Found problems: 85335
Determine the least integer $n$ such that for any set of $n$ lines in the 2D plane, there exists either a subset of $1001$ lines that are all parallel, or a subset of $1001$ lines that are pairwise nonparallel.
[i]Proposed by Samuel Wang[/i]
[hide=Solution][i]Solution.[/i] $\boxed{1000001}$
Since being parallel is a transitive property, we note that in order for this to not exist, there must exist at most $1001$ groups of lines, all pairwise intersecting, with each group containing at most $1001$ lines. Thus, $n = 1000^2 + 1 =
\boxed{1000001}$.[/hide]
Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1) \] is a divisor of $abc-1.$
Let $a,b,c$ be positive real numbers. Prove that \[\frac{2a^2}{b+c} + \frac{2b^2}{c+a} + \frac{2c^2}{a+b} \geq a+b+c\](this is, of course, a joke!)
[b]EDITED with exponent 2 over c[/b]
Compute the number of ordered pairs of integers $(x,y)$ such that $x^2 + y^2 < 2019$ and
$$x^2 + min(x,y) = y^2 + max(x, y) .$$
Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$ for all $n \in N$.
Let $D$ be the midpoint of the hypotenuse $AB$ of a right triangle $ABC$. Let $O_1$ and $O_2$ be the circumcenters of the $ADC$ and $DBC$ triangles, respectively.
a) Prove that $\angle O_1DO_2$ is right.
b) Prove that $AB$ is tangent to the circle of diameter $O_1O_2$ .
Consider triangles in the plane where each vertex has integer coordinates. Such a triangle can be[i] legally transformed[/i] by moving one vertex parallel to the opposite side to a different point with integer coordinates. Show that if two triangles have the same area, then there exists a series of legal transformations that transforms one to the other.
(a) Prove that $(2^{n+1}-1)!$ is divisible by $ \prod_{i=0}^n (2^{n+1-i}-1)^{2^i }$, for every natural number n
(b) Define the sequence ($c_n$) by $c_1=1$ and $c_{n}=\frac{4n-6}{n}c_{n-1}$ for $n\ge 2$. Show that each $c_n$ is an integer.
Let $O$ be the circumcenter and $H$ the orthocenter of an acute-triangle $ABC$. The perpendicular bisector of $AO$ intersects the line $BC$ at point $S$. Let $L$ be the midpoint of $OH$. Prove that $\angle OAH = \angle LSA$.
Rhombus $ ABCD$ is similar to rhombus $ BFDE$. The area of rhombus $ ABCD$ is 24, and $ \angle BAD \equal{} 60^\circ$. What is the area of rhombus $ BFDE$?
[asy]
size(180);
defaultpen(linewidth(0.7)+fontsize(11));
pair A=origin, B=(2,0), C=(3, sqrt(3)), D=(1, sqrt(3)), E=(1, 1/sqrt(3)), F=(2, 2/sqrt(3));
pair point=(3/2, sqrt(3)/2);
draw(B--C--D--A--B--F--D--E--B);
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));
label("$F$", F, dir(point--F));[/asy]
$ \textbf{(A) } 6 \qquad \textbf{(B) } 4\sqrt {3} \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 6\sqrt {3}$
Let $k$ be a positive integer. In the coordinate plane, circle $\omega$ has positive integer radius and is tangent to both axes. Suppose that $\omega$ passes through $(1,1000+k)$. Compute the smallest possible value of $k$.
[i]Proposed by Luke Robitaille
A point $P$ lies on the internal angle bisector of $\angle BAC$ of a triangle $\triangle ABC$. Point $D$ is the midpoint of $BC$ and $PD$ meets the external angle bisector of $\angle BAC$ at point $E$. If $F$ is the point such that $PAEF$ is a rectangle then prove that $PF$ bisects $\angle BFC$ internally or externally.
The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi$. What is the length $AB$?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24$
Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.
The points $D,E$ and $F$ are chosen on the sides $BC,AC$ and $AB$ of triangle $ABC$, respectively. Prove that triangles $ABC$ and $DEF$ have the same centroid if and only if
\[\frac{BD}{DC} = \frac{CE}{EA}=\frac{AF}{FB}\]
Each cell of an $n \times n$ board is colored either black or white. A coloring is called [i]good[/i] if every $2 \times 2$ square contains an even number of black cells, and every cross contains an odd number of black cells. Determine all $n \geqslant 3$ such that, in every good coloring, the four corner cells of the board are the same color.
[b]Note:[/b] Each $2 \times 2$ square contains exactly $4$ cells of the board. Each cross contains exactly $5$ cells of the board.
[asy]
size(5cm);
// Function to draw a filled square centered at a given position
void drawFilledSquare(pair center, real sideLength) {
real halfSide = sideLength / 2;
fill(shift(center) * box((-halfSide, -halfSide), (halfSide, halfSide)), lightgray);
draw(shift(center) * box((-halfSide, -halfSide), (halfSide, halfSide)));
}
// Side length of each square
real sideLength = 1;
// Coordinates for the cross (left shape)
pair[] crossPositions = {
(0, 0), (-1, 0), (1, 0), (0, -1), (0, 1)
};
// Coordinates for the square (right shape)
pair[] squarePositions = {
(3, -0.5), (3, 0.5), (4, -0.5), (4, 0.5)
};
// Draw the cross
for (pair pos : crossPositions) {
drawFilledSquare(pos, sideLength);
}
// Draw the square
for (pair pos : squarePositions) {
drawFilledSquare(pos, sideLength);
}
[/asy]
Suppose $ABCD$ is a convex quadrilateral.Points $P,Q,R$ and $S$ are four points on the line segments $AB,BC,CD$ and $DA$ respectively.The line segments $PR$ and $QS$ meet at $T$.Suppose that each of the quadrilaterals $APTS,BQTP,CRTQ$ and $DSTR$ have an incircle.Prove that the quadrilateral $ABCD$ also has an incircle.
Let $n$ and $k$ be relatively prime positive integers with $k<n$. Each number in the set $M=\{1,2,3,\ldots,n-1\}$ is colored either blue or white. For each $i$ in $M$, both $i$ and $n-i$ have the same color. For each $i\ne k$ in $M$ both $i$ and $|i-k|$ have the same color. Prove that all numbers in $M$ must have the same color.
Let $a, b, c, d \in \mathbb{N^{*}}$ such that the equation \[x^{2}-(a^{2}+b^{2}+c^{2}+d^{2}+1)x+ab+bc+cd+da=0 \] has an integer solution. Prove that the other solution is integer too and both solutions are perfect squares.
Real function $f$ [b]generates[/b] real function $g$ if there exists a natural $k$ such that $f^k=g$ and we show this by $f \rightarrow g$. In this question we are trying to find some properties for relation $\rightarrow$, for example it's trivial that if $f \rightarrow g$ and $g \rightarrow h$ then $f \rightarrow h$.(transitivity)
(a) Give an example of two real functions $f,g$ such that $f\not = g$ ,$f\rightarrow g$ and $g\rightarrow f$.
(b) Prove that for each real function $f$ there exists a finite number of real functions $g$ such that $f \rightarrow g$ and $g \rightarrow f$.
(c) Does there exist a real function $g$ such that no function generates it, except for $g$ itself?
(d) Does there exist a real function which generates both $x^3$ and $x^5$?
(e) Prove that if a function generates two polynomials of degree 1 $P,Q$ then there exists a polynomial $R$ of degree 1 which generates $P$ and $Q$.
Time allowed for this problem was 75 minutes.
Five numbers from a list of nine integers are $7,8,3,5,$ and $9$. The largest possible value of the median of all nine numbers in this list is
$\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7\qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$
On day $1$ of the new year, John Adams and Samuel Adams each drink one gallon of tea. For each positive integer $n$, on the $n$th day of the year, John drinks $n$ gallons of tea and Samuel drinks $n^2$ gallons of tea. After how many days does the combined tea intake of John and Samuel that year first exceed $900$ gallons?
[i]Proposed by Aidan Duncan[/i]
[hide=Solution]
[i]Solution. [/i] $\boxed{13}$
The total amount that John and Samuel have drank by day $n$ is
$$\dfrac{n(n+1)(2n+1)}{6}+\dfrac{n(n+1)}{2}=\dfrac{n(n+1)(n+2)}{3}.$$
Now, note that ourdesired number of days should be a bit below $\sqrt[3]{2700}$. Testing a few values gives $\boxed{13}$ as our answer.
[/hide]
$\frac{(3!)!}{3!} =$
$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 6\qquad \text{(D)}\ 40\qquad \text{(E)}\ 120$
Recall that the [i]Fibonacci numbers[/i] are defined recursively by the equation $F_n = F_{n - 1} + F_{n - 2}$ for every integer $n \ge 2$, with initial values $F_0 = 0$ and $F_1 = 1$. Let $k$ be a positive integer. Say that an integer is $k$-[i]summable[/i] if it is the sum of $k$ Fibonacci numbers (not necessarily distinct).
(a) Prove that every positive integer less than $F_{2k + 3} - 1$ is $k$-summable.
(b) Prove that $F_{2k + 3} - 1$ is not $k$-summable.
In an acute triangle $ABC$, the points $H$, $G$, and $M$ are located on $BC$ in such a way that $AH$, $AG$, and $AM$ are the height, angle bisector, and median of the triangle, respectively. It is known that $HG=GM$, $AB=10$, and $AC=14$. Find the area of triangle $ABC$.