Prove that for every $ n \in \mathbb{N}^*$ exists a multiple of $ n$, having sum of digits equal to $ n$.

In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. If the area of triangle $DOC$ is $4$ and the area of triangle $AOB$ is $36$, compute the minimum possible value of the area of $ABCD$.

Let $a$ and $b$ be two positive real numbers such that $a^{2007} = a + 1$ and $b^{4014} = b + 3a$. Determine whether $a > b$ or $b > a$.

Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} \equal{} b^a$.

The equations are given $$ \begin{array}{c} x^2 + p_1x + q_1 = 0\\ x^2 + p_2x + q_2 = 0\\ x^2 + p_3x + q_3 = 0 \end{array}$$ each two of which have a common root, but all three have no common root. Prove that: 1) $2 (p_1p_2 + p_2p_3 + p_3p_1) - (p_1^2 + p_2^2 + p_3^2) = 4 (q_1 + q_2+ q_3)$ 2) he roots of these equations are rational when the numbers $p_1$, $p_2$ and $p_3$ are rational}.

An $m \times n$ rectangle is tiled with $\frac{mn}{2}$ $1 \times 2$ dominoes. The tiling is such that whenever the rectangle is partitioned into two smaller rectangles, there exists a domino that is part of the interior of both rectangles. Given $mn > 2$, what is the minimum possible value of $mn$? For instance, the following tiling of a $4 \times 3$ rectangle doesn’t work because we can partition along the line shown, but that doesn’t necessarily mean other $4 \times 3$ tilings don’t work. [img]https://cdn.artofproblemsolving.com/attachments/d/3/cb1fed407e45463950542b3cc64185892afdc5.png[/img]

Last summer $30\%$ of the birds living on Town Lake were geese, $25\%$ were swans, $10\%$ were herons, and $35\%$ were ducks. What percent of the birds that were not swans were geese? $ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 50\qquad\textbf{(E)}\ 60 $

Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that \begin{align*} a^2 + b^2 + 3 &= 4ab\\ c^2 + d^2 + 3 &= 4cd\\ 4c^3 - 3c &= a \end{align*} [i]Travis Hance.[/i]

A quadrilateral has all integer sides lengths, a perimeter of $26$, and one side of length $4$. What is the greatest possible length of one side of this quadrilateral? $\textbf{(A)}~9\qquad\textbf{(B)}~10\qquad\textbf{(C)}~11\qquad\textbf{(D)}~12\qquad\textbf{(E)}~13$

Suppose that $ f(x),\ g(x)$ are differential fuctions and their derivatives are continuous. Find $ f(x),\ g(x)$ such that $ f(x)\equal{}\frac 12\minus{}\int_0^x \{f'(t)\plus{}g(t)\}\ dt\ \ g(x)\equal{}\sin x\minus{}\int_0^{\pi} \{f(t)\minus{}g'(t)\}\ dt$.

Let a set $S$ of $n$ points be called [i]cool[/i] if: [list] [*] All points lie in a plane [*] No three points are collinear [*] There exists a triangle with three distinct vertices in $S$ such that the triangle contains another point in $S$ strictly inside it [/list] Define $g(S)$ for a cool set $S$ to be the sum of the number of points strictly inside each triangle with three distinct vertices in $S$. Let $f(n)$ be the minimal possible value of $g(S)$ across all cool sets of size $n$. Find \[ f(4) + \dots + f(2020) \pmod{1000}\] [i]Proposed by Timothy Qian[/i]

Circles $S_1{}$ and $S_2{}$ intersect in $M{}$ and $N{}$. Line $l$ intersects the circles in points $A,B\in S_1$ and $C,D\in S_2$. Prove that $\angle AMC=\angle BND$ and $\angle ANC=\angle BMD$ if the order of points on line $l$ is: [b]a)[/b] $A,C,B,D;\quad$ [b]b)[/b] $A,C,D,B.$

Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$

Find all $n$-tuples of real numbers $(x_1, x_2, \dots, x_n)$ such that, for every index $k$ with $1\leq k\leq n$, the following holds: \[ x_k^2=\sum\limits_{\substack{i < j \\ i, j\neq k}} x_ix_j \] [i]Proposed by Oriol Solé[/i]

Let $ n$ be a positive integer, let the sets $ A_{1},A_{2},\cdots,A_{n \plus{} 1}$ be non-empty subsets of the set $ \{1,2,\cdots,n\}.$ prove that there exist two disjoint non-empty subsets of the set $ \{1,2,\cdots,n \plus{} 1\}$: $ \{i_{1},i_{2},\cdots,i_{k}\}$ and $ \{j_{1},j_{2},\cdots,j_{m}\}$ such that $ A_{i_{1}}\cup A_{i_{2}}\cup\cdots\cup A_{i_{k}} \equal{} A_{j_{1}}\cup A_{j_{2}}\cup\cdots\cup A_{j_{m}}$.

Now Wendy wanders over and joins Dr. Lisi and her younger siblings. Thinking she knows everything there is about how to work with arithmetic series, she nearly turns right around to walk back home when Dr. Lisi poses a more challenging problem. "Suppose I select two distinct terms at random from the $2008$ term sequence. What's the probability that their product is positive?" If $a$ and $b$ are relatively prime positive integers such that $a/b$ is the probability that the product of the two terms is positive, find the value of $a+b$.

How many different ways are there to write 2004 as a sum of one or more positive integers which are all "aproximately equal" to each other? Two numbers are called aproximately equal if their difference is at most 1. The order of terms does not matter: two ways which only differ in the order of terms are not considered different.

A cube with side length $2$ is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube?

Two sides of the cyclic quadrilateral $ABCD$ are known: $AB = a$, $BC = b$. A point $K$ is taken on the side $CD$ so that $CK = m$. A circle passing through $B$, $K$ and $D$ intersects line $DA$ at a point $M$, different from $D$. Find $AM$.

[b]p1.[/b] Find all triples of positive integers such that the sum of their reciprocals is equal to one. [b]p2.[/b] Prove that $a(a + 1)(a + 2)(a + 3)$ is divisible by $24$. [b]p3.[/b] There are $20$ very small red chips and some blue ones. Find out whether it is possible to put them on a large circle such that (a) for each chip positioned on the circle the antipodal position is occupied by a chip of different color; (b) there are no two neighboring blue chips. [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].

$(CZS 5)$ A convex quadrilateral $ABCD$ with sides $AB = a, BC = b, CD = c, DA = d$ and angles $\alpha = \angle DAB, \beta = \angle ABC, \gamma = \angle BCD,$ and $\delta = \angle CDA$ is given. Let $s = \frac{a + b + c +d}{2}$ and $P$ be the area of the quadrilateral. Prove that $P^2 = (s - a)(s - b)(s - c)(s - d) - abcd \cos^2\frac{\alpha +\gamma}{2}$

In a plane, we are given $ 100 $ circles with radius $ 1 $ so that the area of any triangle whose vertices are circumcenters of those circles is at most $ 100 $. Prove that one may find a line that intersects at least $ 10 $ circles.

In the following diagram, $AB = 1$. The radius of the circle with center $C$ can be expressed as $\frac{p}{q}$. Determine $p+q$. [i]2022 CCA Math Bonanza Lightning Round 3.2[/i]

Sophia writes an algorithm to solve the graph isomorphism problem. Given a graph $G=(V,E)$, her algorithm iterates through all permutations of the set $\{v_1, \dots, v_{|V|}\}$, each time examining all ordered pairs $(v_i,v_j)\in V\times V$ to see if an edge exists. When $|V|=8$, her algorithm makes $N$ such examinations. What is the largest power of two that divides $N$?

Let $A_1A_2 . . . A_{12}$ be a regular dodecagon. Equilateral triangles $\vartriangle A_1A_2B_1$, $\vartriangle A_2A_3B_2$, $. . . $, and $\vartriangle A_{12}A_1B_{12}$ are drawn such that points $B_1$, $B_2$,$ . . . $, and B_{12} lie outside dodecagon $A_1A_2 . . . A_{12}$. Then, equilateral triangles $\vartriangle A_1A_2C_1$, $\vartriangle A_2A_3C_2$, $. . .$ , and $\vartriangle A_{12}A_1C_{12}$ are drawn such that points $C_1$, $C_2$, $. . .$ , and $C_{12}$ lie inside dodecagon $A_1A_2 . . . A_{12}$. Compute the ratio of the area of dodecagon $B_1B_2 . . . B_{12}$ to the area of dodecagon $C_1C_2 . . . C_{12}$.