Found problems: 85335
2012 AMC 10, 8
The sums of three whole numbers taken in pairs are $12$, $17$, and $19$. What is the middle number?
$ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 $
2007 Croatia Team Selection Test, 1
Find integral solutions to the equation \[(m^{2}-n^{2})^{2}=16n+1.\]
1996 Baltic Way, 14
The graph of the function $f(x)=x^n+a_{n-1}x_{n-1}+\ldots +a_1x+a_0$ (where $n>1$) intersects the line $y=b$ at the points $B_1,B_2,\ldots ,B_n$ (from left to right), and the line $y=c\ (c\not= b)$ at the points $C_1,C_2,\ldots ,C_n$ (from left to right). Let $P$ be a point on the line $y=c$, to the right to the point $C_n$. Find the sum
\[\cot (\angle B_1C_1P)+\ldots +\cot (\angle B_nC_nP) \]
2014 Harvard-MIT Mathematics Tournament, 22
Let $\omega$ be a circle, and let $ABCD$ be a quadrilateral inscribed in $\omega$. Suppose that $BD$ and $AC$ intersect at a point $E$. The tangent to $\omega$ at $B$ meets line $AC$ at a point $F$, so that $C$ lies between $E$ and $F$. Given that $AE=6$, $EC=4$, $BE=2$, and $BF=12$, find $DA$.
2005 Portugal MO, 6
Prove that there is a unique function $f: N\to N$, that verifies $$f(a + b)f(a - b) = f(a^2)$$, for any $a, b\in N$ such that $a > b$.
2023 Chile Classification NMO Juniors, 1
There are 10 numbers on a board. The product of any four of them is divisible by 30.
Prove that at least one of the numbers on the board is divisible by 30.
2020 Online Math Open Problems, 22
Let $ABC$ be a scalene triangle with incenter $I$ and symmedian point $K$. Furthermore, suppose that $BC = 1099$. Let $P$ be a point in the plane of triangle $ABC$, and let $D$, $E$, $F$ be the feet of the perpendiculars from $P$ to lines $BC$, $CA$, $AB$, respectively. Let $M$ and $N$ be the midpoints of segments $EF$ and $BC$, respectively. Suppose that the triples $(M,A,N)$ and $(K,I,D)$ are collinear, respectively, and that the area of triangle $DEF$ is $2020$ times the area of triangle $ABC$. Compute the largest possible value of $\lceil AB+AC\rceil$.
[i]Proposed by Brandon Wang[/i]
2022 Korea Junior Math Olympiad, 3
For a given odd prime number $p$, define $f(n)$ the remainder of $d$ divided by $p$, where $d$ is the biggest divisor of $n$ which is not a multiple of $p$. For example when $p=5$, $f(6)=1, f(35)=2, f(75)=3$. Define the sequence $a_1, a_2, \ldots, a_n, \ldots$ of integers as the followings:
[list]
[*]$a_1=1$
[*]$a_{n+1}=a_n+(-1)^{f(n)+1}$ for all positive integers $n$.
[/list]
Determine all integers $m$, such that there exist infinitely many positive integers $k$ such that $m=a_k$.
2020 ISI Entrance Examination, 2
Let $a$ be a fixed real number. Consider the equation
$$
(x+2)^{2}(x+7)^{2}+a=0, x \in R
$$
where $R$ is the set of real numbers. For what values of $a$, will the equ have exactly one double-root?
2018 South East Mathematical Olympiad, 7
There are $24$ participants attended a meeting. Each two of them shook hands once or not. A total of $216$ handshakes occured in the meeting. For any two participants who have shaken hands, at most $10$ among the rest $22$ participants have shaken hands with exactly one of these two persons. Define a [i]friend circle[/i] to be a group of $3$ participants in which each person has shaken hands with the other two. Find the minimum possible value of friend circles.
2016 Indonesia TST, 2
Determine all triples of real numbers $(x, y, z)$ which satisfy the following system of equations:
\[ \begin{cases} x+y+z=0 \\ x^3+y^3+z^3 = 90 \\ x^5+y^5+z^5=2850. \end{cases} \]
2010 India IMO Training Camp, 7
Let $ABCD$ be a cyclic quadrilaterla and let $E$ be the point of intersection of its diagonals $AC$ and $BD$. Suppose $AD$ and $BC$ meet in $F$. Let the midpoints of $AB$ and $CD$ be $G$ and $H$ respectively. If $\Gamma $ is the circumcircle of triangle $EGH$, prove that $FE$ is tangent to $\Gamma $.
2017 Czech-Polish-Slovak Junior Match, 1
Find the largest integer $n \ge 3$ for which there is a $n$-digit number $\overline{a_1a_2a_3...a_n}$ with non-zero digits
$a_1, a_2$ and $a_n$, which is divisible by $\overline{a_2a_3...a_n}$.
2010 Junior Balkan Team Selection Tests - Moldova, 7
In the triangle $ABC$ with $| AB | = c, | BC | = a, | CA | = b$ the relations hold simultaneously
$$a \ge max \{ b, c, \sqrt{bc}\}, \sqrt{(a - b) (a + c)} + \sqrt{(a - c) (a + b) } \ge 2\sqrt{a^2-bc}$$
Prove that the triangle $ABC$ is isosceles.
2008 AIME Problems, 2
Square $ AIME$ has sides of length $ 10$ units. Isosceles triangle $ GEM$ has base $ EM$, and the area common to triangle $ GEM$ and square $ AIME$ is $ 80$ square units. Find the length of the altitude to $ EM$ in $ \triangle GEM$.
2020 AMC 12/AHSME, 15
In the complex plane, let $A$ be the set of solutions to $z^3 - 8 = 0$ and let $B$ be the set of solutions to $z^3 - 8z^2 - 8z + 64 = 0$. What is the greatest distance between a point of $A$ and a point of $B?$
$\textbf{(A) } 2\sqrt{3} \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 2\sqrt{21} \qquad \textbf{(E) } 9 + \sqrt{3}$
2025 AIME, 7
The twelve letters $A$,$B$,$C$,$D$,$E$,$F$,$G$,$H$,$I$,$J$,$K$, and $L$ are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is $AB$, $CJ$, $DG$, $EK$, $FL$, $HI$. The probability that the last word listed contains $G$ is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
1993 All-Russian Olympiad, 2
The integers from $1$ to $1993$ are written in a line in some order. The following operation is performed with this line: if the first number is $k$ then the first $k$ numbers are rewritten in reverse order. Prove that after some finite number of these operations, the first number in the line of numbers will be $1$.
2003 National Olympiad First Round, 32
The function $f$ satisfies $f(x)+3f(1-x)=x^2$ for every real $x$. If $S=\{x \mid f(x)=0 \}$, which one is true?
$\textbf{(A)}$ $S$ is an infinite set.
$\textbf{(B)}$ $\{0,1\} \subset S$
$\textbf{(C)}$ $S=\phi$
$\textbf{(D)}$ $S = \{(3+\sqrt 3)/2, (3-\sqrt 3)/2\}$
$\textbf{(E)}$ None of above
2013 Regional Competition For Advanced Students, 3
For non-negative real numbers $a,$ $b$ let $A(a, b)$ be their arithmetic mean and $G(a, b)$ their geometric mean. We consider the sequence $\langle a_n \rangle$ with $a_0 = 0,$ $a_1 = 1$ and $a_{n+1} = A(A(a_{n-1}, a_n), G(a_{n-1}, a_n))$ for $n > 0.$
(a) Show that each $a_n = b^2_n$ is the square of a rational number (with $b_n \geq 0$).
(b) Show that the inequality $\left|b_n - \frac{2}{3}\right| < \frac{1}{2^n}$ holds for all $n > 0.$
2015 ASDAN Math Tournament, 18
Andrew takes a square sheet of paper $ABCD$ of side length $1$ and folds a kite shape. To do this, he takes the corners at $B$ and $D$ and folds the paper such that both corners now rest at a point $E$ on $AC$. This fold results in two creases $CF$ and $CG$, respectively, where $F$ lies on $AB$ and $G$ lies on $AD$. Compute the length of $FG$.
2014 Junior Balkan Team Selection Tests - Moldova, 4
A set $A$ contains $956$ natural numbers between $1$ and $2014$, inclusive. Prove that in the set $A$ there are two numbers $a$ and $b$ such that $a + b$ is divided by $19$.
2022 Baltic Way, 17
Let $n$ be a positive integer such that the sum of its positive divisors is at least $2022n$. Prove that $n$ has at least $2022$ distinct prime factors.
2012 Flanders Math Olympiad, 3
(a) Show that for any angle $\theta$ and for any natural number $m$:
$$| \sin m\theta| \le m| \sin \theta|$$
(b) Show that for all angles $\theta_1$ and $\theta_2$ and for all even natural numbers $m$:
$$| \sin m \theta_2 - \sin m \theta_1| \le m| \sin (\theta_2 - \theta_1)|$$
(c) Show that for every odd natural number $m$ there are two angles, resp. $\theta_1$ and $\theta_2$, exist for which the inequality in (b) is not valid.
2003 India National Olympiad, 1
Let $P$ be an interior point of an acute-angled triangle $ABC$. The line $BP$ meets the line $AC$ at $E$, and the line $CP$ meets the line $AB$ at $F$. The lines $AP$ and $EF$ intersect each other at $D$. Let $K$ be the foot of the perpendicular from the point $D$ to the line $BC$. Show that the line $KD$ bisects the angle $\angle EKF$.