Found problems: 85335
Call a natural number $n{}$ [i]interesting[/i] if any natural number not exceeding $n{}$ can be represented as the sum of several (possibly one) pairwise distinct positive divisors of $n{}$.
[list=a]
[*]Find the largest three-digit interesting number.
[*]Prove that there are arbitrarily large interesting numbers other than the powers of two.
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[i]Proposed by N. Agakhanov[/i]
Compute the number of values of $x$ in the interval $[-11\pi,-2\pi]$ that satisfy $\tfrac{5\cos(x)+4}{5\sin(x)+3}=0$.
$\overline{AB}$ is a diameter of a circle. Tangents $\overline{AD}$ and $\overline{BC}$ are drawn so that $\overline{AC}$ and $\overline{BD}$ intersect in a point on the circle. If $\overline{AD}=a$ and $\overline{BD}=b$, $a \not= b$, the diameter of the circle is:
$\textbf{(A)}\ |a-b|\qquad
\textbf{(B)}\ \frac{1}{2}(a+b)\qquad
\textbf{(C)}\ \sqrt{ab} \qquad
\textbf{(D)}\ \frac{ab}{a+b}\qquad
\textbf{(E)}\ \frac{1}{2}\frac{ab}{a+b}$
Let $ z = \frac{1}{2}(\sqrt{2} + i\sqrt{2}) $. The sum $$ \sum_{k = 0}^{13} \dfrac{1}{1 - ze^{k \cdot \frac{i\pi}{7}}} $$
can be written in the form $ a - bi $. Find $ a + b $.
Let $I$ be the centre of the inscribed circle of the non-isosceles triangle $ABC$, and let the circle touch the sides $BC, CA, AB$ at the points $A_1, B_1, C_1$ respectively. Prove that the centres of the circumcircles of $\vartriangle AIA_1,\vartriangle BIB_1$ and $\vartriangle CIC_1$ are collinear.
Suppose you have $27$ identical unit cubes colored such that $3$ faces adjacent to a vertex are red and the other $3$ are colored blue. Suppose further that you assemble these $27$ cubes randomly into a larger cube with $3$ cubes to an edge (in particular, the orientation of each cube is random). The probability that the entire cube is one solid color can be written as $\frac{1}{2^n}$ for some positive integer $n$. Find $n$.
Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$.
Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$
A graph $G$ with $n$ vertex is called [i]good [/i] if every vertex could be labelled with distinct positive integers which are less than or equal $\lfloor \frac{n^2}{4} \rfloor$ such that there exists a set of nonnegative integers $D$ with the following property: there exists an edge between $2$ vertices if and only if the difference of their labels is in $D$.
Show that there exists a positive integer $N$ such that for every $n \ge N$, there exist a not-good graph with $n$ vertices.
An acute-angled unequal triangle $ABC$ is drawn with its circumcircle and circumcenter $O$. The incenter $I$ is also marked. Using only a ruler (without divisions), construct the symedian (a line symmetrical to the median relative to the corresponding bisector) of the triangle, drawing no more than four lines.
Carter and Vivian decide to spend their afternoon listing pairs of real numbers, $(a, b).$ Carter wants to find all $(a, b)$ such that $(a, b)$ lie within a circle of radius $6$ centered at $(6, 6).$ Vivian hates circles and would rather find all $(a, b)$ such that $a,$ $b,$ and $6$ can be the side lengths of a triangle. If Carter randomly chooses an $(a, b)$ that satisfies his conditions, then the probability that the pair also satisfies Vivian's conditions can be written in the form $\tfrac{p}{q} + \tfrac{r}{s\pi},$ where $p,$ $q,$ $r,$ and $s$ are positive integers, $p$ and $q$ are relatively prime, and $r$ and $s$ are relatively prime. Find $p + q + r + s.$
Let $A$ be a real $n \times n$ matrix and suppose that for every positive integer $m$ there exists a real symmetric matrix $B$ such that
$$2021B = A^m+B^2.$$
Prove that $|\text{det} A| \leq 1$.
Let $E$ be the set of $n^2+1$ closed intervals on the real axis. Prove that there exists a subset of $n+1$ intervals that are monotonically increasing with respect to inclusion, or a subset of $n+1$ intervals none of which contains any other interval from the subset.
Prove that for arbitary positive integer $ n\geq 4$, there exists a permutation of the subsets that contain at least two elements of the set $ G_{n} \equal{} \{1,2,3,\cdots,n\}$: $ P_{1},P_{2},\cdots,P_{2^n \minus{} n \minus{} 1}$ such that $ |P_{i}\cap P_{i \plus{} 1}| \equal{} 2,i \equal{} 1,2,\cdots,2^n \minus{} n \minus{} 2.$
Let $d_1, d_2, ... , d_k$ be the positive divisors of $n = 1990!$. Show that $\sum \frac{d_i}{\sqrt{n}} = \sum \frac{\sqrt{n}}{d_i}$.
Find all real $x$ that satisfy the equation $$\frac{1}{x+1}+\frac{1}{x+2}=\frac{1}{x}$$
[i]2015 CCA Math Bonanza Lightning Round #2.2[/i]
For a positive integer $n$, if $n$ is a product of two different primes and $n \equiv 2 \pmod 3$, then $n$ is called "special number." For example, $14, 26, 35, 38$ is only special numbers among positive integers $1$ to $50$. Prove that for any finite set $S$ with special numbers, there exist two sets $A, B$ such that
[list]
[*] $A \cap B = \emptyset, A \cup B = S$
[*] $||A| - |B|| \leq 1$
[*] For all primes $p$, the difference between number of elements in $A$ which is multiple of $p$ and number of elements in $B$ which is multiple of $p$ is less than or equal to $1$.
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An ant traverses between vertices on a unit cube such that at each vertex, it uniformly at random chooses an adjacent vertex to travel to. What is the expected distance travelled by the ant until it returns to its starting vertex?
A rectangular parallelepiped has dimensions $a,b,c$ that satisfy the relation $3a+4b+10c=500$, and the length of the main diagonal $20\sqrt5$. Find the volume and the total area of the surface of the parallelepiped.
Are there infinitely many different natural numbers $a_1,a_2, a_3,...$ so that for every integer $k$ only finitely many of the numbers $a_1 + k$,$a_2 + k$,$a_3 + k$,$...$ are numbers prime?
Square $CASH$ and regular pentagon $MONEY$ are both inscribed in a circle. Given that they do not share a vertex, how many intersections do these two polygons have?
On a semicircle with diameter $AB$ and centre $S$, points $C$ and $D$ are given such that point $C$ belongs to arc $AD$. Suppose $\angle CSD = 120^\circ$. Let $E$ be the point of intersection of the straight lines $AC$ and $BD$ and $F$ the point of intersection of the straight lines $AD$ and $BC$. Prove that $EF=\sqrt{3}AB$.
Show that for every integer $n \geq 2$ the number
$$a=n^{5n-1}+n^{5n-2}+n^{5n-3}+n+1$$
is a composite number.
Call a positive integer $m$ $\textit{good}$ if there exist integers $a, b, c$ satisfying $m=a^3+2b^3+4c^3-6abc$. Show that there exists a positive integer $n<2024$, such that for infinitely many primes $p$, the number $np$ is $\textit{good}$.
Given a quadrilateral, the midpoints $A, B, C, D$ of its consecutive sides, and the midpoints of its diagonals, $P$ and $Q$. Prove that $\vartriangle BCP = \vartriangle ADQ$.
In table
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC9hLzBjNjFlZWFjM2ZlOTQzMTk2YTdkMzQ2MjJiYzYyMWFlN2Y0ZGZlLnBuZw==&rn=dGFibGljYWEucG5n[/img]
$10$ numbers are circled, in every row and every column exactly one. Prove that among them, there are at least two equal