Found problems: 85335
2007 Pan African, 2
Let $A$, $B$ and $C$ be three fixed points, not on the same line. Consider all triangles $AB'C'$ where $B'$ moves on a given straight line (not containing $A$), and $C'$ is determined such that $\angle B'=\angle B$ and $\angle C'=\angle C$. Find the locus of $C'$.
2006 Tournament of Towns, 4
Quadrilateral $ABCD$ is a cyclic, $AB = AD$. Points $M$ and $N$ are chosen on sides $BC$ and $CD$ respectfully so that $\angle MAN =1/2 (\angle BAD)$. Prove that $MN = BM + ND$.
[i](5 points)[/i]
2017 South East Mathematical Olympiad, 4
For any positive integer $n$, let $D_n$ denote the set of all positive divisors of $n$, and let $f_i(n)$ denote the size of the set
$$F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}$$where $i = 0, 1, 2, 3$.
Determine the smallest positive integer $m$ such that $f_0(m) + f_1(m) - f_2(m) - f_3(m) = 2017$.
2015 CCA Math Bonanza, L1.2
Let $ABCDEF$ be a regular hexagon with side length $2$. Calculate the area of $ABDE$.
[i]2015 CCA Math Bonanza Lightning Round #1.2[/i]
1995 South africa National Olympiad, 2
Find all pairs $(m,n)$ of natural numbers with $m<n$ such that $m^2+1$ is a multiple of $n$ and $n^2+1$ is a multiple of $m$.
2003 China Team Selection Test, 2
Denote by $\left(ABC\right)$ the circumcircle of a triangle $ABC$.
Let $ABC$ be an isosceles right-angled triangle with $AB=AC=1$ and $\measuredangle CAB=90^{\circ}$. Let $D$ be the midpoint of the side $BC$, and let $E$ and $F$ be two points on the side $BC$.
Let $M$ be the point of intersection of the circles $\left(ADE\right)$ and $\left(ABF\right)$ (apart from $A$).
Let $N$ be the point of intersection of the line $AF$ and the circle $\left(ACE\right)$ (apart from $A$).
Let $P$ be the point of intersection of the line $AD$ and the circle $\left(AMN\right)$.
Find the length of $AP$.
1998 China Team Selection Test, 3
For a fixed $\theta \in \lbrack 0, \frac{\pi}{2} \rbrack$, find the smallest $a \in \mathbb{R}^{+}$ which satisfies the following conditions:
[b]I. [/b] $\frac{\sqrt a}{\cos \theta} + \frac{\sqrt a}{\sin \theta} >
1$.
[b]II.[/b] There exists $x \in \lbrack 1 - \frac{\sqrt a}{\sin \theta},
\frac{\sqrt a}{\cos \theta} \rbrack$ such that $\lbrack (1 -
x)\sin \theta - \sqrt{a - x^2 \cos^{2} \theta} \rbrack^{2} +
\lbrack x\cos \theta - \sqrt{a - (1 - x)^2 \sin^{2} \theta}
\rbrack^{2} \leq a$.
1976 Polish MO Finals, 4
The diagonals of some quadrilateral with sides $a,b,c,d$ are perpendicular. Prove that the diagonals of any other quadrilateral with sides $a,b,c,d $ also are perpendicular
2016 AMC 12/AHSME, 10
Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?
$\textbf{(A) }1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$
2016 BMT Spring, 3
Α half-mile long train is traveling at a speed of $90$ miles per hour. As it enters a $1$ mile long tunnel, Steve starts running from the back of the train to the front of the train at a speed of $10$ miles per hour. When Steve is out of the tunnel, he stops running. How far along the train has Steve run in miles?
2008 Harvard-MIT Mathematics Tournament, 6
Determine the number of non-degenerate rectangles whose edges lie completely on the grid lines of the following figure.
$ \begin{tabular}{|c|c|c|c|c|c|} \hline & & & & & \\
\hline & & & & & \\
\hline & & \multicolumn{1}{c}{} & & & \\
\cline{1 \minus{} 2}\cline{5 \minus{} 6} & & \multicolumn{1}{c}{} & & & \\
\hline & & & & & \\
\hline & & & & & \\
\hline \end{tabular}$
2002 China Team Selection Test, 1
Circle $ O$ is inscribed in a trapzoid $ ABCD$, $ \angle{A}$ and $ \angle{B}$ are all acute angles. A line through $ O$ intersects $ AD$ at $ E$ and $ BC$ at $ F$, and satisfies the following conditions:
(1) $ \angle{DEF}$ and $ \angle{CFE}$ are acute angles.
(2) $ AE\plus{}BF\equal{}DE\plus{}CF$.
Let $ AB\equal{}a$, $ BC\equal{}b$, $ CD\equal{}c$, then use $ a,b,c$ to express $ AE$.
Mathley 2014-15, 5
Triangle $ABC$ has incircle $(I)$ and $P,Q$ are two points in the plane of the triangle. Let $QA,QB,QC$ meet $BA,CA,AB$ respectively at $D,E,F$. The tangent at $D$, other than $BC$, of the circle $(I)$ meets $PA$ at $X$. The points $Y$ and $Z$ are defined in the same manner. The tangent at $X$, other than $XD$, of the circle $(I)$ meets $ (I)$ at $U$. The points $V,W$ are defined in the same way. Prove that three lines $(AU,BV,CW)$ are concurrent.
Tran Quang Hung, Dean of the Faculty of Science, Thanh Xuan, Hanoi.
2017 Auckland Mathematical Olympiad, 5
A rectangle $ABCD$ is given. On the side $AB$, n different points are chosen strictly between $A$ and $B$. Similarly, $m$ different points are chosen on the side $AD$ between $A$ and $D$. Lines are drawn from the points parallel to the sides. How many rectangles are formed in this way?
An example of a particular rectangle $ABCD$ is shown with a shaded one rectangle that may be formed in this way.
[img]https://cdn.artofproblemsolving.com/attachments/e/4/f7a04300f0c846fb6418d12dc23f5c74b54242.png[/img]
2025 239 Open Mathematical Olympiad, 7
Point $M$ is the midpoint of side $BC$ of an acute—angled triangle $ABC$. The point $U$ is symmetric to the orthocenter $ABC$ relative to its circumcenter. The point $S$ inside triangle $ABC$ is such that $US = UM$. Prove that $SA + SB + SC + AM < AB + BC + CA$.
2013 Today's Calculation Of Integral, 867
Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$
1962 All-Soviet Union Olympiad, 13
Given are $a_0,a_1, ... , a_n$, satisfying $a_0=a_n = 0$, and $a_{k-1} - 2a_k+a_{k+1}\ge 0$ for $k=0, 1, ... , n-1$. Prove that all the numbers are negative or zero.
2002 Junior Balkan Team Selection Tests - Moldova, 11
Simultaneously from the same point of a circular route and in the same direction for two hours two bodies move evenly. The first body performs a complete rotation three minutes faster than the second body and exceeds it every $9$ minutes and $20$ seconds. Whenever the first body will overtake the other the second exactly at the starting point?
2020 USA TSTST, 3
We say a nondegenerate triangle whose angles have measures $\theta_1$, $\theta_2$, $\theta_3$ is [i]quirky[/i] if there exists integers $r_1,r_2,r_3$, not all zero, such that
\[r_1\theta_1+r_2\theta_2+r_3\theta_3=0.\]
Find all integers $n\ge 3$ for which a triangle with side lengths $n-1,n,n+1$ is quirky.
[i]Evan Chen and Danielle Wang[/i]
1984 Tournament Of Towns, (073) 4
Six musicians gathered at a chamber music festival . At each scheduled concert some of these musicians played while the others listened as members of the audience . What is the least number of such concerts which would need to be scheduled in order to enable each musician to listen , as a member of the audience, to all the other musicians?
(Canadian origin)
2020 Federal Competition For Advanced Students, P1, 4
Determine all positive integers $N$ such that
$$2^N-2N$$
is a perfect square.
(Walther Janous)
2020 CMIMC Team, 1
In a game of ping-pong, the score is $4-10$. Six points later, the score is $10-10$. You remark that it was impressive that I won the previous $6$ points in a row, but I remark back that you have won $n$ points in a row. What the largest value of $n$ such that this statement is true regardless of the order in which the points were distributed?
2020 Saint Petersburg Mathematical Olympiad, 3.
$BB_1$ is the angle bisector of $\triangle ABC$, and $I$ is its incenter. The perpendicular bisector of segment $AC$ intersects the circumcircle of $\triangle AIC$ at $D$ and $E$. Point $F$ is on the segment $B_1C$ such that $AB_1=CF$.Prove that the four points $B, D, E$ and $F$ are concyclic.
2022 Saudi Arabia BMO + EGMO TST, 2.2
Given is an acute triangle $ABC$ with $BC < CA < AB$. Points $K$ and $L$ lie on segments $AC$ and $AB$ and satisfy $AK = AL = BC$. Perpendicular bisectors of segments $CK$ and $BL$ intersect line $BC$ at points $P$ and $Q$, respectively. Segments $KP$ and $LQ$ intersect at $M$. Prove that $CK + KM = BL + LM$.
VMEO II 2005, 5
Let $a,b$ be positive integers . How many integers numbers can be written in the form $ap+bq$, where $p,q$ are nonnegative integers and $p+q\le{2005}$ ?