Found problems: 85335
2008 China Team Selection Test, 2
Let $ x,y,z$ be positive real numbers, show that $ \frac {xy}{z} \plus{} \frac {yz}{x} \plus{} \frac {zx}{y} > 2\sqrt [3]{x^3 \plus{} y^3 \plus{} z^3}.$
2016 Saudi Arabia BMO TST, 2
A circle with center $O$ passes through points $A$ and $C$ and intersects the sides $AB$ and $BC$ of triangle $ABC$ at points $K$ and $N$, respectively. The circumcircles of triangles $ABC$ and $KBN$ meet at distinct points $B$ and $M$. Prove that $\angle OMB = 90^o$.
2022 HMNT, 10
There are 21 competitors with distinct skill levels numbered 1, 2,..., 21. They participate in a ping-pong tournament as follows. First, a random competitor is chosen to be "active", while the rest are "inactive." Every round, a random inactive competitor is chosen to play against the current active one. The player with the higher skill will win and become (or remain) active, while the loser will be eliminated from the tournament. The tournament lasts for 20 rounds, after which there will only be one player remaining. Alice is the competitor with skill 11. What is the expected number of games that she will get to play?
2016 CMIMC, 6
In parallelogram $ABCD$, angles $B$ and $D$ are acute while angles $A$ and $C$ are obtuse. The perpendicular from $C$ to $AB$ and the perpendicular from $A$ to $BC$ intersect at a point $P$ inside the parallelogram. If $PB=700$ and $PD=821$, what is $AC$?
2012 Flanders Math Olympiad, 2
Let $n$ be a natural number. Call $a$ the smallest natural number you need to subtract from $n$ to get a perfect square. Call $b$ the smallest natural number that you must add to $n$ to get a perfect square. Prove that $n - ab$ is a perfect square.
1966 All Russian Mathematical Olympiad, 080
Given a triangle $ABC$. Consider all the tetrahedrons $PABC$ with $PH$ -- the smallest of all tetrahedron's heights. Describe the set of all possible points $H$.
2017 Princeton University Math Competition, A2/B4
Suppose $z^{3}=2+2i$, where $i=\sqrt{-1}$. The product of all possible values of the real part of $z$ can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2019 Saudi Arabia JBMO TST, 3
Are there positive integers $a, b, c$, such that the numbers $a^2bc+2, b^2ca+2, c^2ab+2$ be perfect squares?
2019 AMC 10, 6
A positive integer $n$ satisfies the equation $(n+1)! + (n+2)! = n! \cdot 440$. What is the sum of the digits of $n$?
$\textbf{(A) }2\qquad\textbf{(B) }5\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }15$
2009 Tournament Of Towns, 4
Let $ABCD$ be a rhombus. $P$ is a point on side $ BC$ and $Q$ is a point on side $CD$ such that $BP = CQ$. Prove that centroid of triangle $APQ$ lies on the segment $BD.$
[i](6 points)[/i]
2016 CHMMC (Fall), 7
Let $f(x) = \frac{1}{1-\frac{3x}{16}}$. Consider the sequence $\{ 0, f(0), f(f(0)), f^3(0), \dots \}$ Find the smallest $L$ such that $f^n(0) \leq L$ for all $n$. If the sequence is unbounded, write none as your answer.
2000 Harvard-MIT Mathematics Tournament, 24
At least how many moves must a knight make to get from one corner of a chessboard to the opposite corner?
2018 Polish Junior MO First Round, 6
Positive integers $k, m, n$ satisfy the equation $m^2 + n = k^2 + k$. Show that $m \le n$.
1990 Austrian-Polish Competition, 6
$p(x)$ is a polynomial with integer coefficients. The sequence of integers $a_1, a_2, ... , a_n$ (where $n > 2$) satisfies $a_2 = p(a_1), a_3 = p(a_2), ... , a_n = p(a_{n-1}), a_1 = p(a_n)$. Show that $a_1 = a_3$.
2013 AMC 10, 3
Square $ ABCD $ has side length $ 10 $. Point $ E $ is on $ \overline{BC} $, and the area of $ \bigtriangleup ABE $ is $ 40 $. What is $ BE $?
$\textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 8 \qquad $
[asy]
pair A,B,C,D,E;
A=(0,0);
B=(0,50);
C=(50,50);
D=(50,0);
E = (30,50);
draw(A--B);
draw(B--E);
draw(E--C);
draw(C--D);
draw(D--A);
draw(A--E);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
label("A",A,SW);
label("B",B,NW);
label("C",C,NE);
label("D",D,SE);
label("E",E,N);
[/asy]
2002 IMO Shortlist, 6
Let $A$ be a non-empty set of positive integers. Suppose that there are positive integers $b_1,\ldots b_n$ and $c_1,\ldots,c_n$ such that
- for each $i$ the set $b_iA+c_i=\left\{b_ia+c_i\colon a\in A\right\}$ is a subset of $A$, and
- the sets $b_iA+c_i$ and $b_jA+c_j$ are disjoint whenever $i\ne j$
Prove that \[{1\over b_1}+\,\ldots\,+{1\over b_n}\leq1.\]
2012 Gheorghe Vranceanu, 1
For which natural numbers $ n $ the floor of the number $ \frac{n^3+8n^2+1}{3n} $ is prime?
[i]Gabriel Popa[/i]
2022 Malaysia IMONST 2, 2
The following list shows every number for which more than half of its digits are digits $2$, in increasing order:
$$2, 22, 122, 202, 212, 220, 221, 222, 223, 224, \dots$$
If the $n$th term in the list is $2022$, what is $n$?
2014 BMT Spring, P1
Suppose that $a,b,c,d$ are non-negative real numbers such that $a^2+b^2+c^2+d^2=2$ and $ab+bc+cd+da=1$. Find the maximum value of $a+b+c+d$ and determine all equality cases.
2022 Miklós Schweitzer, 2
Original in Hungarian; translated with Google translate; polished by myself.
Let $n$ be a positive integer. Suppose that the sum of the matrices $A_1, \dots, A_n\in \Bbb R^{n\times n}$ is the identity matrix, but
$\sum\nolimits_{i = 1}^n\alpha_i A_i$ is singular whenever at least one of the coefficients $\alpha_i \in \Bbb R$ is zero.
a) Show that $\sum\nolimits_{i = 1}^n\alpha_i A_i$ is nonsingular if $\alpha_i\neq 0$ for all $i$.
b) Show that if the matrices $A_i$ are symmetric, then all of them have rank $1$.
1979 Swedish Mathematical Competition, 2
Find rational $x$ in $(3,4)$ such that $\sqrt{x-3}$ and $\sqrt{x+1}$ are rational.
LMT Team Rounds 2021+, 6
For all $y$, define cubic $f_y (x)$ such that $f_y (0) = y$, $f_y (1) = y +12$, $f_y (2) = 3y^2$, $f_y (3) = 2y +4$. For all $y$, $f_y(4)$ can be expressed in the form $ay^2 +by +c$ where $a,b,c$ are integers. Find $a +b +c$.
1980 Bundeswettbewerb Mathematik, 1
Let $a$ and $b$ be integers. Prove that if $\sqrt[3]{a}+\sqrt[3]{b}$ is a rational number, then both $a$ and $b$ are perfect cubes.
1997 Brazil National Olympiad, 1
Given $R, r > 0$. Two circles are drawn radius $R$, $r$ which meet in two points. The line joining the two points is a distance $D$ from the center of one circle and a distance $d$ from the center of the other. What is the smallest possible value for $D+d$?
2008 AMC 10, 4
Suppose that $ \frac{2}{3}$ of $ 10$ bananas are worth as much as $ 8$ oranges. How many oranges are worth as much is $ \frac{1}{2}$ of $ 5$ bananas?
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ \frac{5}{2} \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ \frac{7}{2} \qquad
\textbf{(E)}\ 4$