Found problems: 1148
Graph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trinomial $y = x^2$ in points $A$, $B$. The intersection of tangents to $G_2$ in points $A$, $B$ is point $C$. If $C \in G_1$, find all possible values of $p$.
Determine all $f: \mathbb{R}\rightarrow\mathbb{R}$ for which
\[ 2\cdot f(x)-g(x)=f(y)-y \textrm{ and } f(x)\cdot g(x) \geq x+1. \]
Find all pairs $(p,q)$ of real numbers such that $p+q=1998$ and the solutions of the equation $x^2+px+q=0$ are integers.
It is known that for quadratic polynomials $P(x)=x^2+ax+b$ and $Q(x)=x^2+cx+d$ the equation $P(Q(x))=Q(P(x))$ does not have real roots. Prove that $b \neq d$.
Find all the real values of $ \lambda$ for which the system of equations $ x\plus{}y\plus{}z\plus{}v\equal{}0$ and $ \left(xy\plus{}yz\plus{}zv\right)\plus{}\lambda\left(xz\plus{}xv\plus{}yv\right)\equal{}0$, has a unique real solution.
Let $P(x)=x^2+bx+c$. Suppose $P(P(1))=P(P(-2))=0$ and $P(1)\neq P(-2)$. Then $P(0)=$
[list=1]
[*] $-\frac{5}{2}$
[*] $-\frac{3}{2}$
[*] $-\frac{7}{4}$
[*] $\frac{6}{7}$
[/list]
A circle radius $320$ is tangent to the inside of a circle radius $1000$. The smaller circle is tangent to a diameter of the larger circle at a point $P$. How far is the point $P$ from the outside of the larger circle?
Define $\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}$. Find the smallest integer $x$ such that $f(x)\ge50\sqrt{x}$.
(Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".)
Prove that the equation $x^{2}+p|x| = qx - 1 $ has 4 distinct real solutions if and only if $p+|q|+2<0$
($p$ and $q$ are two real parameters).
6. Consider a triangle $ABC$ with $1 < \frac{AB}{AC} < \frac{3}{2}$. Let $M$ and $N$, respectively, be variable points of the sides $AB$ and $AC$, different from $A$, such that $\frac{MB}{AC} - \frac{NC}{AB} = 1$. Show that circumcircle of triangle $AMN$ pass through a fixed point different from $A$.
Find all positive integers $n$ such that $4^n+6^n+9^n$ is a square.
[i]David Yang, Alex Zhu.[/i]
Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$ \[ a \cos^2{x}+b \cos{x}+c=0. \] Using the numbers $a,b,c$ form a quadratic equation in $\cos{2x}$ whose roots are the same as those of the original equation. Compare the equation in $\cos{x}$ and $\cos{2x}$ for $a=4$, $b=2$, $c=-1$.
Two different positive numbers $ a$ and $ b$ each differ from their reciprocals by 1. What is $ a \plus{} b$?
\[ \textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } \sqrt {5} \qquad \textbf{(D) } \sqrt {6} \qquad \textbf{(E) } 3
\]
Let $f(x)=x+\cfrac{1}{2x+\cfrac{1}{2x+\cfrac{1}{2x+\cdots}}}$. Find $f(99)f^\prime (99)$.
A round table has radius $ 4$. Six rectangular place mats are placed on the table. Each place mat has width $ 1$ and length $ x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $ x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $ x$?
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draw(Circle((0,0),4));
path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle;
draw(mat);
draw(rotate(60)*mat);
draw(rotate(120)*mat);
draw(rotate(180)*mat);
draw(rotate(240)*mat);
draw(rotate(300)*mat);
label("$x$",(-2.687,0),E);
label("$1$",(-3.187,1.5513),S);[/asy]$ \textbf{(A)}\ 2\sqrt {5} \minus{} \sqrt {3} \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ \frac {3\sqrt {7} \minus{} \sqrt {3}}{2} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {5 \plus{} 2\sqrt {3}}{2}$
Let $P(n)$ be a quadratic trinomial with integer coefficients. For each positive integer $n$, the number $P(n)$ has a proper divisor $d_n$, i.e., $1<d_n<P(n)$, such that the sequence $d_1,d_2,d_3,\ldots$ is increasing. Prove that either $P(n)$ is the product of two linear polynomials with integer coefficients or all the values of $P(n)$, for positive integers $n$, are divisible by the same integer $m>1$.
If the discriminant of $ ax^2\plus{}2bx\plus{}c\equal{}0$ is zero, then another true statement about $ a$, $ b$, and $ c$ is that:
$ \textbf{(A)}\ \text{they form an arithmetic progression} \\
\textbf{(B)}\ \text{they form a geometric progression} \\
\textbf{(C)}\ \text{they are unequal} \\
\textbf{(D)}\ \text{they are all negative numbers} \\
\textbf{(E)}\ \text{only b is negative and a and c are positive}$
Find all positive integers $n$ that are quadratic residues modulo all primes greater than $n$.
A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and, on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,000$th number in the sequence?
$ \textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996507\qquad\textbf{(C)}\ 996508\qquad\textbf{(D)}\ 996509\qquad\textbf{(E)}\ 996510 $
Simon and Garfunkle play in a round-robin golf tournament. Each player is awarded one point for a victory, a half point for a tie, and no points for a loss. Simon beat Garfunkle in the first game by a record margin as Garfunkle sent a shot over the bridge and into troubled waters on the final hole. Garfunkle went on to score $8$ total victories, but no ties at all. Meanwhile, Simon wound up with exactly $8$ points, including the point for a victory over Garfunkle. Amazingly, every other player at the tournament scored exactly $n$. Find the sum of all possible values of $n$.
In trapezoid $ABCD,$ leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD},$ and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB=\sqrt{11}$ and $AD=\sqrt{1001},$ find $BC^2.$
Let $f(x)=ax^2+bx+c$, where $a,b,c$ are nonnegative real numbers, not all equal to zero. Prove that $f(xy)^2\le f(x^2)f(y^2)$ for all real numbers $x,y$.
Call admissible a set $A$ of integers that has the following property:
If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$.
Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers.
[i]Proposed by Warut Suksompong, Thailand[/i]