Found problems: 85335
1996 Swedish Mathematical Competition, 3
For every positive integer $n$, we define the function $p_n$ for $x\ge 1$ by
$$p_n(x) = \frac12 \left(\left(x+\sqrt{x^2-1}\right)^n+\left(x-\sqrt{x^2-1}\right)^n\right).$$
Prove that $p_n(x) \ge 1$ and that $p_{mn}(x) = p_m(p_n(x))$.
2017 AMC 8, 10
A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected?
$\textbf{(A) }\frac{1}{10}\qquad\textbf{(B) }\frac{1}{5}\qquad\textbf{(C) }\frac{3}{10}\qquad\textbf{(D) }\frac{2}{5}\qquad\textbf{(E) }\frac{1}{2}$
1990 AIME Problems, 11
Someone observed that $6! = 8 \cdot 9 \cdot 10$. Find the largest positive integer $n$ for which $n!$ can be expressed as the product of $n - 3$ consecutive positive integers.
1989 AIME Problems, 12
Let $ABCD$ be a tetrahedron with $AB=41$, $AC=7$, $AD=18$, $BC=36$, $BD=27$, and $CD=13$, as shown in the figure. Let $d$ be the distance between the midpoints of edges $AB$ and $CD$. Find $d^{2}$.
[asy]
pair C=origin, D=(4,11), A=(8,-5), B=(16,0);
draw(A--B--C--D--B^^D--A--C);
draw(midpoint(A--B)--midpoint(C--D), dashed);
label("27", B--D, NE);
label("41", A--B, SE);
label("7", A--C, SW);
label("$d$", midpoint(A--B)--midpoint(C--D), NE);
label("18", (7,8), SW);
label("13", (3,9), SW);
pair point=(7,0);
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));[/asy]
2018 Saudi Arabia BMO TST, 2
Suppose that $2018$ numbers $1$ and $-1$ are written around a circle. For every two adjacent numbers, their product is taken. Suppose that the sum of all $2018$ products is negative. Find all possible values of sum of $2018$ given numbers.
2007 JBMO Shortlist, 1
$\boxed{\text{G1}}$ Let $M$ be interior point of the triangle $ABC$ with <BAC=70and <ABC=80 If <ACM=10 and <CBM=20.Prove that $AB=MC$
2023 Azerbaijan Senior NMO, 1
The teacher calculates and writes on the board all the numbers $a^b$ that satisfy the condition $n = a\times b$ for the natural number $n.$ Here $a$ and $b$ are natural numbers. Is there a natural number $n$ such that each of the numbers $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ is the last digit of one of the numbers written by the teacher on the board? Justify your opinion.
2008 Alexandru Myller, 2
Let $ A,B,S $ be three $ 3\times 3 $ complex matrices with $ B=S^{-1}AS , $ and $ S $ nonsingular. Show:
$$ \text{tr} \left( B^2\right) +2\text{tr}(C(B)) = \left(\text{tr} (A)\right)^2 , $$
where $ C(B) $ is the cofactor of $ B. $
[i]Mihai Haivas[/i]
PEN J Problems, 1
Let $n$ be an integer with $n \ge 2$. Show that $\phi(2^{n}-1)$ is divisible by $n$.
2009 District Olympiad, 1
Let $ f:[0,\infty )\longrightarrow [0,\infty ) $ a nonincreasing function that satisfies the inequality:
$$ \int_0^x f(t)dt <1,\quad\forall x\ge 0. $$ Prove the following affirmations:
[b]a)[/b] $ \exists \lim_{x\to\infty} \int_0^x f(t)dt \in\mathbb{R} . $
[b]b)[/b] $ \lim_{x\to\infty} xf(x) =0. $
2013 Moldova Team Selection Test, 2
Find all pairs of real numbers $(x,y)$ satisfying
$\left\{\begin{array}{rl}
2x^2+xy &=1 \\
\frac{9x^2}{2(1-x)^4}&=1+\frac{3xy}{2(1-x)^2}
\end{array}\right.$
2018 Bulgaria EGMO TST, 2
A country has $100$ cities and $n$ airplane companies which take care of a total of $2018$ two-way direct flights between pairs of cities. There is a pair of cities such that one cannot reach one from the other with just one or two flights. What is the largest possible value of $n$ for which between any two cities there is a route (a sequence of flights) using only one of the airplane companies?
2012 USAMTS Problems, 1
Fill in each of the ten boxes with a 3-digit number so that the following conditions are satisfied.
[list=1]
[*]Every number has three distinct digits that sum to $15$. $0$ may not be a leading digit. One digit of each number has been given to you.
[*]No two numbers in any pair of boxes use the same three digits. For example, it is not allowed for two different boxes to have the numbers $456$ and $645$.
[*]Two boxes joined by an arrow must have two numbers that share an equal hundreds digit, tens digit, or ones digit. Also, the smaller number must point to the larger.[/list]
You do not need to prove that your configuration is the only one possible; you merely need to find a configuration that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)
[asy]
size(200);
defaultpen(linewidth(0.8));
path arrow;
pair squares[]={(2,4),(6,4),(10,4),(0,0),(4,0),(8,0),(12,0),(2,-4),(6,-4),(10,-4)};
pair horizarrows[]={(4,4),(2,0),(6,0),(10,0),(4,-4),(8,-4)};
bool isLeft[]={false,false,true,false,false,false};
pair diagarrows[]={(1,2),(7,2),(9,2),(1,-2),(5,-2),(11,-2)};
bool isDown[]={true,false,true,false,false,true};
for(int i=0;i<=9;i=i+1)
{
draw(box(squares[i]-(1,1),squares[i]+(1,1)));
label("$"+(string)i+"$",squares[i]);
}
for(int j=0;j<=5;j=j+1)
{
if(isLeft[j])
arrow=(horizarrows[j].x-1,horizarrows[j].y)--(horizarrows[j].x+1,horizarrows[j].y);
else
arrow=(horizarrows[j].x+1,horizarrows[j].y)--(horizarrows[j].x-1,horizarrows[j].y);
draw(arrow,BeginArrow(size=7));
}
for(int k=0;k<=5;k=k+1)
{
if(isDown[k])
arrow=(diagarrows[k].x-1/3,diagarrows[k].y-1)--(diagarrows[k].x+1/3,diagarrows[k].y+1);
else
arrow=(diagarrows[k].x-1/3,diagarrows[k].y+1)--(diagarrows[k].x+1/3,diagarrows[k].y-1);
draw(arrow,BeginArrow(size=7));
}
[/asy]
1966 IMO Longlists, 23
Three faces of a tetrahedron are right triangles, while the fourth is not an obtuse triangle.
[i](a) [/i]Prove that a necessary and sufficient condition for the fourth face to be a right triangle is that at some vertex exactly two angles are right.
[i](b)[/i] Prove that if all the faces are right triangles, then the volume of the tetrahedron equals one -sixth the product of the three smallest edges not belonging to the same face.
2004 Iran Team Selection Test, 3
Suppose that $ ABCD$ is a convex quadrilateral. Let $ F \equal{} AB\cap CD$, $ E \equal{} AD\cap BC$ and $ T \equal{} AC\cap BD$. Suppose that $ A,B,T,E$ lie on a circle which intersects with $ EF$ at $ P$. Prove that if $ M$ is midpoint of $ AB$, then $ \angle APM \equal{} \angle BPT$.
2020 USMCA, 1
Let $\mathcal P$ be a finite set of squares on an infinite chessboard. Kelvin the Frog notes that $\mathcal P$ may be tiled with only $1 \times 2$ dominoes, while Alex the Kat notes that $\mathcal P$ may be tiled with only $2 \times 1$ dominoes. The dominoes cannot be rotated in each tiling. Prove that the area of $\mathcal P$ is a multiple of 4.
LMT Speed Rounds, 21
If $a \diamondsuit b = \vert a - b \vert \cdot \vert b - a \vert$ then find the value of $1 \diamondsuit (2 \diamondsuit (3 \diamondsuit (4 \diamondsuit 5)))$.
[i]Proposed by Muztaba Syed[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{9}$
$a\diamondsuit b = (a-b)^2$. This gives us an answer of $\boxed{9}$.
[/hide]
2023 CCA Math Bonanza, I15
Triangle $ABC$ has side lengths $AB=7, BC=8, CA=9.$ Define $M,N,P$ to be the midpoints of sides $BC,CA,AB,$ respectively. The circumcircles of $\triangle APN$ and $\triangle ABM$ intersect at another point $K.$ Find $NK.$
[i]Individual #15[/i]
2019 Romanian Master of Mathematics, 4
Prove that for every positive integer $n$ there exists a (not necessarily convex) polygon with no three collinear vertices, which admits exactly $n$ diffferent triangulations.
(A [i]triangulation[/i] is a dissection of the polygon into triangles by interior diagonals which have no common interior points with each other nor with the sides of the polygon)
2018 District Olympiad, 4
a) Consider the positive integers $a, b, c$ so that $a < b < c$ and $a^2+b^2 = c^2$. If $a_1 = a^2$, $a_2 = ab$, $a_3 = bc$, $a_4 = c^2$, prove that $a_1^2+a_2^2+a_3^2=a_4^2$ and $a_1 < a_2 < a_3 < a_4$.
b) Show that for any $n \in N$, $n\ge 3$, there exist the positive integers $a_1, a_2,..., a_n$ so that $a_1^2+a_2^2+...+ a_{n-1}^2=a_n^2$ and $a_1 < a_2 < ...< a_{n-1} < a_n$
2016 CMIMC, 5
Determine the sum of the positive integers $n$ such that there exist primes $p,q,r$ satisfying $p^{n} + q^{2} = r^{2}$.
Novosibirsk Oral Geo Oly VIII, 2019.7
The square was cut into acute -angled triangles. Prove that there are at least eight of them.
2019-2020 Winter SDPC, 8
Let $ABC$ be a triangle with circumcircle $\Gamma$. If the internal angle bisector of $\angle A$ meets $BC$ and $\Gamma$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $A$ and $D$ tangent to $BC$, let the external angle bisector of $\angle A$ meet $\Gamma$ at $F$, and let $FO_1$ meet $\Gamma$ at some point $P \neq F$. Show that the circumcircle of $DEP$ is tangent to $BC$.
2009 China Team Selection Test, 2
Let $ n,k$ be given positive integers satisfying $ k\le 2n \minus{} 1$. On a table tennis tournament $ 2n$ players take part, they play a total of $ k$ rounds match, each round is divided into $ n$ groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer $ m \equal{} f(n,k)$ such that no matter how the tournament processes, we always find $ m$ players each of pair of which didn't match each other.
2012 Iran MO (3rd Round), 5
Two fixed lines $l_1$ and $l_2$ are perpendicular to each other at a point $Y$. Points $X$ and $O$ are on $l_2$ and both are on one side of line $l_1$. We draw the circle $\omega$ with center $O$ and radius $OY$. A variable point $Z$ is on line $l_1$. Line $OZ$ cuts circle $\omega$ in $P$. Parallel to $XP$ from $O$ intersects $XZ$ in $S$. Find the locus of the point $S$.
[i]Proposed by Nima Hamidi[/i]