Found problems: 85335
Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$.
The set $ G$ is defined by the points $ (x,y)$ with integer coordinates, $ 3\le|x|\le7$, $ 3\le|y|\le7$. How many squares of side at least $ 6$ have their four vertices in $ G$?
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draw((-8,0)--(8.5,0),Arrow(HookHead,1mm));
draw((0,-8)--(0,8.5),Arrow(HookHead,1mm));
int i,j;
for (i=-7;i<8;++i) {
for (j=-7;j<8;++j) {
if (((-7 <= i) && (i <= -3)) || ((3 <= i) && (i<= 7))) { if (((-7 <= j) && (j <= -3)) || ((3 <= j) && (j<= 7))) { fill(shift(i,j)*p,black); }}}} draw((-7,-.2)--(-7,.2),black+0.5bp);
draw((-3,-.2)--(-3,.2),black+0.5bp);
draw((3,-.2)--(3,.2),black+0.5bp);
draw((7,-.2)--(7,.2),black+0.5bp);
draw((-.2,-7)--(.2,-7),black+0.5bp);
draw((-.2,-3)--(.2,-3),black+0.5bp);
draw((-.2,3)--(.2,3),black+0.5bp);
draw((-.2,7)--(.2,7),black+0.5bp);
label("$-7$",(-7,0),S);
label("$-3$",(-3,0),S);
label("$3$",(3,0),S);
label("$7$",(7,0),S);
label("$-7$",(0,-7),W);
label("$-3$",(0,-3),W);
label("$3$",(0,3),W);
label("$7$",(0,7),W);[/asy]$ \textbf{(A)}\ 125\qquad \textbf{(B)}\ 150\qquad \textbf{(C)}\ 175\qquad \textbf{(D)}\ 200\qquad \textbf{(E)}\ 225$
On a triangle $ABC$, points $D$, $E$, $F$ are given on the segments $BC$, $AC$, $AB$ respectively such that $DE \parallel AB$, $DF \parallel AC$ and $\frac{BD}{DC}=\frac{AB^2}{AC^2}$ holds. Let the circumcircle of $AEF$ meet $AD$ at $R$ and the line that is tangent to the circumcircle of $ABC$ at $A$ at $S$ again. Let the line $EF$ intersect $BC$ at $L$ and $SR$ at $T$. Prove that $SR$ bisects $AB$ if and only if $BS$ bisects $TL$.
There are $8!= 40320$ eight-digit positive integers that use each of the digits 1, 2, 3, 4, 5, 6, 7, 8 exactly once. Let N be the number of these integers that are divisible by $22$. Find the difference between $N$ and 2025.
In triangle $ABC$, the angles $\alpha=\angle A$ and $\beta=\angle B$ are acute. The isosceles triangle $ACD$ and $BCD$ with the bases $AC$ and $BC$ and $\angle ADC=\beta$, $\angle BEC=\alpha$ are constructed in the exterior of the triangle $ABC$. Let $O$ be the circumcenter of $\triangle ABC$. Prove that $DO+EO$ equals the perimeter of triangle $ABC$ if and only if $\angle ACB$ is right.
Integers $a, b$ satisfy the following property: the line $y = 2x + ab$ passes through all intersection points of the two parabolas given by \[y = x^2 + 2x + a, \quad y = 2x^2 +bx,\] which intersect at least once. How many such $(a, b)$ satisfy $|ab| \leq 100$?
[i]Proposed by Justin Hsieh[/i]
Which of the following numbers is a perfect square?
$\textbf{(A) }4^45^56^6\qquad\textbf{(B) }4^45^66^5\qquad\textbf{(C) }4^55^46^6\qquad\textbf{(D) }4^65^46^5\qquad\textbf{(E) }4^65^56^4$
Find the positive integer $n$ such that the least common multiple of $n$ and $n - 30$ is $n + 1320$.
A certain class of $32$ pupils is organised into $33$ clubs , so that each club contains $3$ pupils and no two clubs have the same composition. Prove that there are two clubs which have exactly one common member.
For a positive integer $n$, numbers $2n+1$ and $3n+1$ are both perfect squares. Is it possible for $5n+3$ to be prime?
Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.
In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\varphi$ be the non-obtuse angle of the rhombus. Prove that $\varphi \le \max \{ \angle BAC, \angle ABC \}$.
Cyrus the frog jumps 2 units in a direction, then 2 more in another direction. What is the probability that he lands less than 1 unit away from his starting position?
(I forgot answer choices)
Two players play the following game: there are two heaps of tokens, and they take turns to pick some tokens from them. The winner of the game is the player who takes away the last token. If the number of tokens in the two heaps are $A$ and $B$ at a given moment, the player whose turn it is can take away a number of tokens that is a multiple of $A$ or a multiple of $B$ from one of the heaps.
Find those pair of integers $(k,n)$ for which the second player has a winning strategy, if the initial number of tokens is $k$ in the first heap and $n$ in the second heap.
[i]Proposed by Dömötör Pálvölgyi, Budapest[/i]
Suppose that we have a $2\times5$ grid, and we wish to write $0$'s and $1$'s inside such that for any $2\times2$ sub-block, the $\textit{determinant}$ is $0$. The determinant of a $2\times2$ block $\textstyle\begin{bmatrix}a&b\\c&d\end{bmatrix}$ is $ad-bc$. For example, the following is a valid configuration:
[center]<see attached>[/center]
However, the following is not valid because the last $2\times2$ sub-block has determinant $1$:
[center]<see attached>[/center]
How many such valid $2\times5$ configurations are there?
Given several nonnegative integers (repetitions allowed), the allowed operation is to choose a positive integer $a$ and replace each number $b$ greater than or equal to $a$ by $b-a$ (the numbers $a$ , if any, are replaced by $0$). Initially, the integers from $1$ are written on the blackboard until $2013$ inclusive. After a few operations the numbers on the board have a sum equal to $10$. Determine what the numbers that remained on the board could be. Find all the possibilities.
In an acute triangle $ABC$ with $\overline{AB} > \overline{AC}$, let $D, E, F$ be the feet of the altitudes from $A, B, C$, respectively. Let $P$ be an intersection of lines $EF$ and $BC$, and let $Q$ be a point on the segment $BD$ such that $\angle QFD = \angle EPC$. Let $O, H$ denote the circumcenter and the orthocenter of triangle $ABC$, respectively. Suppose that $OH$ is perpendicular to $AQ$. Prove that $P, O, H$ are collinear.
Given vertex $A$ and $A$-excircle $\omega_A$ . Construct all possible triangles such that circumcenter of $\triangle ABC$ coincide with centroid of the triangle formed by tangent points of $\omega_A$ and triangle sides.
[i]Proposed by Seyed Reza Hosseini Dolatabadi, Pooya Esmaeil Akhondy[/i] [b]Rated 4[/b]
Set $n = 425425$. Let $S$ be the set of proper divisors of $n$. Compute the remainder when $$ \sum_{k\in S} \phi (k) {2n/k \choose n/k}$$ is divided by $2n$, where $\phi (x)$ is the number of positive integers at most $x$ that are relatively prime to it.
Find the largest real number $a$ such that \[\left\{ \begin{array}{l}
x - 4y = 1 \\
ax + 3y = 1\\
\end{array} \right.
\] has an integer solution.
Show that \[\cos(56^{\circ}) \cdot \cos(2 \cdot 56^{\circ}) \cdot \cos(2^2\cdot 56^{\circ})\cdot . . . \cdot \cos(2^{23}\cdot 56^{\circ}) = \frac{1}{2^{24}} .\]
Two concentric circles have radii $1$ and $4$. Six congruent circles form a ring where each of the six circles is tangent to the two circles adjacent to it as shown. The three lightly shaded circles are internally tangent to the circle with radius $4$ while the three darkly shaded circles are externally tangent to the circle with radius $1$. The radius of the six congruent circles can be written $\textstyle\frac{k+\sqrt m}n$, where $k,m,$ and $n$ are integers with $k$ and $n$ relatively prime. Find $k+m+n$.
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filldraw(circle(dir(90 + i*120)*(4-r),r),gray);
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for(int j=0;j<=2;j=j+1)
{
filldraw(circle(dir(30+j*120)*(1+r),r),darkgray);
}
[/asy]
Let $A,B$ be two circles in the plane with $B$ inside $A$. Assume that $A$ has radius $3$, $B$ has radius $1$, $P$ is a point on $A$, $Q$ is a point on $B$, and $A$ and $B$ touch so that $P$ and $Q$ are the same point. Suppose that $A$ is kept fixed and $B$ is rolled once round the inside of $A$ so that $Q$ traces out a curve starting and finishing at $P$. What is the area enclosed by this curve?
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOS84LzkwMDBjOTAwODk5M2QyM2IxMGUxZGE5OTI1NWU1ZDYwMDkyYTUwLnBuZw==&rn=VlRSTUMgMjAxMC5wbmc=[/img]
A positive integer has 1000 digits (decimal system), all but at most one of them being the digit $5$. Show that this number isn't a perfect square.
Let $ R$ denote a non-negative rational number. Determine a fixed set of integers $ a,b,c,d,e,f$, such that for [i]every[/i] choice of $ R$, \[ \left| \frac{aR^2\plus{}bR\plus{}c}{dR^2\plus{}eR\plus{}f}\minus{}\sqrt[3]{2}\right| < \left|R\minus{}\sqrt[3]{2}\right|.\]