Found problems: 85335
2016 Argentina National Olympiad Level 2, 5
For each pair $a, \,b$ of coprime natural numbers, let $d_{a,\,b}$ be the greatest common divisor of $51a + b$ and $a + 51b$. Find the maximum possible value of $d_{a,\,b}$.
2007 Hungary-Israel Binational, 3
Let $ AB$ be the diameter of a given circle with radius $ 1$ unit, and let $ P$ be a given point on $ AB$. A line through $ P$ meets the circle at points $ C$ and $ D$, so a convex quadrilateral $ ABCD$ is formed. Find the maximum possible area of the quadrilateral.
2023 Federal Competition For Advanced Students, P2, 1
Given is a nonzero real number $\alpha$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy$$ for all $x, y \in \mathbb{R}$.
2023 Paraguay Mathematical Olympiad, 1
In the following sequence of numbers, each term, starting with the third, is obtained by adding three times the previous term plus twice the previous term to the previous one:
$$a_1, a_2, 78, a_4, a_5, 3438, a_7, a_8,…$$
As seen in the sequence, the third term is $78$ and the sixth term is $3438$. What is the value of the term $a_7$?
1958 Miklós Schweitzer, 10
[b]10.[/b] Prove that the function
$f(x)= \int_{-\infty}^{\infty} \left (\frac{\sin\theta}{\theta} \right )^{2k}\cos (2x\theta) d\theta$
where $k$ is a positive integer, satisfies the following conditions:
[b](i)[/b] $f(x)=0$ if $\mid x \mid \geq k$ and $f(x) \geq 0$ elsewhere;
[b](ii)[/b] in interval $(l,l+1)$ $(l= -k, -k+1, \dots , k-1)$ the function $f(x)$ is a polynomial of degree $2k-1$ at most. [b](R. 7)[/b]
1998 Switzerland Team Selection Test, 6
Find all prime numbers $p$ for which $p^2 +11$ has exactly six positive divisors.
Durer Math Competition CD 1st Round - geometry, 2015.D4
The altitude of the acute triangle $ABC$ drawn from $A$ , intersects the side $BC$ at $A_1$ and the circumscribed circle at $A_2$ (different from $A$). Similarly, we get the points $B_1$, $B_2$, $C_1$, $C_2$. Prove that
$$\frac{AA_2}{AA_1}+\frac{BB_2}{BB_1}+\frac{CC_2}{CC_1}= 4.$$
2020 Yasinsky Geometry Olympiad, 2
An equilateral triangle $BDE$ is constructed on the diagonal $BD$ of the square $ABCD$, and the point $C$ is located inside the triangle $BDE$. Let $M$ be the midpoint of $BE$. Find the angle between the lines $MC$ and $DE$.
(Dmitry Shvetsov)
2006 May Olympiad, 4
Let $ABCD$ be a trapezoid of bases $AB$ and $CD$ . Let $O$ be the intersection point of the diagonals $AC$ and $BD$. If the area of the triangle $ABC$ is $150$ and the area of the triangle $ACD$ is $120$, calculate the area of the triangle $BCO$.
1959 Polish MO Finals, 6
Given a triangle in which the sides $ a $, $ b $, $ c $ form an arithmetic progression and the angles also form an arithmetic progression. Find the ratios of the sides of this triangle.
1961 AMC 12/AHSME, 3
If the graphs of $2y+x+3=0$ and $3y+ax+2=0$ are to meet at right angles, the value of $a$ is:
${{ \textbf{(A)}\ \pm \frac{2}{3} \qquad\textbf{(B)}\ -\frac{2}{3}\qquad\textbf{(C)}\ -\frac{3}{2} \qquad\textbf{(D)}\ 6}\qquad\textbf{(E)}\ -6} $
2021 Lotfi Zadeh Olympiad, 1
In the inscribed quadrilateral $ABCD$, $P$ is the intersection point of diagonals and $M$ is the midpoint of arc $AB$. Prove that line $MP$ passes through the midpoint of segment $CD$, if and only if lines $AB, CD$ are parallel.
2018 Iran Team Selection Test, 6
$a_1,a_2,\ldots,a_n$ is a sequence of positive integers that has at least $\frac {2n}{3}+1$ distinct numbers and each positive integer has occurred at most three times in it. Prove that there exists a permutation $b_1,b_2,\ldots,b_n$ of $a_i $'s such that all the $n$ sums $b_i+b_{i+1}$ are distinct ($1\le i\le n $ , $b_{n+1}\equiv b_1 $)
[i]Proposed by Mohsen Jamali[/i]
Ukraine Correspondence MO - geometry, 2015.8
On the sides $BC, AC$ and $AB$ of the equilateral triangle $ABC$ mark the points $D, E$ and $F$ so that $\angle AEF = \angle FDB$ and $\angle AFE = \angle EDC$. Prove that $DA$ is the bisector of the angle $EDF$.
2022 CCA Math Bonanza, L2.3
Given that the height of a greater sage grouse flying through the air is defined by the function $64x-x^2$ for $0<x<64$, what is the first time at which the bird reaches a height of 903?
[i]2022 CCA Math Bonanza Lightning Round 2.3[/i]
2010 Tuymaada Olympiad, 2
For a given positive integer $n$, it's known that there exist $2010$ consecutive positive integers such that none of them is divisible by $n$ but their product is divisible by $n$. Prove that there exist $2004$ consecutive positive integers such that none of them is divisible by $n$ but their product is divisible by $n$.
1997 AMC 12/AHSME, 15
Medians $ BD$ and $ CE$ of triangle $ ABC$ are perpendicular, $ BD \equal{} 8$, and $ CE \equal{} 12$. The area of triangle $ ABC$ is
[asy]defaultpen(linewidth(.8pt));
dotfactor=4;
pair A = origin;
pair B = (1.25,1);
pair C = (2,0);
pair D = midpoint(A--C);
pair E = midpoint(A--B);
pair G = intersectionpoint(E--C,B--D);
dot(A);dot(B);dot(C);dot(D);dot(E);dot(G);
label("$A$",A,S);label("$B$",B,N);label("$C$",C,S);label("$D$",D,S);label("$E$",E,NW);label("$G$",G,NE);
draw(A--B--C--cycle);
draw(B--D);
draw(E--C);
draw(rightanglemark(C,G,D,3));[/asy]$ \textbf{(A)}\ 24\qquad \textbf{(B)}\ 32\qquad \textbf{(C)}\ 48\qquad \textbf{(D)}\ 64\qquad \textbf{(E)}\ 96$
2022 Czech-Polish-Slovak Junior Match, 1
Let $n\ge 3$. Suppose $a_1, a_2, ... , a_n$ are $n$ distinct in pairs real numbers.
In terms of $n$, find the smallest possible number of different assumed values by the following $n$ numbers:
$$a_1 + a_2, a_2 + a_3,..., a_{n- 1} + a_n, a_n + a_1$$
2012-2013 SDML (High School), 6
Naoki's favorite positive integer $n$ is a two-digit number with distinct digits. It also has the property that when it is divided by $10$, $12$, and $14$, the remainder has a units digit of one. What is the value of $n$?
2007 AMC 10, 2
Define $ a@b \equal{} ab \minus{} b^{2}$ and $ a\#b \equal{} a \plus{} b \minus{} ab^{2}$. What is $ \frac {6@2}{6\#2}$?
$ \textbf{(A)}\ \minus{} \frac {1}{2}\qquad \textbf{(B)}\ \minus{} \frac {1}{4}\qquad \textbf{(C)}\ \frac {1}{8}\qquad \textbf{(D)}\ \frac {1}{4}\qquad \textbf{(E)}\ \frac {1}{2}$
2016 JBMO Shortlist, 3
Find all the pairs of integers $ (m, n)$ such that $ \sqrt {n +\sqrt {2016}} +\sqrt {m-\sqrt {2016}} \in \mathbb {Q}.$
1982 IMO Shortlist, 20
Let $ABCD$ be a convex quadrilateral and draw regular triangles $ABM, CDP, BCN, ADQ$, the first two outward and the other two inward. Prove that $MN = AC$. What can be said about the quadrilateral $MNPQ$?
2009 Argentina Iberoamerican TST, 2
Let $ a$ and $ k$ be positive integers. Let $ a_i$ be the sequence defined by
$ a_1 \equal{} a$ and
$ a_{n \plus{} 1} \equal{} a_n \plus{} k\pi(a_n)$
where
$ \pi(x)$ is the product of the digits of $ x$ (written in base ten)
Prove that we can choose $ a$ and $ k$ such that the infinite sequence $ a_i$ contains exactly $ 100$ distinct terms
2019 PUMaC Geometry B, 8
Let $ABCD$ be a trapezoid such that $AB||CD$ and let $P=AC\cap BD,AB=21,CD=7,AD=13,[ABCD]=168.$ Let the line parallel to $AB$ through $P$ intersect the circumcircle of $BCP$ in $X.$ Circumcircles of $BCP$ and $APD$ intersect at $P,Y.$ Let $XY\cap BC=Z.$ If $\angle ADC$ is obtuse, then $BZ=\frac{a}{b},$ where $a,b$ are coprime positive integers. Compute $a+b.$
2024 Junior Balkan Team Selection Tests - Romania, P4
Let $ABC$ be a triangle. An arbitrary circle which passes through the points $B,C$ intersects the sides $AC,AB$ for the second time in $D,E$ respectively. The line $BD$ intersects the circumcircle of the triangle $AEC$ at $P{}$ and $Q{}$ and the line $CE$ intersects the circumcircle of the triangle $ABD$ at $R{}$ and $S{}$ such that $P{}$ is situated on the segment $BD{}$ and $R{}$ lies on the segment $CE.$ Prove that:
[list=a]
[*]The points $P,Q,R$ and $S{}$ are concyclic.
[*]The triangle $APQ$ is isosceles.
[/list]
[i]Petru Braica[/i]