Found problems: 85335
Let $a, b, c, d, e, f, g, h, i$ be distinct integers from $1$ to $9$. The minimum possible positive value of $$\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}$$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
Find all positive real numbers $c$ such that the graph of $f\text{ : }\mathbb{R}\to\mathbb{R}$ given by $f(x)=x^3-cx$ has the property that the circle of curvature at any local extremum is centered at a point on the $x$-axis.
A $\vartriangle ABC$ is given with $\angle A = 70^o$. The angle bisectors of $\vartriangle ABC$ intersect at $I$. Suppose that $CA + AI=BC$. Find, with proof, the value of $\angle B$.
Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other?
Find all three-digit numbers whose remainders after division by $11$ give quotient, equal to the sum of the squares of its digits.
Find all real solutions to the equation
\[| | | | | | |x^2 -x - 1| - 3| - 5| - 7| - 9| - 11| - 13| = x^2 - 2x - 48.\]
Find all integers $ n\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$), there exists an integer polynomial $ f(x)$ such that congruence equation $ f(x)\equiv 0 (mod n)$ exactly has $ k$ roots $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$
Let $A=[a_{ij}]_{n\times n}$ be a matrix with nonnegative entries such that
$$\sum_{i=1}^n\sum_{j=1}^na_{ij}=n.$$
(a) Prove that $|\det A|\le1$.
(b) If $|\det A|=1$ and $\lambda\in\mathbb C$ is an arbitrary eigenvalue of $A$, show that $|\lambda|=1$.
Determine the largest $k$ such that for all competitive graph with $2019$ points, if the difference between in-degree and out-degree of any point is less than or equal to $k$, then this graph is strongly connected.
Let $P$ and $Q$ be the points on the diagonal $BD$ of a quadrilateral $ABCD$ such that $BP = PQ = QD$. Let $AP$ and $BC$ meet at $E$, and let $AQ$ meet $DC$ at $F$.
(a) Prove that if $ABCD$ is a parallelogram, then $E$ and $F$ are the midpoints of the corresponding sides.
(b) Prove the converse of (a).
The number $ 25^{64}\cdot64^{25}$ is the square of a positive integer $ N$. In decimal representation, the sum of the digits of $ N$ is
$ \textbf{(A)}\ 7 \qquad
\textbf{(B)}\ 14 \qquad
\textbf{(C)}\ 21 \qquad
\textbf{(D)}\ 28 \qquad
\textbf{(E)}\ 35$
The vertical mast located on the tower can be seen at the greatest angle from a point on the ground whose distance from the mast axis is $ a $; this angle equals the given angle $ \alpha $. Calculate the height of the tower and the height of the mast.
Evaluate the integral:
\[\int_{\frac{\pi^2}{4}}^{4\pi^2}\sin(\sqrt{x})dx.\]
Sam Wang decides to evaluate an expression of the form $x +2 \cdot 2+ y$. However, he unfortunately reads each ’plus’ as a ’times’ and reads each ’times’ as a ’plus’. Surprisingly, he still gets the problem correct. Find $x + y$.
[i]Proposed by Edwin Zhao[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{4}$
We have $x+2*2+y=x \cdot 2+2 \cdot y$. When simplifying, we have $x+y+4=2x+2y$, and $x+y=4$.
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Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression?
Given line $AB$ and point $M$. Find all lines in space passing through $M$ at distance $d$.
Let $ABC$ be an isosceles triangle with $CA=CB$ and $\angle ACB=20^{\circ}$. Let $D$ be a point on side $CA$ such that $\angle ADB=30^{\circ}$. Show that $AB=CD$.
Positive integers $x,y,z$ satisfy $x^3+xy+x^2+xz+y+z=301$. Compute $y+z-x$.
[i]2015 CCA Math Bonanza Individual Round #12[/i]
Prove that circles constructed on the sides of a convex quadrilateral as diameters completely cover this quadrilateral.
Let $I_1, I_2, I_3$ be three open intervals of $\mathbb{R}$ such that none is contained in another. If $I_1\cap I_2 \cap I_3$ is non-empty, then show that at least one of these intervals is contained in the union of the other two.
Consider the polynomials $P(x)=x^{6}-x^{5}-x^{3}-x^{2}-x$ and $Q(x)=x^{4}-x^{3}-x^{2}-1.$ Given that $z_{1},z_{2},z_{3},$ and $z_{4}$ are the roots of $Q(x)=0,$ find $P(z_{1})+P(z_{2})+P(z_{3})+P(z_{4}).$
Given integer $n\geq 2$ and positive real number $a$, find the smallest real number $M=M(n,a)$, such that for any positive real numbers $x_1,x_2,\cdots,x_n$ with $x_1 x_2\cdots x_n=1$, the following inequality holds:
\[\sum_{i=1}^n \frac {1}{a+S-x_i}\leq M\]
where $S=\sum_{i=1}^n x_i$.
Prove the inequality $$\frac{a^2}{b+c-a}+\frac{b^2}{a+c-b}+\frac{c^2}{a+b-c} \geq 3\sqrt{3}R$$ in triangle $ABC$ where $a$, $b$ and $c$ are sides of triangle and $R$ radius of circumcircle of $ABC$
Alice and Bob play the following "point guessing game." First, Alice marks an equilateral triangle $ABC$ and a point $D$ on segment $BC$ satisfying $BD=3$ and $CD=5$. Then, Alice chooses a point $P$ on line $AD$ and challenges Bob to mark a point $Q\neq P$ on line $AD$ such that $\frac{BQ}{QC}=\frac{BP}{PC}$. Alice wins if and only if Bob is unable to choose such a point. If Alice wins, what are the possible values of $\frac{BP}{PC}$ for the $P$ she chose?
Suppose that the sum of the squares of two complex numbers $x$ and $y$ is 7 and the sum of the cubes is 10. What is the largest real value that $x + y$ can have?