Found problems: 85335
Ukrainian TYM Qualifying - geometry, X.13
A paper square is bent along the line $\ell$, which passes through its center, so that a non-convex hexagon is formed. Investigate the question of the circle of largest radius that can be placed in such a hexagon.
III Soros Olympiad 1996 - 97 (Russia), 9.6
Let $ABC$ be an isosceles right triangle with hypotenuse $AB$, $D$ be some point in the plane such that $2CD = AB$ and point $C$ inside the triangle $ABD$. We construct two rays with a start in $C$, intersecting $AD$ and $BD$ and perpendicular to them. On the first one, intersecting $AD$, we will plot the segment $CK = AD$, and on the second one - $CM = BD$. Prove that points $M$, $D$ and $K$ lie on the same line.
2009 India IMO Training Camp, 4
Let $ \gamma$ be circumcircle of $ \triangle ABC$.Let $ R_a$ be radius of circle touching $ AB,AC$&$ \gamma$ internally.Define $ R_b,R_c$ similarly.
Prove That $ \frac {1}{aR_a} \plus{} \frac {1}{bR_b} \plus{} \frac {1}{cR_c} \equal{} \frac {s^2}{rabc}$.
2017 CCA Math Bonanza, T8
A group of $25$ CCA students decide they want to go to Disneyland, which is $105$ miles away. To save some time, they rent a bus with capacity $10$ people which can travel up to $60$ miles per hour. On the other hand, a student will run up to $9$ miles per hour. However, because a complicated plan of getting on and off the bus may be confusing to some students, a student may only board the bus once. What is the least number of minutes it will take for all students to reach Disneyland?
Note: both the bus and students may travel backwards.
[i]2017 CCA Math Bonanza Team Round #8[/i]
2001 Saint Petersburg Mathematical Olympiad, 11.6
Find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that for any $x,y$ the following is true:
$$f(x+y+f(y))=f(x)+2y$$
[I]proposed by F. Petrov[/i]
2014 Junior Balkan Team Selection Tests - Romania, 2
Let $x_1, x_2 \ldots , x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p+x_q+x_r$ (with $1\le p<q<r\le 5$) are equal to $0$, then $x_1=x_2=\cdots=x_5=0$.
2004 France Team Selection Test, 2
Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$.
Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$.
[i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$.
Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.
2007 Indonesia TST, 3
On each vertex of a regular $ n\minus{}$gon there was a crow. Call this as initial configuration. At a signal, they all flew by and after a while, those $ n$ crows came back to the $ n\minus{}$gon, one crow for each vertex. Call this as final configuration. Determine all $ n$ such that: there are always three crows such that the triangle they formed in the initial configuration and the triangle they formed in the final configuration are both right-angled triangle.
1989 Romania Team Selection Test, 3
Let $F$ be the boundary and $M,N$ be any interior points of a triangle $ABC$. Consider the function $f_{M,N}: F \to R$ defined by $f_{M,N}(P) = MP^2 +NP^2$ and let $\eta_{M,N}$ be the number of points $P$ for which $f{M,N}$ attains its minimum.
(a) Prove that $1 \le \eta_{M,N} \le 3$.
(b) If $M$ is fixed, find the locus of $N$ for which $\eta_{M,N} > 1$.
(c) Prove that the locus of $M$ for which there are points $N$ such that $\eta_{M,N} = 3$ is the interior of a tangent hexagon.
2018 Bulgaria JBMO TST, 3
Prove for all positive real numbers $m,n,p,q$ that
$$\frac{m}{t+n+p+q} + \frac{n}{t+p+q+m} + \frac{p}{t+q+m+n} + \frac{q}{t+m+n+p} \geq \frac{4}{5},$$
where $t=\frac{m+n+p+q}{2}.$
2017 HMNT, 4
An equiangular hexagon has side lengths $1, 1, a, 1, 1, a$ in that order. Given that there exists a circle that intersects the hexagon at $12$ distinct points, we have $M < a < N$ for some real numbers $M$ and $N$. Determine the minimum possible value of the ratio $\frac{N}{M}$ .
2023 ISL, N8
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$,
\[
f^{bf(a)}(a+1)=(a+1)f(b).
\]
1997 Estonia Team Selection Test, 2
Prove that for all positive real numbers $a_1,a_2,\cdots a_n$ \[\frac{1}{\frac{1}{1+a_1}+\frac{1}{1+a_2}+\cdots +\frac{1}{1+a_n}}-\frac{1}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots +\frac{1}{a_n}}\geq \frac{1}{n}\] When does the inequality hold?
2021 Sharygin Geometry Olympiad, 10-11.7
Let $I$ be the incenter of a right-angled triangle $ABC$, and $M$ be the midpoint of hypothenuse $AB$. The tangent to the circumcircle of $ABC$ at $C$ meets the line passing through $I$ and parallel to $AB$ at point $P$. Let $H$ be the orthocenter of triangle $PAB$. Prove that lines $CH$ and $PM$ meet at the incircle of triangle $ABC$.
2010 Contests, 3
On a line $l$ there are three different points $A$, $B$ and $P$ in that order. Let $a$ be the line through $A$ perpendicular to $l$, and let $b$ be the line through $B$ perpendicular to $l$. A line through $P$, not coinciding with $l$, intersects $a$ in $Q$ and $b$ in $R$. The line through $A$ perpendicular to $BQ$ intersects $BQ$ in $L$ and $BR$ in $T$. The line through $B$ perpendicular to $AR$ intersects $AR$ in $K$ and $AQ$ in $S$.
(a) Prove that $P$, $T$, $S$ are collinear.
(b) Prove that $P$, $K$, $L$ are collinear.
[i](2nd Benelux Mathematical Olympiad 2010, Problem 3)[/i]
2021 CMIMC Integration Bee, 14
$$\int_0^\infty \frac{\sin(20x)\sin(21x)}{x^2}\,dx$$
[i]Proposed by Connor Gordon and Vlad Oleksenko[/i]
2018 Peru MO (ONEM), 4
4) A $100\times 200$ board has $k$ black cells. An operations consists of choosing a $2\times 3$ or $3\times 2$ sub-board having exactly $5$ black cells and painting of black the remaining cell. Find the least value of $k$ for which exists an initial distribution of the black cells such that after some operations the board is completely black.
1994 Baltic Way, 8
Show that for any integer $a\ge 5$ there exist integers $b$ and $c$, $c\ge b\ge a$, such that $a,b,c$ are the lengths of the sides of a right-angled triangle.
2024 Israel National Olympiad (Gillis), P2
A positive integer $x$ satisfies the following:
\[\{\frac{x}{3}\}+\{\frac{x}{5}\}+\{\frac{x}{7}\}+\{\frac{x}{11}\}=\frac{248}{165}\]
Find all possible values of
\[\{\frac{2x}{3}\}+\{\frac{2x}{5}\}+\{\frac{2x}{7}\}+\{\frac{2x}{11}\}\]
where $\{y\}$ denotes the fractional part of $y$.
2013 BMT Spring, 7
Let $ABC$ be a triangle with $BC = 5$, $CA = 3$, and $AB = 4$. Variable points $P, Q$ are on segments $AB$, $AC$, respectively such that the area of $APQ$ is half of the area of $ABC$. Let $x$ and $y$ be the lengths of perpendiculars drawn from the midpoint of $PQ$ to sides $AB$ and $AC$, respectively. Find the range of values of $2y + 3x$.
2017 Math Prize for Girls Problems, 2
In the figure below, $BDEF$ is a square inscribed in $\triangle ABC$. If $\frac{AB}{BC} = \frac{4}{5}$, what is the area of $BDEF$ divided by the area of $\triangle ABC$?
[asy]
unitsize(20);
pair A = (0, 3);
pair B = (0, 0);
pair C = (4, 0);
draw(A -- B -- C -- cycle);
real w = 12.0 / 7;
pair D = (w, 0);
pair E = (w, w);
pair F = (0, w);
draw(D -- E -- F);
dot(Label("$A$", A, NW), A);
dot(Label("$B$", B, SW), B);
dot(Label("$C$", C, SE), C);
dot(Label("$D$", D, S), D);
dot(Label("$E$", E, NE), E);
dot(Label("$F$", F, W), F);
[/asy]
2001 IMO, 4
Let $n$ be an odd integer greater than 1 and let $c_1, c_2, \ldots, c_n$ be integers. For each permutation $a = (a_1, a_2, \ldots, a_n)$ of $\{1,2,\ldots,n\}$, define $S(a) = \sum_{i=1}^n c_i a_i$. Prove that there exist permutations $a \neq b$ of $\{1,2,\ldots,n\}$ such that $n!$ is a divisor of $S(a)-S(b)$.
LMT Team Rounds 2021+, B3
Aidan rolls a pair of fair, six sided dice. Let$ n$ be the probability that the product of the two numbers at the top is prime. Given that $n$ can be written as $a/b$ , where $a$ and $b$ are relatively prime positive integers, find $a +b$.
[i]Proposed by Aidan Duncan[/i]
2018 AMC 8, 14
Let $N$ be the greatest five-digit number whose digits have a product of $120$. What is the sum of the digits of $N$?
$\textbf{(A) }15\qquad\textbf{(B) }16\qquad\textbf{(C) }17\qquad\textbf{(D) }18\qquad\textbf{(E) }20$
2019 Ramnicean Hope, 2
Let $ P,Q,R $ be the intersections of the medians $ AD,BE,CF $ of a triangle $ ABC $ with its circumcircle, respectively. Show that $ ABC $ is equilateral if $ \overrightarrow{DP} +\overrightarrow{EQ} +\overrightarrow{FR} =0. $
[i]Dragoș Lăzărescu[/i]