Find the number of subsets of $\{1, 2,... , 2100\}$ such that each has sum of the elements giving a remainder of $3$ when divided by $7$.

Find the remainder when the number of positive divisors of the value $$(3^{2020}+3^{2021})(3^{2021}+3^{2022})(3^{2022}+3^{2023})(3^{2023}+3^{2024})$$ is divided by $1000$. [i]Proposed by pog[/i]

The solution set to the inequality $\sqrt{\log_2 x-1}+\frac{1}{2}\log_{\frac{1}{2}}x^3+2>0$ is $\text{(A)}[2,3)\qquad\text{(B)}(2,3]\qquad\text{(C)}[2,4)\qquad\text{(D)}(2,4]$

The number $N$ consists only $2's$ and $1's$ in its [b]decimal representation[/b].We know that,after deleting digits from N,we can get any number consisting $9999$- $1's$ and $one$ - $2's$ in its [b]decimal representation[/b].[b][u]Find the least number of digits in the decimal representation of N[/u][/b]

Prove that : $\frac{e-1}{e}<\int_0^1 e^{-x^2}dx<\frac{\pi}{4}.$

Determine the prime numbers $p$ for which the number $a = 7^p - p - 16$ is a perfect square. Lucian Petrescu

Let $ABC$ be a triangle such that $\angle BAC = 45^{\circ}$. Let $H,O$ be the orthocenter and circumcenter of $ABC$, respectively. Let $\omega$ be the circumcircle of $ABC$ and $P$ the point on $\omega$ such that the circumcircle of $PBH$ is tangent to $BC$. Let $X$ and $Y$ be the circumcenters of $PHB$ and $PHC$ respectively. Let $O_1,O_2$ be the circumcenters of $PXO$ and $PYO$ respectively. Prove that $O_1$ and $O_2$ lie on $AB$ and $AC$, respectively.

Triangle $ABC$ is a triangle with side lengths $13$, $14$, and $15$. A point $Q$ is chosen uniformly at random in the interior of $\triangle{ABC}$. Choose a random ray (of uniformly random direction) with endpoint $Q$ and let it intersect the perimeter of $\triangle{ABC}$ at $P$. What is the expected value of $QP^2$? [i]2018 CCA Math Bonanza Tiebreaker Round #4[/i]

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]B16 / G11[/b] Let triangle $ABC$ be an equilateral triangle with side length $6$. If point $D$ is on $AB$ and point $E$ is on $BC$, find the minimum possible value of $AD + DE + CE$. [b]B17 / G12[/b] Find the smallest positive integer $n$ with at least seven divisors. [b]B18 / G13[/b] Square $A$ is inscribed in a circle. The circle is inscribed in Square $B$. If the circle has a radius of $10$, what is the ratio between a side length of Square $A$ and a side length of Square $B$? [b]B19 / G14[/b] Billy Bob has $5$ distinguishable books that he wants to place on a shelf. How many ways can he order them if he does not want his two math books to be next to each other? [b]B20 / G15[/b] Six people make statements as follows: Person $1$ says “At least one of us is lying.” Person $2$ says “At least two of us are lying.” Person $3$ says “At least three of us are lying.” Person $4$ says “At least four of us are lying.” Person $5$ says “At least five of us are lying.” Person $6$ says “At least six of us are lying.” How many are lying? [u]Set 5[/u] [b]B21 / G16[/b] If $x$ and $y$ are between $0$ and $1$, find the ordered pair $(x, y)$ which maximizes $(xy)^2(x^2 - y^2)$. [b]B22 / G17[/b] If I take all my money and divide it into $12$ piles, I have $10$ dollars left. If I take all my money and divide it into $13$ piles, I have $11$ dollars left. If I take all my money and divide it into $14$ piles, I have $12$ dollars left. What’s the least amount of money I could have? [b]B23 / G18[/b] A quadratic equation has two distinct prime number solutions and its coefficients are integers that sum to a prime number. Find the sum of the solutions to this equation. [b]B24 / G20[/b] A regular $12$-sided polygon is inscribed in a circle. Gaz then chooses $3$ vertices of the polygon at random and connects them to form a triangle. What is the probability that this triangle is right? [b]B25 / G22[/b] A book has at most $7$ chapters, and each chapter is either $3$ pages long or has a length that is a power of $2$ (including $1$). What is the least positive integer $n$ for which the book cannot have $n$ pages? [u]Set 6[/u] [b]B26 / G26[/b] What percent of the problems on the individual, team, and guts rounds for both divisions have integer answers? [b]B27 / G27[/b] Estimate $12345^{\frac{1}{123}}$. [b]B28 / G28[/b] Let $O$ be the center of a circle $\omega$ with radius $3$. Let $A, B, C$ be randomly selected on $\omega$. Let $M$, $N$ be the midpoints of sides $BC$, $CA$, and let $AM$, $BN$ intersect at $G$. What is the probability that $OG \le 1$? [b]B29 / G29[/b] Let $r(a, b)$ be the remainder when $a$ is divided by $b$. What is $\sum^{100}_{i=1} r(2^i , i)$? [b]B30 / G30[/b] Bongo flips $2023$ coins. Call a run of heads a sequence of consecutive heads. Say a run is maximal if it isn’t contained in another run of heads. For example, if he gets $HHHT T HT T HHHHT H$, he’d have maximal runs of length $3, 1, 4, 1$. Bongo squares the lengths of all his maximal runs and adds them to get a number $M$. What is the expected value of $M$? - - - - - - [b]G19[/b] Let $ABCD$ be a square of side length $2$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AD$. Let the intersection of $BN$ and $CM$ be $E$. Find the area of quadrilateral $NECD$. [b]G21[/b] Quadrilateral $ABCD$ has $\angle A = \angle D = 60^o$. If $AB = 8$, $CD = 10$, and $BC = 3$, what is length $AD$? [b]G23[/b] $\vartriangle ABC$ is an equilateral triangle of side length $x$. Three unit circles $\omega_A$, $\omega_B$, and $\omega_C$ lie in the plane such that $\omega_A$ passes through $A$ while $\omega_B$ and $\omega_C$ are centered at $B$ and $C$, respectively. Given that $\omega_A$ is externally tangent to both $\omega_B$ and $\omega_C$, and the center of $\omega_A$ is between point $A$ and line $\overline{BC}$, find $x$. [b]G24[/b] For some integers $n$, the quadratic function $f(x) = x^2 - (6n - 6)x - (n^2 - 12n + 12)$ has two distinct positive integer roots, exactly one out of which is a prime and at least one of which is in the form $2^k$ for some nonnegative integer $k$. What is the sum of all possible values of $n$? [b]G25[/b] In a triangle, let the altitudes concur at $H$. Given that $AH = 30$, $BH = 14$, and the circumradius is $25$, calculate $CH$ PS. You should use hide for answers. Rest problems have been posted [url=https://artofproblemsolving.com/community/c3h3132167p28376626]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f: \mathbb{Q}\rightarrow \mathbb{R}$ be a function such that \[|f(x)-f(y)|\leq (x-y)^{2}\] for all $x,y \in\mathbb{Q}$. Prove that $f$ is constant.

Morteza and Amir Reza play the following game. First each of them independently roll a dice $100$ times in a row to construct a $100$-digit number with digits $1,2,3,4,5,6$ then they simultaneously shout a number from $1$ to $100$ and write down the corresponding digit to the number other person shouted in their $100$ digit number. If both of the players write down $6$ they both win otherwise they both loose. Do they have a strategy with wining chance more than $\frac{1}{36}$? [i]Proposed by Morteza Saghafian[/i]

Anna and Bob play the following game. In the beginning, Bob writes down the numbers $1, 2, ... , 2022$ on a piece of paper, such that half of the numbers are on the left and half on the right. Furthermore, we assume that the $1011$ numbers on both sides are written in some order. After Bob does this, Anna has the opportunity to swap the positions of the two numbers lying on different sides of the paper if they have different parity. Anna wins if, after finitely many moves, all odd numbers end up on the left, in increasing order, and all even ones end up on the right, in increasing order. Can Bob write down a arrangement of numbers for which Anna cannot win? For example, Bob could write down numbers in the following way: $$4, 2, 5, 7, 9, ... , 2021\,\,\,\,\,\,\,\,\,\,,\, \,\,\,\,\,\,\,\,\,\,,\, 3, 1, 6, 8, 10, ... , 2022$$ Then Anna could swap the numbers $1, 4$ and then swap $2, 3$ to win. However, if Anna swapped the pairs $3, 4$ and $1, 2$, the resulting numbers on the left and on the right would not be in increasing order, and hence Anna would not win.

Find all 4-tuples $(a,b,c,n)$ of naturals such that $n^a + n^b = n^c$

An isosceles triangle has a base with length $12$ and the altitude to the base has length $18$. Find the area of the region of points inside the triangle that are a distance of at most 3 from that altitude.

Give an example of four pairwise distinct natural numbers $a$, $b$, $c$ and $d$ such that $$a^2 + b^3 + c^4 = d^5.$$

Let $ABC$ be an acute triangle. Tangents on the circumscribed circle of triangle $ABC$ at points $B$ and $C$ intersect at point $T$. Let $D$ and $E$ be a foot of the altitudes from $T$ onto $AB$ and $AC$ and let $M$ be the midpoint of $BC$. Prove: A) Prove that $M$ is the orthocenter of the triangle $ADE$. B) Prove that $TM$ cuts $DE$ in half.