Found problems: 85335
Let $ABCD$ be a quadrilateral with $AD=BC$ and $\widehat{A}+\widehat{B}=120^{0}$. Let us draw equilateral $ACP,DCQ,DBR$ away from $AB$ . Prove that the points $P,Q,R$ are collinear.
Compute the number of ordered quadruples $(a,b,c,d)$ of distinct positive integers such that $\displaystyle \binom{\binom{a}{b}}{\binom{c}{d}}=21$.
[i]Proposed by Luke Robitaille[/i]
Show that there is an unique group $ G $ (up to isomorphism) of order $ 1986 $ which has the property that there is at most one subgroup of it having order $ n, $ for every natural number $ n. $
A honeybug crawls along the honeycombs with the unite length of their hexagons. He has moved from the node $A$ to the node $B$ along the shortest possible trajectory. Prove that the half of his way he moved in one direction.
Jeff writes down the two-digit base-$10$ prime $\overline{ab_{10}}$. He realizes that if he misinterprets the number as the base $11$ number $\overline{ab_{11}}$ or the base $12$ number $\overline{ab_{12}}$, it is still a prime. What is the least possible value of Jeff’s number (in base $10$)?
[i]Proposed byMuztaba Syed[/i]
Find all functions $f: \mathbb{N} \rightarrow \mathbb {N}$ such that for all positive integers $m, n$, $f(m+n)\mid f(m)+f(n)$ and $f(m)f(n) \mid f(mn)$.
On the sides (excluding its endpoints) $ AB,BC,CD,DA $ of a parallelogram consider the points $ M,N,P,Q, $ respectively, such that $ \overrightarrow{AP} +\overrightarrow{AN} +\overrightarrow{CQ} +\overrightarrow{CM} = 0. $ Show that $ QN, PM,AC $ are concurrent.
[i]Adrian Ivan[/i]
Let $S$ be the set of all natural numbers with the property: the sum of the biggest three divisors of number $n$, different from $n$, is bigger than $n$. Determine the largest natural number $k$, which divides any number from $S$.
(A natural number is a positive integer)
Let $\mathbb{N}^2$ denote the set of ordered pairs of positive integers. A finite subset $S$ of $\mathbb{N}^2$ is [i]stable[/i] if whenever $(x,y)$ is in $S$, then so are all points $(x',y')$ of $\mathbb{N}^2$ with both $x'\leq x$ and $y'\leq y$.
Prove that if $S$ is a stable set, then among all stable subsets of $S$ (including the empty set and $S$ itself), at least half of them have an even number of elements.
[i]Ashwin Sah and Mehtaab Sawhney[/i]
Let $f : \left[ a, b\right]\rightarrow\mathbb{R}$ be continuous on $\left[ a, b\right]$ and differentiable on $\left( a, b\right)$. Suppose that $f$ has infinitely many zeros, but there is no $x\in \left( a, b\right)$ with $f(x)=f'(x)=0$.
(a) Prove that $f(a)f(b)=0$.
(b) Give an example of such a function on $\left[ 0, 1\right]$.
(Proposed by Alexandr Bolbot, Novosibirsk State University)
Positive integer $n>2$ is called [i]good[/i] if there exist $n$ distinct points on plane($X_1, \ldots, X_n$), such that for all $1 \leq i \leq n$ vectors $X_iX_1, \ldots, X_iX_n$ can be partitioned into two groups with equal sums.
Find all [i]good[/i] numbers
Prove that the application $ \mathbb{R}\ni x\mapsto 2x+ \{ x\} $ and its inverse are bijective and continuous.
50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine.
[i]Proposed by A. Khrabrov, incorrect translation from Hungarian[/i]
Investigate the path of a light ray in a plane medium with refractive index $n(y)=y^4-y^2+1$ using Snell's law $n(y)\sin\alpha = \text{const}$, where $\alpha$ is the angle made by the ray with the $y$-axis.
Inside an acute angle is a circle. Investigate the possibility of constructing with only a compass and a ruler, a tangent to this circle that the point of contact will bisect the segment of the tangent that is cut off by the sides of the angle.
Let $n$ be a positive integer. A Steiner tree associated with a finite set $S$ of points in the Euclidean $n$-space is a finite collection $T$ of straight-line segments in that space such that any two points in $S$ are joined by a unique path in $T$ , and its length is the sum of the segment lengths. Show that there exists a Steiner tree of length $1+(2^{n-1}-1)\sqrt{3}$ associated with the vertex set of a unit $n$-cube.
[hide=instructions]Time limit: 20 minutes.
Fill in the crossword above with answers to the problems below.
Notice that there are three directions instead of two. You are probably used to "down" and "across," but this crossword has "1," $e^{4\pi i/3}$, and $e^{5\pi i/3}$. You can think of these labels as complex numbers pointing in the direction to fill in the spaces. In other words "1" means "across", $e^{4\pi i/3}$ means "down and to the left," and $e^{5\pi i/3}$ means "down and to the right."
To fill in the answer to, for example, $12$ across, start at the hexagon labeled $12$, and write the digits, proceeding to the right along the gray line. (Note: $12$ across has space for exactly $5$ digits.)
Each hexagon is worth one point, and must be filled by something from the set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Note that $\pi$ is not in the set, and neither is $i$, nor $\sqrt2$, nor $\heartsuit$,etc.
None of the answers will begin with a $0$.
"Concatenate $a$ and $b$" means to write the digits of $a$, followed by the digits of $b$. For example, concatenating $10$ and $3$ gives $103$. (It's not the same as concatenating $3$ and $10$.)
Calculators are allowed!
THIS SHEET IS PROVIDED FOR YOUR REFERENCE ONLY. DO NOT TURN IN THIS SHEET. TURN IN THE OFFICIAL ANSWER SHEET PROVIDED TO THE TEAM. OTHERWISE YOU WILL GET A SCORE OF ZERO! ZERO! ZERO! AND WHILE SOMETIMES "!" MEANS FACTORIAL, IN THIS CASE IT DOES NOT.
Good luck, and have fun![/hide]
[img]https://cdn.artofproblemsolving.com/attachments/b/f/f7445136e40bf4889a328da640f0935b2b8b82.png[/img]
[u][b][i]Across[/i][/b][/u] (1)
[b]A 3.[/b] (3 digits) Suppose you draw $5$ vertices of a convex pentagon (but not the sides!). Let $N$ be the number of ways you can draw at least $0$ straight line segments between the vertices so that no two line segments intersect in the interior of the pentagon. What is $N - 64$? (Note what the question is asking for! You have been warned!)
[b]A 5.[/b] (3 digits) Among integers $\{1, 2,..., 10^{2012}\}$, let $n$ be the number of numbers for which the sum of the digits is divisible by $5$. What are the first three digits (from the left) of $n$?
[b]A 6.[/b] (3 digits) Bob is punished by his math teacher and has to write all perfect squares, one after another. His teacher's blackboard has space for exactly $2012$ digits. He can stop when he cannot fit the next perfect square on the board. (At the end, there might be some space left on the board - he does not write only part of the next perfect square.) If $n^2$ is the largest perfect square he writes, what is $n$?
[b]A 8. [/b](3 digits) How many positive integers $n$ are there such that $n \le 2012$, and the greatest common divisor of $n$ and $2012$ is a prime number?
[b]A 9.[/b] (4 digits) I have a random number machine generator that is very good at generating integers between $1$ and $256$, inclusive, with equal probability. However, right now, I want to produce a random number between $1$ and $n$, inclusive, so I do the following:
$\bullet$ I use my machine to generate a number between $1$ and $256$. Call this $a$.
$\bullet$ I take a and divide it by $n$ to get remainder $r$. If $r \ne 0$, then I record $r$ as the randomly generated number. If $r = 0$, then I record $n$ instead.
Note that this process does not necessarily produce all numbers with equal probability, but that is okay. I apply this process twice to generate two numbers randomly between $1$ and $10$. Let $p$ be the probability that the two numbers are equal. What is $p \cdot 2^{16}$?
[b]A 12.[/b] (5 digits) You and your friend play the following dangerous game. You two start off at some point $(x, y)$ on the plane, where $x$ and $y$ are nonnegative integers.
When it is player $A$'s turn, A tells his opponent $B$ to move to another point on the plane. Then $A$ waits for a while. If $B$ is not eaten by a tiger, then $A$ moves to that point as well.
From a point $(x, y)$ there are three places $A$ can tell $B$ to walk to: leftwards to $(x - 1, y)$, downwards to $(x, y-1)$, and simultaneously downwards and leftwards to $(x-1, y-1)$. However, you cannot move to a point with a negative coordinate.
Now, what was this about being eaten by a tiger? There is a tiger at the origin, which will eat the first person that goes there! Needless to say, you lose if you are eaten.
Consider all possible starting points $(x, y)$ with $0 \le x \le 346$ and $0 \le y \le 346$, and $x$ and $y$ are not both zero. Also suppose that you two play strategically, and you go first (i.e., by telling your friend where to go). For how many of the starting points do you win?
[b][u][i]Down and to the left [/i][/u][/b] $e^{4\pi i/3}$
[b]DL 2.[/b] (2 digits) ABCDE is a pentagon with $AB = BC = CD = \sqrt2$, $\angle ABC = \angle BCD = 120$ degrees, and $\angle BAE = \angle CDE = 105$ degrees. Find the area of triangle $\vartriangle BDE$. Your answer in its simplest form can be written as $\frac{a+\sqrt{b}}{c}$ , where where $a, b, c$ are integers and $b$ is square-free. Find $abc$.
[b]DL 3.[/b] (3 digits) Suppose $x$ and $y$ are integers which satisfy $$\frac{4x^2}{y^2} + \frac{25y^2}{x^2} =
\frac{10055}{x^2} +\frac{4022}{y^2} +\frac{2012}{x^2y^2}- 20. $$ What is the maximum possible value of $xy -1$?
[b]DL 5.[/b] (3 digits) Find the area of the set of all points in the plane such that there exists a square centered around the point and having the following properties:
$\bullet$ The square has side length $7\sqrt2$.
$\bullet$ The boundary of the square intersects the graph of $xy = 0$ at at least $3$ points.
[b]DL 8.[/b] (3 digits) Princeton Tiger has a mom that likes yelling out math problems. One day, the following exchange between Princeton and his mom occurred:
$\bullet$ Mom: Tell me the number of zeros at the end of $2012!$
$\bullet$ PT: Huh? $2012$ ends in $2$, so there aren't any zeros.
$\bullet$ Mom: No, the exclamation point at the end was not to signify me yelling. I was not asking about $2012$, I was asking about $2012!$.
What is the correct answer?
[b]DL 9.[/b] (4 digits) Define the following:
$\bullet$ $A = \sum^{\infty}_{n=1}\frac{1}{n^6}$
$\bullet$ $B = \sum^{\infty}_{n=1}\frac{1}{n^6+1}$
$\bullet$ $C = \sum^{\infty}_{n=1}\frac{1}{(n+1)^6}$
$\bullet$ $D = \sum^{\infty}_{n=1}\frac{1}{(2n-1)^6}$
$\bullet$ $E = \sum^{\infty}_{n=1}\frac{1}{(2n+1)^6}$
Consider the ratios $\frac{B}{A}, \frac{C}{A}, \frac{D}{A} , \frac{E}{A}$. Exactly one of the four is a rational number. Let that number be $r/s$, where $r$ and $s$ are nonnegative integers and $gcd \,(r, s) = 1$. Concatenate $r, s$.
(It might be helpful to know that $A = \frac{\pi^6}{945}$ .)
[b]DL 10.[/b] (3 digits) You have a sheet of paper, which you lay on the xy plane so that its vertices are at $(-1, 0)$, $(1, 0)$, $(1, 100)$, $(-1, 100)$. You remove a section of the bottom of the paper by cutting along the function $y = f(x)$, where $f$ satisfies $f(1) = f(-1) = 0$. (In other words, you keep the bottom two vertices.)
You do this again with another sheet of paper. Then you roll both of them into identical cylinders, and you realize that you can attach them to form an $L$-shaped elbow tube.
We can write $f\left( \frac13 \right)+f\left( \frac16 \right) = \frac{a+\sqrt{b}}{\pi c}$ , where $a, b, c$ are integers and $b$ is square-free. $Find a+b+c$.
[b]DL 11.[/b] (3 digits) Let
$$\Xi (x) = 2012(x - 2)^2 + 278(x - 2)\sqrt{2012 + e^{x^2-4x+4}} + 1392 + (x^2 - 4x + 4)e^{x^2-4x+4}$$
find the area of the region in the $xy$-plane satisfying:
$$\{x \ge 0 \,\,\, and x \le 4 \,\,\, and \,\,\, y \ge 0 \,\,\, and \,\,\, y \le \sqrt{\Xi(x)}\}$$
[b]DL 13.[/b] (3 digits) Three cones have bases on the same plane, externally tangent to each other. The cones all face the same direction. Two of the cones have radii of $2$, and the other cone has a radius of $3$. The two cones with radii $2$ have height $4$, and the other cone has height $6$. Let $V$ be the volume of the tetrahedron with three of its vertices as the three vertices of the cones and the fourth vertex as the center of the base of the cone with height $6$. Find $V^2$.
[b][u][i]Down and to the right[/i][/u][/b] $e^{5\pi i/3}$
[b]DR 1.[/b] (2 digits) For some reason, people in math problems like to paint houses. Alice can paint a house in one hour. Bob can paint a house in six hours. If they work together, it takes them seven hours to paint a house. You might be thinking "What? That's not right!" but I did not make a mistake.
When Alice and Bob work together, they get distracted very easily and simultaneously send text messages to each other. When they are texting, they are not getting any work done.
When they are not texting, they are painting at their normal speeds (as if they were working alone). Carl, the owner of the house decides to check up on their work. He randomly picks a time during the seven hours. The probability that they are texting during that time can be written as $r/s$, where r and s are integers and $gcd \,(r, s) = 1$. What is $r + s$?
[b]DR 4.[/b] (3 digits) Let $a_1 = 2 +\sqrt2$ and $b_1 =\sqrt2$, and for $n \ge 1$, $a_{n+1} = |a_n - b_n|$ and $b_{n+1} = a_n + b_n$. The minimum value of $\frac{a^2_n+a_nb_n-6b^2_n}{6b^2_n-a^2_n}$ can be written in the form $a\sqrt{b} - c$, where $a, b, c$ are integers and $b$ is square-free. Concatenate $c, b, a$ (in that order!).
[b]DR 7.[/b] (3 digits) How many solutions are there to $a^{503} + b^{1006} = c^{2012}$, where $a, b, c$ are integers and $|a|$,$|b|$, $|c|$ are all less than $2012$?
PS. You should use hide for answers.
Let $S$ be a set of size $11$. A random $12$-tuple $(s_1, s_2, . . . , s_{12})$ of elements of $S$ is chosen uniformly at random. Moreover, let $\pi : S \to S$ be a permutation of $S$ chosen uniformly at random. The probability that $s_{i+1}\ne \pi (s_i)$ for all $1 \le i \le 12$ (where $s_{13} = s_1$) can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Compute $a$.
The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at $P$. The point $Q$ is chosen on the segment $BC$ so that $PQ$ is perpendicular to $AC$. Prove that the line joining the centres of the circumcircles of triangles $APD$ and $BQD$ is parallel to $AD$.
Given any positive integer, we can write the integer in base $12$ and add together the digits of its base $12$ representation. We perform this operation on the number $ 7^{6^{5^{4^{3^{2^{1}}}}}}$ repeatedly until a single base $12$ digit remains. Find this digit.
In the four figures, which one has the largest area?
$\text{(A)}\triangle ABC: \angle A=\frac{\pi}{3},\angle B=\frac{\pi}{4},|AC|=\sqrt2$
$\text{(B)}$trapezium: two diagonals are $\sqrt2$ and $\sqrt3$, intersection angle is $\frac{5\pi}{12}$.
$\text{(C)}$Circle: with a radius of $1$.
$\text{(D)}$Square: the length of a diagonal is $2.5$.
Bernado and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernado. Whenever Bernado receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernado. The winner is the last person who produces a number less than 1000. Let $N$ be the smallest initial number that results in a win for Bernado. What is the sum of the digits of $N$?
$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$
Given a scalene acute triangle $ ABC$ with $ AC>BC$ let $ F$ be the foot of the altitude from $ C$. Let $ P$ be a point on $ AB$, different from $ A$ so that $ AF\equal{}PF$. Let $ H,O,M$ be the orthocenter, circumcenter and midpoint of $ [AC]$. Let $ X$ be the intersection point of $ BC$ and $ HP$. Let $ Y$ be the intersection point of $ OM$ and $ FX$ and let $ OF$ intersect $ AC$ at $ Z$. Prove that $ F,M,Y,Z$ are concyclic.
On the side \(AC\) of an acute-angled triangle \(ABC\), arbitrary points \(D\) and \(E\) are chosen. The circumcircles of triangles \(BDC\) and \(BEA\) intersect the sides \(BA\) and \(BC\) respectively for the second time at points \(F\) and \(G\). Point \(O\) is the circumcenter of \(\triangle BFG\). Prove that \(OD = OE\).
[i]Proposed by Anton Trygub[/i]
Let $a, b, c, d$ be positive reals with $a - c = b - d > 0$. Show that
$$\frac{ab}{cd} \ge \left(\frac{\sqrt{a} +\sqrt{b}}{\sqrt{c}+\sqrt{d}}\right)^4$$