Found problems: 85335
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]
2013 Saudi Arabia BMO TST, 7
Ayman wants to color the cells of a $50 \times 50$ chessboard into black and white so that each $2 \times 3$ or $3 \times 2$ rectangle contains an even number of white cells. Determine the number of ways Ayman can color the chessboard.
2013 Tournament of Towns, 6
There are fi ve distinct real positive numbers. It is known that the total sum of their squares and the total sum of their pairwise products are equal.
(a) Prove that we can choose three numbers such that it would not be possible to make a triangle with sides' lengths equal to these numbers.
(b) Prove that the number of such triples is at least six (triples which consist of the same numbers in different order are considered the same).
2020 Junior Macedonian National Olympiad, 1
Let $S$ be the set of all positive integers $n$ such that each of the numbers $n + 1$, $n + 3$, $n + 4$, $n + 5$, $n + 6$, and $n + 8$ is composite. Determine the largest integer $k$ with the following property: For each $n \in S$ there exist at least $k$ consecutive composite integers in the set
{$n, n +1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9$}.