Found problems: 85335
Let $AB$ be a diameter of the circle $\Omega$ with center $O{}$. The points $C, D, X$ and $Y{}$ are chosen on $\Omega$ so that the segments $CX$ and $DX$ intersect the segment $AB$ at points symmetric with respect to $O{}$, and $XY\parallel AB$. Let the lines $AB{}$ and $CD{}$ intersect at the point $E$. Prove that the tangent to $\Omega$ through $Y{}$ passes through $E{}$.
Consider $\triangle OAB$ with acute angle $AOB$. Thorugh a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range over
a) the side $AB$;
b) the interior of $\triangle OAB$.
Triangle $ABC$ is inscribed in a circle $\omega$. Let the bisector of angle $A$ meet $\omega$ at $D$ and $BC$ at $E$. Let the reflections of $A$ across $D$ and $C$ be $D'$ and $C'$, respectively. Suppose that $\angle A = 60^o$, $AB = 3$, and $AE = 4$. If the tangent to $\omega$ at $A$ meets line $BC$ at $P$, and the circumcircle of APD' meets line $BC$ at $F$ (other than $P$), compute $FC'$.
Let be the sequence $ \left( I_n \right)_{n\ge 1} $ defined as $ I_n=\int_0^1 \frac{x^n}{\sqrt{x^{2n} +1}} dx . $
[b]a)[/b] Show that $ \left( I_n \right)_{n\ge 1} $ converges to $ 0. $
[b]b)[/b] Calculate $ \lim_{m\to\infty } m\cdot I_m. $
[b]c)[/b] Prove that the sequence $ \left( n\left( -n\cdot I_n +\lim_{m\to\infty } m\cdot I_m \right) \right)_{n\ge 1} $ is convergent.
Let $\triangle ABC$ be a triangle with sides $a,b,c$ and corresponding angles $A,B,C$ (so $a=BC$ and $A=\angle BAC$ etc.). Suppose that $4A+3C=540^\circ$. Prove that $(a-b)^2(a+b)=bc^2$.
Prove that for any positive integer $n$ there exist coprime numbers $a$ and $b$ such that for all $1 \leq k \leq n$ numbers $a+k$ and $b+k$ are not coprime.
Prove that there are no positive integers $a$ and $b$ such that for all different primes $p$ and $q$ greater than $1000$, the number $ap+bq$ is also prime.
One needs to ffll the cells of an $n\times n$ table ($n > 1$) with distinct integers from $1$ to $n^2$ so that every two consecutive integers are placed in cells that share a side, while every two integers with the same remainder if divided by $n$ are placed in distinct rows and distinct columns. For which $n$ is this possible?
(Alexandr Gribalko)
Suppose $\triangle ABC$ has circumcircle $\Gamma$, circumcentre $O$ and orthocentre $H$. Parallel lines $\alpha, \beta, \gamma$ are drawn through the vertices $A, B, C$, respectively. Let $\alpha ', \beta ', \gamma '$ be the reflections of $\alpha, \beta, \gamma$ in the sides $BC, CA, AB$, respectively.
$(a)$ Show that $\alpha ', \beta ', \gamma '$ are concurrent if and only if $\alpha, \beta, \gamma$ are parallel to the Euler line $OH$.
$(b)$ Suppose that $\alpha ', \beta ' , \gamma '$ are concurrent at the point $P$ . Show that $\Gamma$ bisects $OP$ .
Given a triangle $ABC$. The circle $\omega_1$ passes through the vertex $B$ and touches the side $AC$ at the point $A$, and the circle $\omega_2$ passes through the vertex $C$ and touches the side $AB$ at the point $A$. The circles $\omega_1$ and $\omega_2$ intersect a second time at the point $D$. The line $AD$ intersects the circumcircle of the triangle $ABC$ at point $E$. Prove that $D$ is the midpoint of $AE$..
Triangle $ABC$ is given. Circle $ \omega $ passes through $B$, touch $AC$ in $D$ and intersect sides $AB$ and $BC$ at $P$ and $Q$ respectively. Line $PQ$ intersect $BD$ and $AC$ at $M$ and $N$ respectively. Prove that $ \omega $, circumcircle of $DMN$ and circle, touching $PQ$ in $M$ and passes through B, intersects in one point.
Prove that $ \left\{ (x,y)\in\mathbb{C}^2 |x^2+y^2=1 \right\} =\{ (1,0)\}\cup \left\{ \left( \frac{z^2-1}{z^2+1} ,\frac{2z}{z^2+1} \right) | z\in\mathbb{C}\setminus \{\pm \sqrt{-1}\} \right\} . $
Fermat determines that since his final exam counts as two tests, he only needs to score a $28$ on it for his test average to be $70$. If he gets a perfect $100$ on the final exam, his average will be $88$. What is the lowest score Fermat can receive on his final and still have an average of $80$?
$\text{(A) }60\qquad\text{(B) }66\qquad\text{(C) }68\qquad\text{(D) }70\qquad\text{(E) }72$
Let $ABCD$ be a convex quadrilateral such that the line $BD$ bisects the angle $ABC.$ The circumcircle of triangle $ABC$ intersects the sides $AD$ and $CD$ in the points $P$ and $Q,$ respectively. The line through $D$ and parallel to $AC$ intersects the lines $BC$ and $BA$ at the points $R$ and $S,$ respectively. Prove that the points $P, Q, R$ and $S$ lie on a common circle.
Given a parallelogram with area $1$ and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the $8$ lines formed we have a octagon inside of the parallelogram. Determine the area of this octagon
Let $\mathcal M$ be the set of all polynomial functions $f$ of degree at most 3 such that \[\forall x\in[-1,1]:\ |f(x)|\le 1.\] Denote $a$ the (possibly zero) coefficient of $f$ at $x^3.$ Show that there is a positive number $k$ such that \[\forall f\in\mathcal M:\ |a|\le k\] and find the least $k$ with this property.
$ (C)$ and $(C')$ are two circles which intersect in $A$ and $B$. $(D)$ is a line that moves and passes through $A$, intersecting $(C)$ in P and $(C')$ in P'.
Prove that the bisector of $[PP']$ passes through a non-moving point.
Given $k \ge 1,$ let $p_k$ denote the $k$-th smallest prime number. If $N$ is the number of ordered $4$-tuples $(a,b,c,d)$ of positive integers satisfying $abcd=\prod_{k=1}^{2023} p_k$ with $a<b$ and $c<d,$ find $N \pmod{1000}.$
Given that $(2\cos^2{7.5}-\cos{75}-1)^2$ can be expressed as $\frac{p}{q}$, what is $p+q$?
[i]2022 CCA Math Bonanza Tiebreaker Round #3[/i]
Find all functions $ f :\mathbb{Z}\mapsto\mathbb{Z} $ such that following conditions holds:
$a)$ $f(n) \cdot f(-n)=f(n^2)$ for all $n\in\mathbb{Z}$
$b)$ $f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}$
The circle $k'$ rolls along the inside of circle $k$, the radius of $k$ is twice the radius of $k'$. Describe the path of a point on $k$..
(F.Nilov) Two arcs with equal angular measure are constructed on the medians $ AA'$ and $ BB'$ of triangle $ ABC$ towards vertex $ C$. Prove that the common chord of the respective circles passes through $ C$.
Let $P(x) = 1-x+x^2-x^3+\dots+x^{18}-x^{19}$ and $Q(x)=P(x-1)$. What is the coefficient of $x^2$ in polynomial $Q$?
$
\textbf{(A)}\ 840
\qquad\textbf{(B)}\ 816
\qquad\textbf{(C)}\ 969
\qquad\textbf{(D)}\ 1020
\qquad\textbf{(E)}\ 1140
$
Find the number of ordered pairs $(x,y)$ of integers with $0 \le x < 2023$ and $0 \le y < 2023$ such that $y^3 \equiv x^2 \pmod{2023}.$
Which one of the following combinations of given parts does not determine the indicated triangle?
$ \textbf{(A)}\ \text{base angle and vertex angle; isosceles triangle}$
$ \textbf{(B)}\ \text{vertex angle and the base; isosceles triangle}$
$ \textbf{(C)}\ \text{the radius of the circumscribed circle; equilateral triangle}$
$ \textbf{(D)}\ \text{one arm and the radius of the inscribed circle; right triangle}$
$ \textbf{(E)}\ \text{two angles and a side opposite one of them; scalene triangle}$