Found problems: 85335
In a right triangle the square of the hypotenuse is equal to twice the product of the legs. One of the acute angles of the triangle is:
$ \textbf{(A)}\ 15^{\circ} \qquad\textbf{(B)}\ 30^{\circ} \qquad\textbf{(C)}\ 45^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 75^{\circ} $
The country has $n \ge 3$ airports, some pairs of which are connected by bidirectional flights. Every day, the government closes the airport with the strictly highest number of flights going out of it. What is the maximum number of days this can continue?
[i]Proposed by Fedir Yudin[/i]
Let $a_1,a_2,a_3,\ldots,a_{250}$ be real numbers such that $a_1=2$ and
$$a_{n+1}=a_n+\frac{1}{a_n^2}$$
for every $n=1,2, \ldots, 249$. Let $x$ be the greatest integer which is less than
$$\frac{1}{a_1}+\frac{1}{a_2}+\ldots+\frac{1}{a_{250}}$$
How many digits does $x$ have?
[i]Proposed by Miroslav Marinov, Bulgaria[/i]
We colour every square of a $4\times 19$ chess board with one of the colours red, green and blue. Prove that however this colouring is done, we can always find two horizontal rows and two vertical columns such that the $4$ squares on the intersections of these lines all have the same colour.
Let $\Omega$ be the circumcircle of the triangle $ABC$. The points $M_a,M_b$ and $M_c$ are the midpoints of the sides $BC, CA$ and $AB{}$, respectively. Let $A_l, B_l$ and $C_l$ be the intersection points of $\Omega$ with the rays $M_cM_b, M_aM_c$ and $M_bM_a$ respectively. Similarly, let $A_r, B_r$ and $C_r$ be the intersection points of $\Omega$ with the rays $M_bM_c, M_cM_a$ and $M_aM_b$ respectively. Prove that the mean of the areas of the triangles $A_lB_lC_l$ and $A_rB_rC_r$ is not less than the area of the triangle $ABC$.
[i]Proposed by L. Shatunov and T. Kazantseva[/i]
Prove that for each natural number $m \ge 2$, there is a natural number $n$ such that $3^m$ divides $n^3 + 17$ but $3^{m+1}$ does not divide it.
Let $a$ and $b$ be positive integers and $c$ be a positive real number satisfying
$$\frac{a + 1}{b + c}=\frac{b}{a}.$$ Prove that $c \ge 1$ holds.
(Karl Czakler)
A positive six-digit integer begins and ends in $8$, and is also the product of three consecutive even numbers. What is the sum of the three even numbers?
Pentagon $ABCDE$ is given with the following conditions:
(a) $\angle CBD + \angle DAE = \angle BAD = 45^o$, $\angle BCD + \angle DEA = 300^o$
(b) $\frac{BA}{DA} =\frac{ 2\sqrt2}{3}$ , $CD =\frac{ 7\sqrt5}{3} $, and $DE = \frac{15\sqrt2}{4}$
(c) $AD^2 \cdot BC = AB \cdot AE \cdot BD$
Compute $BD$.
Let $ABC$ be a triangle with side lengths $AB=6, AC=7,$ and $BC=8.$ Let $H$ be the orthocenter of $\triangle ABC$ and $H'$ be the reflection of $H$ across the midpoint $M$ of $BC.$ $\tfrac{[ABH']}{[ACH']}$ can be expressed as $\frac{p}{q}$. Find $p+q$.
[i]2022 CCA Math Bonanza Individual Round #14[/i]
Let $n$ be a positive integer, $n\geq 2$. Let $M=\{0,1,2,\ldots n-1\}$. For an integer nonzero number $a$ we define the function $f_{a}: M\longrightarrow M$, such that $f_{a}(x)$ is the remainder when dividing $ax$ at $n$. Find a necessary and sufficient condition such that $f_{a}$ is bijective. And if $f_{a}$ is bijective and $n$ is a prime number, prove that $a^{n(n-1)}-1$ is divisible by $n^{2}$.
for any positive integer $n$ greater than $1$, show that
\[2^n<\binom{2n}{n}<\frac{2^n}{\prod\limits_{i=0}^{n-1} \left(1-\frac{i}{n}\right)}\]
There are two identical clocks on the wall, one showing the current Moscow time and the other showing current local time. The minimum distance between the ends of their hour hands equals $m$ and the maximum distance equals $M$. Find the distance between the centres of the clocks.
(S Fomin, Leningrad)
Let $ n$ be a positive integer and $ a_{1}, \ldots, a_{n}$ be arbitrary integers. Suppose that a function $ f: \mathbb{Z}\to \mathbb{R}$ satisfies $ \sum_{i=1}^{n}f(k+a_{i}l) = 0$ whenever $ k$ and $ l$ are integers and $ l \ne 0$. Prove that $ f = 0$.
If, in a triangle of sides $a, b, c$, the incircle has radius $\frac{b+c-a}{2}$, what is the magnitude of $\angle A$?
Let $s(m)$ denote the sum of (decimal) digits of a positive integer $m$. Prove that for every integer $n > 1$ not equal to $10$ there is a unique integer $f(n) \ge 2$ such that $s(k)+s(f(n)-k) = n$ for all integers $k$ with $0 < k < f(n)$.
Let $N = 11\cdots 122 \cdots 25$, where there are $n$ $1$s and $n+1$ $2$s. Show that $N$ is a perfect square.
If $x_i\geq0(i=1,2,\cdots,n)$, and
$$\sum_{i=1}^n x_i^2 + 2\sum_{1 \leq k < j \leq n} \sqrt{\frac{k}{j}}x_kx_j = 1.$$
Find the maximum and minumum value of $\sum_{i=1}^n x_i$.
Let $r$ be a nonzero real number. The values of $z$ which satisfy the equation \[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9r^8+10r^4+1) = 0 \] are plotted on the complex plane (i.e. using the real part of each root as the x-coordinate
and the imaginary part as the y-coordinate). Show that the area of the convex quadrilateral with these points as vertices is independent of $r$, and find this area.
Given a $2n \times 2m$ table $(m,n \in \mathbb{N})$ with one of two signs ”+” or ”-” in each of its cells. A union of all the cells of some row and some column is called a cross. The cell on the intersectin of this row and this column is called the center of the cross. The following procedure we call a transformation of the table: we mark all cells which contain ”−” and then, in turn, we replace the signs in all cells of the crosses which centers are marked by the opposite signs. (It is easy to see that the order of the choice of the crosses doesn’t matter.) We call a table attainable if it can be obtained from some table applying such transformations one time.
Find the number of all attainable tables.
Let $D$ and $E$ be points on respectively $[AB]$ and $[AC]$ of $\triangle ABC$ where $|AB|=|AC|$, $m(\widehat{BAC})=40^\circ$. Let $F$ be a point on $BC$ such that $C$ is between $B$ and $F$. If $|BE|=|CF|$, $|AD|=|AE|$, and $m(\widehat{BEC})=60^\circ$, then what is $m(\widehat{DFB})$ ?
$ \textbf{(A)}\ 45^\circ
\qquad\textbf{(B)}\ 40^\circ
\qquad\textbf{(C)}\ 35^\circ
\qquad\textbf{(D)}\ 30^\circ
\qquad\textbf{(E)}\ 25^\circ
$
Let $ABC$ be an acute triangle and $\Gamma$ be its circumcircle. Line $\ell$ is tangent to $\Gamma$ at $A$ and let $D$ and $E$ be distinct points on $\ell$ such that $AD = AE$. Suppose that $B$ and $D$ lie on the same side of line $AC$. The circumcircle $\Omega_1$ of $\vartriangle ABD$ meets $AC$ again at $F$. The circumcircle $\Omega_2$ of $\vartriangle ACE$ meets $AB$ again at $G$. The common chord of $\Omega_1$ and $\Omega_2$ meets $\Gamma$ again at $H$. Let $K$ be the reflection of $H$ across line $BC$ and let $L$ be the intersection of $BF$ and $CG$. Prove that $A, K$ and $L$ are collinear.
In a triangle $ABC, X$ and $Y$ are points on the segments $AB$ and $AC$, respectively, such that $AX : XB = 1 : 2$ and $AY :YC = 2:1$. If the area of triangle $AXY$ is $10$, then what is the area of triangle $ABC$?
Solve the equation $|x-m|+|x+m|=x$ depending on the value of the parameter $m$.
A palindrome between $ 1000$ and $ 10,000$ is chosen at random. What is the probability that it is divisible by $ 7?$
$ \textbf{(A)}\ \dfrac{1}{10} \qquad \textbf{(B)}\ \dfrac{1}{9} \qquad \textbf{(C)}\ \dfrac{1}{7} \qquad \textbf{(D)}\ \dfrac{1}{6}\qquad \textbf{(E)}\ \dfrac{1}{5}$