Found problems: 85335
1992 Taiwan National Olympiad, 1
Let $A,B$ be two points on a give circle, and $M$ be the midpoint of one of the arcs $AB$ . Point $C$ is the orthogonal projection of $B$ onto the tangent $l$ to the circle at $A$. The tangent at $M$ to the circle meets $AC,BC$ at $A',B'$ respectively. Prove that if $\hat{BAC}<\frac{\pi}{8}$ then $S_{ABC}<2S_{A'B'C'}$.
2017 District Olympiad, 2
Let $ E(x,y)=\frac{x}{y} +\frac{x+1}{y+1} +\frac{x+2}{y+2} . $
[b]a)[/b] Solve in $ \mathbb{N}^2 $ the equation $ E(x,y)=3. $
[b]b)[/b] Show that there are infinitely many natural numbers $ n $ such that the equation $ E(x,y)=n $ has at least one solution in $ \mathbb{N}^2. $
2019 All-Russian Olympiad, 4
Let $ABC$ be an acute-angled triangle with $AC<BC.$ A circle passes through $A$ and $B$ and crosses the segments $AC$ and $BC$ again at $A_1$ and $B_1$ respectively. The circumcircles of $A_1B_1C$ and $ABC$ meet each other at points $P$ and $C.$ The segments $AB_1$ and $A_1B$ intersect at $S.$ Let $Q$ and $R$ be the reflections of $S$ in the lines $CA$ and $CB$ respectively. Prove that the points $P,$ $Q,$ $R,$ and $C$ are concyclic.
2010 Canadian Mathematical Olympiad Qualification Repechage, 5
The Fibonacci sequence is de ned by $f_1=f_2=1$ and $f_n=f_{n-1}+f_{n-2}$ for $n\ge 3$. A Pythagorean triangle is a right-angled triangle with integer side lengths. Prove that $f_{2k+1}$ is the hypotenuse of a Pythagorean triangle for every positive integer $k$ with $k\ge 2$
2021 Taiwan TST Round 3, 2
Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$. Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$. Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$, and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$. Rays $PM$ and $QN$ meet at $X$. Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$.
Show that $A,X,Y$ are collinear.
1968 Spain Mathematical Olympiad, 8
We will assume that the sides of a square are reflective and we will designate them with the names of the four cardinal points. Marking a point on the side $N$ , determine in which direction a ray of light should exit (into the interior of the square) so that it returns to it after having undergone $n$ reflections on the side $E$ , another $n$ on the side $W$ , $m$ on the $S$ and $m - 1$ on the $N$, where $n$ and $m$ are known natural numbers. What happens if m and $n$ are not prime to each other? Calculate the length of the light ray considered as a function of $m$ and $n$, and of the length of the side of the square.
2004 District Olympiad, 3
On the tetrahedron $ ABCD $ make the notation $ M,N,P,Q, $ for the midpoints of $ AB,CD,AC, $ respectively, $ BD. $ Additionally, we know that $ MN $ is the common perpendicular of $ AB,CD, $ and $ PQ $ is the common perpendicular of $ AC,BD. $ Show that $ AB=CD, BC=DA, AC=BD. $
1990 AMC 12/AHSME, 23
If $x,y>0$, $\log_yx+\log_xy=\frac{10}{3}$ and $xy=144$, then $\frac{x+y}{2}=$
$ \textbf{(A)}\ 12\sqrt{2} \qquad\textbf{(B)}\ 13\sqrt{3} \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)}\ 30 \qquad\textbf{(E)}\ 36 $
1972 Miklós Schweitzer, 8
Given four points $ A_1,A_2,A_3,A_4$ in the plane in such a way that $ A_4$ is the centroid of the $ \bigtriangleup A_1A_2A_3$,
find a point $ A_5$ in the plane that maximizes the ratio \[ \frac{\min_{1 \leq i < j < k \leq 5}T(A_iA_jA_k)}{\max_{1 \leq i < j < k \leq 5}T(A_iA_jA_k)}.\] ($ T(ABC)$ denotes the area of the triangle $ \bigtriangleup ABC.$ )
[i]J. Suranyi[/i]
2024 Belarusian National Olympiad, 11.5
On the chord $AB$ of the circle $\omega$ points $C$ and $D$ are chosen such that $AC=BD$ and point $C$ lies between $A$ and $D$. On $\omega$ point $E$ and $F$ are marked, they lie on different sides with respect to line $AB$ and lines $EC$ and $FD$ are perpendicular to $AB$. The angle bisector of $AEB$ intersects line $DF$ at $R$
Prove that the circle with center $F$ and radius $FR$ is tangent to $\omega$
[i]V. Kamenetskii, D. Bariev[/i]
2010 HMNT, 4
An ant starts at the point $(1, 0)$. Each minute, it walks from its current position to one of the four adjacent lattice points until it reaches a point $(x, y)$ with $|x| + |y| \le 2$. What is the probability that the ant ends at the point $(1, 1)$?
2011 Princeton University Math Competition, A5
Let
\[f_1(x) = \frac{1}{x}\quad\text{and}\quad f_2(x) = 1 - x\]
Let $H$ be the set of all compositions of the form $h_1 \circ h_2 \circ \ldots \circ h_k$, where each $h_i$ is either $f_1$ or $f_2$. For all $h$ in $H$, let $h^{(n)}$ denote $h$ composed with itself $n$ times. Find the greatest integer $N$ such that $\pi, h(\pi), \ldots, h^{(N)}(\pi)$ are all distinct for some $h$ in $H$.
2002 Germany Team Selection Test, 2
Prove: If $x, y, z$ are the lengths of the angle bisectors of a triangle with perimeter 6, than we have:
\[\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} \geq 1.\]
2014 BAMO, 4
Let $F_1, F_2, F_3 \cdots$ be the Fibonacci sequence, the sequence of positive integers satisfying $$F_1 =F_2=1$$ and $$F_{n+2} = F_{n+1} + F_n$$ for all $n \ge 1$.
Does there exist an $n \ge 1$ such that $F_n$ is divisible by $2014$? Prove your answer.
1997 AMC 8, 7
The area of the smallest square that will contain a circle of radius 4 is
$\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 128$
2012 Ukraine Team Selection Test, 1
Let $a, b, c$ be positive reals. Prove that $\sqrt{2a^2+bc}+\sqrt{2b^2+ac}+\sqrt{2c^2+ab}\ge 3 \sqrt{ab+bc+ca}$
2024 CCA Math Bonanza, L3.4
Regular hexagon $ABCDEF$ has side length $2$. Points $M$ and $N$ lie on $BC$ and $DE$, respectively. Find the minimum possible value of $(AM + MN + NA)^2$.
[i]Lightning 3.4[/i]
2022 JHMT HS, 5
Let $P(x)$ be a quadratic polynomial satisfying the following conditions:
[list]
[*] $P(x)$ has leading coefficient $1$.
[*] $P(x)$ has nonnegative integer roots that are at most $2022$.
[*] the set of the roots of $P(x)$ is a subset of the set of the roots of $P(P(x))$.
[/list]
Let $S$ be the set of all such possible $P(x)$, and let $Q(x)$ be the polynomial obtained upon summing all the elements of $S$. Find the sum of the roots of $Q(x)$.
2014 India PRMO, 10
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 $ABC$?
1991 IMTS, 2
Find all pairs of integers, $n$ and $k$, $2 < k < n$, such that the binomial coefficients
\[\binom{n}{k-1}, \binom{n}{k}, \binom{n}{k+1}\]
form an increasing arithmetic series.
Kvant 2023, M2769
The incircle of the triangle $ABC$ touches the sides $BC, CA$ and $AB{}$ at $D,E$ and $F{}$ respectively. Let the circle $\omega$ touch the segments $CA{}$ and $AB{}$ at $Q{}$ and $R{}$ respectively, and the points $M{}$ and $N{}$ are selected on the segments $AB{}$ and $AC{}$ respectively, so that the segments $CM{}$ and $BN{}$ touch $\omega$. The bisectors of $\angle NBC$ and $\angle MCB$ intersect the segments $DE{}$ and $DF{}$ at $K{}$ and $L{}$ respectively. Prove that the lines $RK{}$ and $QL{}$ intersect on $\omega$.
[i]Proposed by Tran Quang Hung[/i]
2003 AMC 10, 18
What is the sum of the reciprocals of the roots of the equation
\[ \frac {2003}{2004}x \plus{} 1 \plus{} \frac {1}{x} \equal{} 0?
\]
$ \textbf{(A)}\ \minus{}\! \frac {2004}{2003} \qquad \textbf{(B)}\ \minus{} \!1 \qquad \textbf{(C)}\ \frac {2003}{2004} \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \frac {2004}{2003}$
2018 India IMO Training Camp, 1
Let $\Delta ABC$ be an acute triangle. $D,E,F$ are the touch points of incircle with $BC,CA,AB$ respectively. $AD,BE,CF$ intersect incircle at $K,L,M$ respectively. If,$$\sigma = \frac{AK}{KD} + \frac{BL}{LE} + \frac{CM}{MF}$$ $$\tau = \frac{AK}{KD}.\frac{BL}{LE}.\frac{CM}{MF}$$
Then prove that $\tau = \frac{R}{16r}$. Also prove that there exists integers $u,v,w$ such that, $uvw \neq 0$, $u\sigma + v\tau +w=0$.
1977 IMO Longlists, 7
Prove the following assertion: If $c_1,c_2,\ldots ,c_n\ (n\ge 2)$ are real numbers such that
\[ (n-1)(c_1^2+c_2^2+\cdots +c_n^2)=(c_1+c_2+\cdots + c_n)^2,\]
then either all these numbers are nonnegative or all these numbers are nonpositive.
1964 AMC 12/AHSME, 34
If $n$ is a multiple of $4$, the sum $s=1+2i+3i^2+ ... +(n+1)i^{n}$, where $i=\sqrt{-1}$, equals:
$ \textbf{(A)}\ 1+i\qquad\textbf{(B)}\ \frac{1}{2}(n+2) \qquad\textbf{(C)}\ \frac{1}{2}(n+2-ni) \qquad$
$ \textbf{(D)}\ \frac{1}{2}[(n+1)(1-i)+2]\qquad\textbf{(E)}\ \frac{1}{8}(n^2+8-4ni) $