Consider the set of all 6-digit numbers consisting of only three digits, $a,b,c$ where $a,b,c$ are distinct. Suppose the sum of all these numbers is $593999406$. What is the largest remainder when the three digit number $abc$ is divided by $100$?

In a convex quadrilateral $ABCD, AB = 10, BC = 14, CD = 11$ and $DA = 5$. Determine the angle between its diagonals.

Find all functions $ f : R \to R$ that satisfies $$xf(y) - yf(x)= f\left(\frac{y}{x}\right)$$ for all $x, y \in R$.

A container is shaped like a square-based pyramid where the base has side length $23$ centimeters and the height is $120$ centimeters. The container is open at the base of the pyramid and stands in an open field with its vertex pointing down. One afternoon $5$ centimeters of rain falls in the open field partially filling the previously empty container. Find the depth in centimeters of the rainwater in the bottom of the container after the rain.

Evin and Jerry are playing a game with a pile of marbles. On each players' turn, they can remove $2$, $3$, $7$, or $8$ marbles. If they can’t make a move, because there's $0$ or $1$ marble left, they lose the game. Given that Evin goes first and both players play optimally, for how many values of $n$ from $1$ to $1434$ does Evin lose the game? [i]Proposed by Evin Liang[/i] [hide=Solution][i]Solution.[/i] $\boxed{573}$ Observe that no matter how many marbles a one of them removes, the next player can always remove marbles such that the total number of marbles removed is $10$. Thus, when the number of marbles is a multiple of $10$, the first player loses the game. We analyse this game based on the number of marbles modulo $10$: If the number of marbles is $0$ modulo $10$, the first player loses the game If the number of marbles is $2$, $3$, $7$, or $8$ modulo $10$, the first player wins the game by moving to $0$ modulo 10 If the number of marbles is $5$ modulo $10$, the first player loses the game because every move leads to $2$, $3$, $7$, or $8$ modulo $10$ In summary, the first player loses if it is $0$ mod 5, and wins if it is $2$ or $3$ mod $5$. Now we solve the remaining cases by induction. The first player loses when it is $1$ modulo $5$ and wins when it is $4$ modulo $5$. The base case is when there is $1$ marble, where the first player loses because there is no move. When it is $4$ modulo $5$, then the first player can always remove $3$ marbles and win by the inductive hypothesis. When it is $1$ modulo $5$, every move results in $3$ or $4$ modulo $5$, which allows the other player to win by the inductive hypothesis. Thus, Evin loses the game if n is $0$ or $1$ modulo $5$. There are $\boxed{573}$ such values of $n$ from $1$ to $1434$.[/hide]

Let $k$ be a positive integer. Prove that there exist integers $x$ and $y$, neither of which is divisible by $3$, such that $x^2+2y^2 = 3^k$.

In $\triangle{ABC}$ with an obtuse angle at $A$, let $D$ be the foot of the $A$ altitude and $E$ be the foot of the $B$ altitude. If $AC+CD=DB$ and $BC-AE=EC$, compute $\angle{A}$ in degrees. [i]2020 CCA Math Bonanza Team Round #10[/i]

Let $\Omega$ be a unit circle and $A$ be a point on $\Omega$. An angle $0 < \theta < 180^\circ$ is chosen uniformly at random, and $\Omega$ is rotated $\theta$ degrees clockwise about $A$. What is the expected area swept by this rotation?

The headteacher wants to hire a certain number of new teachers in addition to existing teachers. If he hired an additional $10$ teachers, the number of school students would be reduced number per teacher by $5$. However, if the headmaster hired $20$ new teachers, the number of students per teacher would be reduced by $8$. How many students and how many there are teachers in this school? [img]https://cdn.artofproblemsolving.com/attachments/2/8/c0157ff43fd3d92138c87556a0fca2414e8a3f.png[/img]

In the square shown in the figure, find the value of $x$. [img]https://cdn.artofproblemsolving.com/attachments/0/1/4659d5afa5b409d9264924735297d1188b0be3.png[/img]

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Minh enters the numbers from 1 to 81 in a $9\times9$ grid in some order. She calculates the product of the numbers in each row and column. What is the least number of rows and columns that could have a product divisible by 3? $\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }12$

Sides $\overline{AB}$ and $\overline{AC}$ of equilateral triangle $ABC$ are tangent to a circle at points $B$ and $C$ respectively. What fraction of the area of $\triangle ABC$ lies outside the circle? $ \textbf{(A) }\dfrac{4\sqrt{3}\pi}{27}-\frac{1}{3}\qquad \textbf{(B) } \frac{\sqrt{3}}{2}-\frac{\pi}{8}\qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) }\sqrt{3}-\frac{2\sqrt{3}\pi}{9}\qquad \textbf{(E) } \frac{4}{3}-\dfrac{4\sqrt{3}\pi}{27}$

Let $k$ be a fixed odd integer, $k>3$. Prove: There exist infinitely many positive integers $n$, such that there are two positive integers $d_1, d_2$ satisfying $d_1,d_2$ each dividing $\frac{n^2+1}{2}$, and $d_1+d_2=n+k$.

Two polynomials $f(x)$ and $g(x)$ with real coefficients are called similar if there exist nonzero real number a such that $f(x) = q \cdot g(x)$ for all $x \in R$. [b]I.[/b] Show that there exists a polynomial $P(x)$ of degree 1999 with real coefficients which satisfies the condition: $(P(x))^2 - 4$ and $(P'(x))^2 \cdot (x^2-4)$ are similar. [b]II.[/b] How many polynomials of degree 1999 are there which have above mentioned property.

Prove that for all real numbers $x, y$ holds $(x^2 + 1)(y^2 + 1) \ge 2(xy - 1)(x + y)$. For which integers $x, y$ does equality occur?

Consider $h+1$ chess boards. Number the squares of each board from 1 to 64 in such a way that when the perimeters of any two boards of the collection are brought into coincidence in any possible manner, no two squares in the same position have the same number. What is the maximum value of $h?$