints. For which $n{}$ is it possible that the numbers on the sides form a permutation of $1, 2, 3,\ldots , n$?
[i]From the folklore[/i]
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]
Given is a finite set of points $M$ and an equilateral triangle $\Delta$ in the plane. It is known that for any subset $M' \subset M$, which has no more than $9$ points, can be covered by two translations of the triangle $\Delta$. Prove that the entire set $M$ can be covered by two translations of $\Delta$.
Find how many are the numbers of $2013$ digits $d_1d_2…d_{2013}$ with odd digits $d_1,d_2,…,d_{2013}$ such that the sum of $1809$ terms $$d_1 \cdot d_2+d_2\cdot d_3+…+d_{1809}\cdot d_{1810}$$ has remainder $1$ when divided by $4$ and the sum of $203$ terms $$d_{1810}\cdot d_{1811}+d_{1811}\cdot d_{1812}+…+d_{2012}\cdot d_{2013}$$ has remainder $1$ when dividing by $4$.
For a given integer $n \ge 4$ we examine whether there exists a table with three rows and $n$ columns which can be filled by the numbers $1, 2,...,, 3n$ such that
$\bullet$ each row totals to the same sum $z$ and
$\bullet$ each column totals to the same sum $s$.
Prove:
(a) If $n$ is even, such a table does not exist.
(b) If $n = 5$, such a table does exist.
(Gerhard J. Woeginger)
To a sheet of paper, we glue $2011$ “handles” that do not intersect, that is, strips of paper glued to the sheet in your ends. No handle can be twisted. Prove that the surface boundary thus formed has at least two cycles (closed curves). That is, an ant that only walks along the edge of the paper never runs through the entire surface boundary.
For example, the configuration represented in the figure has three cycles: one in dashed lines, one in lines dotted lines and another in a continuous line (this cycle passes under a tab twice).
[img]https://cdn.artofproblemsolving.com/attachments/f/e/121146a240215f241278b3aabde13a67544e7a.png[/img]
Consider the parallelogram $ABCD$. A circle is drawn that passes through $A$ and intersects side $AD$ at $N$, side $AB$ at $M$ and diagonal $AC$ in $P$ such that points $A, M, N, P$ are different. Prove that $$AP\cdot AC = AM \cdot AB + AN \cdot AD.$$
For each positive integer $n$ determine the maximum number of points in space creating the set $A$ which has the following properties:
$1)$ the coordinates of every point from the set $A$ are integers from the range $[0, n]$
$2)$ for each pair of different points $(x_1,x_2,x_3), (y_1,y_2,y_3)$ belonging to the set $A$ it is satisfied at least one of the following inequalities $x_1< y_1, x_2<y_2, x_3<y_3$ and at least one of the following inequalities $x_1>y_1, x_2>y_2,x_3>y_3$.
Compute the largest root of $x^4-x^3-5x^2+2x+6$.
Let $A$ be a subgroup of the symmetric group $S_n$, and $G$ be a normal subgroup of $A$. Show that if $G$ is transitive, then $|A\colon G|\le 5^{n-1}$
(translated by Miklós Maróti)
Let $AB$ be a diameter of a circle and let $C$ be a point on $AB$ with $2\cdot AC=BC$. Let $D$ and $E$ be points on the circle such that $DC\perp AB$ and $DE$ is a second diameter. What is the ratio of the area of $\triangle{DCE}$ to the area of $\triangle{ABD}$?
Crisp All, a basketball player, is [i]dropping dimes[/i] and nickels on a number line. Crisp drops a dime on every positive multiple of $10$, and a nickel on every multiple of $5$ that is not a multiple of $10$. Crisp then starts at $0$. Every second, he has a $\frac{2}{3}$ chance of jumping from his current location $x$ to $x+3$, and a $\frac{1}{3}$ chance of jumping from his current location $x$ to $x+7$. When Crisp jumps on either a dime or a nickel, he stops jumping. What is the probability that Crisp [i]stops on a dime[/i]?
For all integral values of parameter $t$, find all integral solutions $(x,y)$ of the equation
$$ y^2 = x^4-22x^3+43x^2+858x+t^2+10452(t+39)$$ .
Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible.
[i]Proposed by Horst Sewerin, Germany[/i]