Found problems: 85335
Let $G$ be a simple graph with $100$ edges on $20$ vertices. Suppose that we can choose a pair of disjoint edges in $4050$ ways. Prove that $G$ is regular.
Mikko takes a multiple choice test with ten questions. His only goal is to pass the test, and this requires seven points. A correct answer is worth one point, and answering wrong results in the deduction of one point. Mikko knows for sure that he knows the correct answer in the six first questions. For the rest, he estimates that he can give the correct answer to each problem with probability $p, 0 < p < 1$. How many questions Mikko should try?
An HMMT party has $m$ MIT students and $h$ Harvard students for some positive integers $m$ and $h,$ For every pair of people at the party, they are either friends or enemies. If every MIT student has $16$ MIT friends and $8$ Harvard friends, and every Harvard student has $7$ MIT enemies and $10$ Harvard enemies, compute how many pairs of friends there are at the party.
Let $A$ be a set of functions $f : R\to R$.
For all $f_1, f_2 \in A$ there exists a $f_3 \in A$ such that $f_1(f_2(y) - x)+ 2x = f_3(x + y)$ for all $x, y \in R$.
Prove that for all $f \in A$, we have $f(x - f(x))= 0$ for all $x \in R$.
Let \( \triangle ABC \) be a triangle such that \( BC > AC > AB \). A point \( X \) is marked on side \( BC \) such that \( AX = XC \). Let \( Y \) be a point on segment \( AX \) such that \( CY = AB \). Prove that \( \angle CYX = \angle ABC \).
All $n$-digit words from the alphabet $\{0, 1\}$ considered. These $2^n$ words should be in a sequence $w_0, w_1, w_2, ..., w_{2^-1}$ be arranged that $w_m$ from $w_{m-1}$ by changing of a single ornament ($m = 1, 2, 3, ..., 2n-1$). Prove that the following algorithm achievesthis :
a) Start with $w_0 = 000... 00$.
b) Let $w_{m-1} = a_1a_2a_3 ... a_n$ with $a_i \in \{0; 1\}$, $i = 1, 2, 3, ..., n$.
Determine the exponent $e(m)$ of the highest power of two dividing $m$ and set $j = e(m)+1$. In $w_{m-1}$ replace the ornament $a_j$ with $1-aj$. this is now $w_m$.
Find the positive integer $N$ for which there exist reals $\alpha, \beta, \gamma, \theta$ which obey
\begin{align*}
0.1 &= \sin \gamma \cos \theta \sin \alpha, \\
0.2 &= \sin \gamma \sin \theta \cos \alpha, \\
0.3 &= \cos \gamma \cos \theta \sin \beta, \\
0.4 &= \cos \gamma \sin \theta \cos \beta, \\
0.5 &\ge \left\lvert N-100 \cos2\theta \right\rvert.
\end{align*}[i]Proposed by Evan Chen[/i]
Circles $C_1,C_2,$ and $C_3$ have radii $2,3,$ and $6$ respectively. If the fourth circle $C_4$ is the sum of the areas of $C_1,C_2,$ and $C_3,$ compute the radius of $C_4.$
[i]Proposed by Alex Li[/i]
We consider a right trapezoid $ABCD$, in which $AB||CD,AB>CD,AD\perp AB$ and $AD>CD$. The diagonals $AC$ and $BD$ intersect at $O$. The parallel through $O$ to $AB$ intersects $AD$ in $E$ and $BE$ intersects $CD$ in $F$. Prove that $CE\perp AF$ if and only if $AB\cdot CD=AD^2-CD^2$ .
Let $O_1, O_2, ..., O_{2012}$ be $2012$ circles in the plane such that no circle intersects or contains anyother circle and no two circles have the same radius. For each $1\le i < j \le 2012$, let $P_{i,j}$ denotethe point of intersection of the two external tangent lines to $O_i$ and $O_j$, and let $T$ be the set of all $P_{i,j}$ (so $|T|=\binom {2012}{2}= 2023066$). Suppose there exists a subset $S\subset T$ with $|S|= 2021056$ such that all points in $S$ lie on the same line. Prove that all points in $T$ lie on the same line.
Jan rolls a fair six-sided die and calls the result $r$. Then, he picks real numbers $a$ and $b$ between 0 and 1 uniformly at random and independently. If the probability that the polynomial $\tfrac{x^2}{r} - x\sqrt{a} + b$ has a real root can be expressed as simplified fraction $\frac{p}{q}$, find $p$.
Let $T = TNYWR$. Compute the number of ordered triples $(a,b,c)$ such that $a$, $b$, and $c$ are distinct positive integers and $a + b + c = T$.
Given a cyclic quadrilateral $ABCD$, let it's diagonals intersect at the point $O$. Take the midpoints of $AD$ and $BC$ as $M$ and $N$ respectively. Take a point $S$ on the arc $AB$ not containing $C$ or $D$ such that $$\angle SMA=\angle SNB$$ Prove that if the diagonals of the quadrilateral made from the lines $SM$, $SN$, $AB$, and $CD$ intersect at the point $T$, then $S$, $O$, and $T$ are collinear.
Let $m$ and $n$ be positive integers such that $2^n+3^m$ is divisible by $5$. Prove that $2^m+3^n$ is divisible by $5$.
$ABCDE$ is a pentagon whose vertices lie on circle $\omega$ where $\angle DAB=90^{\circ}$. Let $EB$ and $AC$ intersect at $F$, $EC$ meet $BD$ at $G$. $M$ is the midpoint of arc $AB$ on $\omega$, not containing $C$. If $FG\parallel DE\parallel CM$ holds, then what is the value of $\frac{|GE|}{|GD|}$?
Let $a_0, a_1, \ldots , a_k \ (k \geq 1)$ be positive integers. Find all positive integers $y$ such that
\[a_0 | y, (a_0 + a_1) | (y + a1), \ldots , (a_0 + a_n) | (y + a_n).\]
Andrew has a four-digit number whose last digit is $2$. Given that this number is divisible by $9$, determine the number of possible values for this number that Andrew could have. Note that leading zeros are not allowed.
Given a triangle $ABC$ for which $\angle BAC \neq 90^{\circ}$, let $B_1, C_1$ be variable points on $AB,AC$, respectively. Let $B_2,C_2$ be the points on line $BC$ such that a spiral similarity centered at $A$ maps $B_1C_1$ to $C_2B_2$. Denote the circumcircle of $AB_1C_1$ by $\omega$. Show that if $B_1B_2$ and $C_1C_2$ concur on $\omega$ at a point distinct from $B_1$ and $C_1$, then $\omega$ passes through a fixed point other than $A$.
[i]Proposed by Max Jiang[/i]
In $xyz$ space, find the volume of the solid expressed by $x^2+y^2\leq z\le \sqrt{3}y+1.$
Let $f$ be an invertible function defined on the complex numbers such that \[z^2 = f(z + f(iz + f(-z + f(-iz + f(z + \ldots)))))\]
for all complex numbers $z$. Suppose $z_0 \neq 0$ satisfies $f(z_0) = z_0$. Find $1/z_0$.
(Note: an invertible function is one that has an inverse).
A $2003\times 2004$ rectangle consists of unit squares. We consider rhombi formed by four diagonals of unit squares.
What maximum number of such rhombi can be arranged in this rectangle so that no two of them have any common points except vertices?
[i]Proposed by A. Golovanov[/i]
In an island, there exist 7 towns and a railway system which connected some of the towns. Every railway segment connects 2 towns, and in every town there exists at least 3 railway segments that connects the town to another towns. Prove that there exists a route that visits 4 different towns once and go back to the original town. (Example: $ A\minus{}B\minus{}C\minus{}D\minus{}A$)
Let $p$ and $q$ be positive integers such that $p$ is a prime, $p$ divides $q-1$, and $p+q$ divides $p^2+2020q^2$. Find the sum of the possible values of $p$.
[i]Proposed by Andy Xu[/i]
For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as
$$a_{n+1} =
\begin{cases}
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\
a_n + 3 & \text{otherwise.}
\end{cases}
$$
Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$.
[i]Proposed by Stephan Wagner, South Africa[/i]
How many ordered pairs of integers $(x, y)$ satisfy the equation$$x^{2020}+y^2=2y?$$
$\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } \text{infinitely many}$
Suppose $ABCD$ is a trapezoid with $AB\parallel CD$ and $AB\perp BC$. Let $X$ be a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally, and denote $Y$ as the intersection of $AC$ and $BD$. If $AB=10$ and $CD=15$, compute the maximum possible value of $XY$.