Found problems: 85335
The non-isosceles triangle $ABC$ is inscribed in the circle ω. The tangent to this circle at the point $C$ intersects the line $AB$ at the point $D$. Let the bisector of the angle $CDB$ intersect the segments $AC$ and $BC$ at the points $K$ and $L$, respectively. On the side $AB$, the point $M$ is taken such that $AK / BL = AM / BM$. Let the perpendiculars from the point $M$ to the lines $KL$ and $DC$ intersect the lines $AC$ and $DC$ at the points $P$ and $Q$, respectively. Prove that the angle $CQP$ is half of the angle $ACB$.
Let $r_1, r_2, ..., r_{47}$ be the roots of $x^{47} - 1 = 0$. Compute $$\sum^{47}_{i=1}r^{2020}_i .$$
In a convex pentagon, let the perpendicular line from a vertex to the opposite side be called an altitude. Prove that if four of the altitudes are concurrent at a point then the fifth altitude also passes through this point.
The inscribed circle of $\triangle ABC$ touches $BC, AC$ and $AB$ at $D,E$ and $F$ respectively. Denote the perpendicular foots from $F, E$ to $BC$ by $K, L$ respectively. Let the second intersection of these perpendiculars with the incircle be $M, N$ respectively. Show that $\frac{{{S}_{\triangle BMD}}}{{{S}_{\triangle CND}}}=\frac{DK}{DL}$
by Mahdi Etesami Fard
The star shown is symmetric with respect to each of the six diagonals shown. All segments connecting the points $A_1, A_2, . . . , A_6$ with the centre of the star have the length $1$, and all the angles at $B_1, B_2, . . . , B_6$ indicated in the figure are right angles. Calculate the area of the star.
[img]https://1.bp.blogspot.com/-Rso2aWGUq_k/XzcAm4BkAvI/AAAAAAAAMW0/277afcqTfCgZOHshf_6ce2XpinWWR4SZACLcBGAsYHQ/s0/2006%2BMohr%2Bp1.png[/img]
The point $P$ lies inside $\vartriangle ABC$ so that $\vartriangle BPC$ is isosceles, and angle $P$ is a right angle. Furthermore both $\vartriangle BAN$ and $\vartriangle CAM$ are isosceles with a right angle at $A$, and both are outside $\vartriangle ABC$. Show that $\vartriangle MNP$ is isosceles and right-angled.
[img]https://1.bp.blogspot.com/-i9twOChu774/XzcBLP-RIXI/AAAAAAAAMXA/n5TJCOJypeMVW28-9GDG4st5C47yhvTCgCLcBGAsYHQ/s0/2005%2BMohr%2Bp3.png[/img]
In the sequence $\{a_n\}_{n=0}^{\infty}$ we have $a_0=1$, $a_1=2$ and
\[a_{n+1}=a_n+\dfrac{a_{n-1}}{1+a_{n-1}^2} \qquad \forall n \geq 1\]
Prove that
\[52 < a_{1371} < 65\]
Let $p,q$ and $s{}$ be prime numbers such that $2^sq =p^y-1$ where $y > 1.$ Find all possible values of $p.$
At Ignus School there are $425$ students. Of these students $351$ study mathematics, $71$ study Latin, and $203$ study chemistry. There are $199$ students who study more than one of these subjects, and $8$ students who do not study any of these subjects. Find the number of students who study all three of these subjects.
Let $x,y,z$ be positive real numbers, less than $\pi$, such that:
$$\cos x + \cos y + \cos z = 0$$
$$\cos 2x + \cos 2 y + \cos 2z = 0$$
$$\cos 3x + \cos 3y + \cos 3z = 0$$
Find all the values that $\sin x + \sin y + \sin z$ can take.
Two people $A$ and $B$ play the following game: They take from $\{0, 1, 2, 3,..., 1024\}$ alternately $512$, $256$, $128$, $64$, $32$, $16$, $8$, $4$, $2$, $1$, numbers away where $A$ first removes $512$ numbers, $B$ removes $256$ numbers etc. Two numbers $a, b$ remain ($a < b$). $B$ pays $A$ the amount $b - a$. $A$ would like to win as much as possible, $B$ would like to lose as little as possible. What profit does $A$ make if does every player play optimally according to their goals? The result must be justified.
Three travel companies provide transportation between $n$ cities, such that each connection between a pair of cities is covered by one company only. Prove that, for $n \geq 11$, there must exist a round-trip through some four cities, using the services of a same company, while for $n < 11$ this is not anymore necessarily true.
[i]Dan Schwarz[/i]
Let $n\geq 2$ be an integer. For each $k$-tuple of positive integers $a_1, a_2, \ldots, a_k$ such that $a_1+a_2+\cdots +a_k=n$, consider the sums $S_i=1+2+\ldots +a_i$ for $1\leq i\leq k$. Determine, in terms of $n$, the maximum possible value of the product $S_1S_2\cdots S_k$.
[i]Proposed by Misael Pelayo[/i]
Let be an increasing, infinite sequence of natural numbers $ \left( a_n \right)_{n\ge 1} . $
[b]a)[/b] Prove that if $ a_n=n, $ for any natural numbers $ n, $ then
$$ -2+2\sqrt{1+n} <\frac{1}{\sqrt{a_1}} +\frac{1}{\sqrt{a_2}} +\cdots +\frac{1}{\sqrt{a_n}} <2\sqrt n , $$
for any natural numbers $ n. $
[b]b)[/b] Disprove the converse of [b]a).[/b]
[i]Vasile Radu[/i]
Is it possible to arrange the numbers $1, 2, . . . , 60$ in that order, so that the sum of any two numbers between which there is one number, divisible by $2$, the sum of any two numbers between which there are two numbers divisible by $3$, . . . , the sum of any two numbers between which there is are there six numbers, divisible by $7$?
A triangle ABC with $\angle A = 60^o$ is given. Points $M$ and $N$ on $AB$ and $AC$ respectively are such that the circumcenter of $ABC$ bisects segment $MN$. Find the ratio $AN:MB$.
by E.Bakaev
Let be given a positive integer $n \ge 3$. Consider integers $a_1,a_2,...,a_n >1$ with the product equals to $A$ such that: for each $k \in \{1, 2,..., n\}$ then the remainder when $\frac{A}{a_k}$ divided by $a_k$ are all equal to $r$. Prove that $r \le n- 2$
$\{a_{n}\}$ is a sequence of natural numbers satisfying the following inequality for all natural number $n$:
$$(a_{1}+\cdots+a_{n})\left(\frac{1}{a_{1}}+\cdots+\frac{1}{a_{n}}\right)\le{n^{2}}+2019$$
Prove that $\{a_{n}\}$ is constant.
In [i]Terra Brasilis[/i] there are $n$ houses where $n$ goblins live, each in a house. There are one-way routes such that:
- each route joins two houses,
- in each house exactly one route begins,
- in each house exactly one route ends.
If a route goes from house $A$ to house $B$, then we will say that house $B$ is next to house $A$. This relationship is not symmetric, that is: in this situation, not necessarily house $A$ is next to house $B$.
Every day, from day $1$, each goblin leaves the house where he is and arrives at the next house. A legend of [i]Terra Brasilis[/i] says that when all the goblins return to the original position, the world will end.
a) Show that the world will end.
b) If $n = 98$, show that it is possible for elves to build and guide the routes so that the world does not end before $300,000$ years.
Does there exist a positive integer $n$ such that all its digits (in the decimal system) are greather than 5, while all the digits of $n^2$ are less than 5?
Find all real solutions of the equation $x + cos x = 1$.
Let $ ABCD$ be a cyclic quadrilateral and $ O$ be the intersection of diagonal $ AC$ and $ BD$. The circumcircles of triangle $ ABO$ and the triangle $ CDO$ intersect at $ K$. Let $ L$ be a point such that the triangle $ BLC$ is similar to $ AKD$ (in that order). Prove that if $ BLCK$ is a convex quadrilateral, then it has an incircle.
Estimate the number of primes among the first thousand primes divide some term of the sequence
\[2^0+1,2^1+1,2^2+1,2^3+1,\ldots.\]
An estimate of $E$ earns $2^{1-0.02|A-E|}$ points, where $A$ is the actual answer.
[i]2021 CCA Math Bonanza Lightning Round #5.4[/i]
2. Let [i]ABCD [/i]be a square and let $\Gamma$ denote the circle with diameter [i]CD[/i]. A tangent line is drawn to the circle $\Gamma$ from [b][i]B[/i][/b], meeting the circle $\Gamma$ at [i]E[/i] and intersecting the segment [i]AD[/i] at [i]K[/i].
Prove that |[i]AD[/i]| = 4 |[i]KD[/i]|.
Do there exist polynomials $p(x)$ and $q(x)$ with real coefficients such that $p^3(x)-q^2(x)$ is linear but not constant?