Found problems: 85335
Alice creates the graphs $y=|x-a|$ and $y=c-|x-b|$ , where $a,b,c\in\mathbb{R^+}$. She observes that these two graphs and $x$ axis divides the positive side of the plane ($x,y>0$) into two triangles and a quadrilateral. Find the ratio of sums of two triangles' areas to the area of quadrilateral.
[hide=There might be a translation error] In the original statement,it says $XOY$ plane,instead of positive side of the plane. I think these 2 are the same,but I might be wrong [/hide]
Let $ABC$ be an isosceles triangle with $BA=BC$. The points $D, E$ lie on the extensions of $AB, BC$ beyond $B$ such that $DE=AC$. The point $F$ lies on $AC$ is such that $\angle CFE=\angle DEF$. Show that $\angle ABC=2\angle DFE$.
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
(i): For different reals $x,y$, $f(x) \not= f(y)$.
(ii): For all reals $x,y$, $f(x+f(f(-y)))=f(x)+f(f(y))$
What is the maximum number of points that can be placed in the interior of an equilateral triangle of side length $2$ such that the distance between any two points is greater than one?
$\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) }6\qquad\text{(E) }7$
Let $P$ and $Q$ be any points inside a circle $C$ with center $O$ such that $OP=OQ.$ Determine the location of a point $Z$ on $C$ such that $PZ+QZ$ is minimal.
The circle whose diameter is the altitude dropped from the vertex $A$ of the triangle $ABC$ intersects the sides
$AB$ and $AC$ at $D$ and $E$, respectively $(A\ne D, A \ne E)$. Show that the circumcenter of $ABC$ lies on the altitude drawn from the vertex $A$ of the triangle $ADE$, or on its extension.
Determine the smallest natural number $n =>2$ with the property:
For every positive integers $a_1, a_2,. . . , a_n$ the product of all differences $a_j-a_i$,
$1 <=i <j <=n$, is divisible by 2001.
Determine, for any positive real number $a$, the number of solutions $(x,y)$ to the system of equations
$$\begin{cases} |x|+|y|= 1 \\ x^2 + y^2 = a \end{cases}$$
where $x$ and $y$ are real numbers.
Prove that if $a,b,c$ are positive numbers with $abc=1$, then
\[\frac{a}{b} +\frac{b}{c} + \frac{c}{a} \ge a + b + c. \]
If $$s_n = 1 + q + q^2 +... + q^n$$ and $$ S_n = 1 +\frac{1 + q}{2}+ \left( \frac{1 + q}{2}\right)^2 +... + \left( \frac{1 + q}{2}\right)^n,$$ prove that $${n + 1 \choose 1}+{n + 1 \choose 2} s_1 + {n + 1 \choose 3} s_2 + ... + {n + 1 \choose n + 1} s_n = 2^nS_n$$
The function $f$ defined by $\displaystyle f(x)= \frac{ax+b}{cx+d}$. where $a,b,c$ and $d$ are nonzero real numbers, has the properties $f(19)=19, f(97)=97$ and $f(f(x))=x$ for all values except $\displaystyle \frac{-d}{c}$. Find the unique number that is not in the range of $f$.
Let $p \geq 5$ be a prime number. Prove that there exists an integer $a$ with $1 \leq a \leq p-2$ such that neither $a^{p-1}-1$ nor $(a+1)^{p-1}-1$ is divisible by $p^2$.
Compute the side length of the largest cube contained in the region
\[ \{(x, y, z) : x^2+y^2+z^2 \le 25 \text{ and } x, y \ge 0 \} \]
of three-dimensional space.
A teacher asks each of eleven pupils to write a positive integer with at most four digits, each on a separate yellow sticky note. Show that if all the numbers are different, the teacher can always submit two or more of the eleven stickers so that the average of the numbers on the selected notes are not an integer.
Let $p = 10009$ be a prime number. Determine the number of ordered pairs of integers $(x,y)$ such that $1\le x,y \le p$ and $x^3-3xy+y^3+1$ is divisible by $p$.
A sphere is inscribed in tetrahedron $ ABCD$ and is tangent to faces $ BCD,CAD,ABD,ABC$ at points $ P,Q,R,S$ respectively. Segment $ PT$ is the sphere's diameter, and lines $ TA,TQ,TR,TS$ meet the plane $ BCD$ at points $ A',Q',R',S'$. respectively. Show that $ A$ is the center of a circumcircle on the triangle $ S'Q'R'$.
Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying "$1$", so Blair follows by saying "$1$, $2$". Jo then says "$1$, $2$, $3$", and so on. What is the $53$rd number said?
$ \textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8 $
Let $M$ be a point which has coordinates $(p\times 1994,7p\times 1994)$ in the Cartesian plane ($p$ is a prime). Find the number of right-triangles satisfying the following conditions:
(1) all vertexes of the triangle are lattice points, moreover $M$ is on the right-angled corner of the triangle;
(2) the origin ($0,0$) is the incenter of the triangle.
$ABCD$ is a quadrilateral on complex plane whose four vertices satisfy $z^4+z^3+z^2+z+1=0$. Then $ABCD$ is a
[list=1]
[*] Rectangle
[*] Rhombus
[*] Isosceles Trapezium
[*] Square
[/list]
Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him $ 1$ Joule of energy to hop one step north or one step south, and $ 1$ Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with $ 100$ Joules of energy, and hops till he falls asleep with $ 0$ energy. How many different places could he have gone to sleep?
[asy]
//size(100);//local
size(200);
real r1=2;
pair
O=(0,0),
D=(.5,.5*sqrt(3)),
C=(D.x+.5*3,D.y),
B,
B_prime=endpoint(arc(D, 3, 0,-2));
B=B_prime;
path
c1=circle(O, r1);
pair C=midpoint(D--B_prime);
path arc2=arc(B_prime, 6/2, 158.25,250);
draw(c1);
draw(O--D);
draw(D--C);
draw(C--B_prime);
pair A=beginpoint(arc2);
draw(B_prime--A);
//dot(O^^D^^C^^A);
//dot(B_prime);
label("\scriptsize{$O$}",O,.6dir(D--O));
label("\scriptsize{$C$}",C,.5dir(-55));
label("\scriptsize{$D$}", D,.2NW);
//label("\scriptsize{$B$}",B,S);
label("\scriptsize{$B$}", B_prime, .5*dir(D--B_prime));
label("\scriptsize{$A$}",A,.5dir(NE));
label("\tiny{2}", O--D, .45*LeftSide);
label("\tiny{3}", D--C, .45*LeftSide);
label("\tiny{6}", B_prime--A, .45*RightSide);
label("\tiny{3}", waypoint(C--B_prime,.1), .45*N);
//Credit to Klaus-Anton for the diagram[/asy]
In the adjoining figure, $AB$ is tangent at $A$ to the circle with center $O$; point $D$ is interior to the circle; and $DB$ intersects the circle at $C$. If $BC=DC=3$, $OD=2$, and $AB=6$, then the radius of the circle is
$\textbf{(A) }3+\sqrt{3}\qquad\textbf{(B) }15/\pi\qquad\textbf{(C) }9/2\qquad\textbf{(D) }2\sqrt{6}\qquad \textbf{(E) }\sqrt{22}$
Find all integer pairs $(a, b)$ such that $ab+a-3b=5$
$ABCD$ is a rectangle. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ respectively so that $KL$ is parallel to $MN$, and $KM$ is perpendicular to $LN$. Show that the intersection of $KM$ and $LN$ lies on $BD$.
Let $n \geq 3$ be a positive integers and $p$ be a prime number such that $p > 6^{n-1} - 2^n + 1$. Let $S$ be the set of $n$ positive integers with different residues modulo $p$. Show that there exists a positive integer $c$ such that there are exactly two ordered triples $(x,y,z) \in S^3$ with distinct elements, such that $x-y+z-c$ is divisible by $p$.
Each of the digits 3, 5, 6, 7, and 8 is placed one to a box in
the diagram. If the two digit number is subtracted from the
three digit number, what is the smallest difference?
$\textbf{(A)}\ 269 \qquad \textbf{(B)}\ 278 \qquad \textbf{(C)}\ 484 \qquad \textbf{(D)}\ 271 \qquad \textbf{(E)}\ 261$