Found problems: 85335
1998 AMC 12/AHSME, 5
If $2^{1998} - 2^{1997} - 2^{1996} + 2^{1995} = k \cdot 2^{1995}$, what is the value of $k$?
$\text{(A)} \ 1 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ 3 \qquad \text{(D)} \ 4 \qquad \text{(E)} \ 5$
2009 ISI B.Math Entrance Exam, 10
Given odd integers $a,b,c$ prove that the equation $ax^2+bx+c=0$ cannot have a solution $x$ which is a rational number.
1997 India National Olympiad, 2
Show that there do not exist positive integers $m$ and $n$ such that \[ \dfrac{m}{n} + \dfrac{n+1}{m} = 4 . \]
2019 China Second Round Olympiad, 1
In acute triangle $\triangle ABC$, $M$ is the midpoint of segment $BC$. Point $P$ lies in the interior of $\triangle ABC$ such that $AP$ bisects $\angle BAC$. Line $MP$ intersects the circumcircles of $\triangle ABP,\triangle ACP$ at $D,E$ respectively. Prove that if $DE=MP$, then $BC=2BP$.
2005 Alexandru Myller, 1
Let $A,B\in M_2(\mathbb Z)$ s.t. $AB=\begin{pmatrix}1&2005\\0&1\end{pmatrix}$. Prove that there is a matrix $C\in M_2(\mathbb Z)$ s.t. $BA=C^{2005}$.
[i]Dinu Serbanescu[/i]
2008 Estonia Team Selection Test, 2
Let $ABCD$ be a cyclic quadrangle whose midpoints of diagonals $AC$ and $BD$ are $F$ and $G$, respectively.
a) Prove the following implication: if the bisectors of angles at $B$ and $D$ of the quadrangle intersect at diagonal $AC$ then $\frac14 \cdot |AC| \cdot |BD| = | AG| \cdot |BF| \cdot |CG| \cdot |DF|$.
b) Does the converse implication also always hold?
III Soros Olympiad 1996 - 97 (Russia), 11.7
On the plane there are two circles $a$ and $b$ and a line $\ell$ perpendicular to the line passing through the centers of these circles. It is known that there are $4$ unequal circles, each of which touches $a$, $b$ and $\ell$. Find the radius of the smallest of these four circles if the radii of the other three are $2$, $3$ and $6$. Also find the ratio of the radii of the circles $a$ and $b$.
2006 India IMO Training Camp, 1
Find all triples $(a,b,c)$ such that $a,b,c$ are integers in the set $\{2000,2001,\ldots,3000\}$ satisfying $a^2+b^2=c^2$ and $\text{gcd}(a,b,c)=1$.
2024 Korea - Final Round, P2
For a positive integer $n(\geq 2)$, there are $2n$ candies. Alice distributes $2n$ candies into $4n$ boxes $B_1, B_2, \dots, B_{4n}.$ Bob checks the number of candies that Alice puts in each box. After this, Bob chooses exactly $2n$ boxes $B_{k_1}, B_{k_2}, \dots, B_{k_{2n}}$ out of $4n$ boxes that satisfy the following condition, and takes all the candies.
(Condition) $k_i - k_{i - 1}$ is either $1$ or $3$ for each $i = 1, 2, \dots, 2n$, and $k_{2n} = 4n$. ($k_0 = 0$)
Alice takes all the candies in the $2n$ boxes that Bob did not choose. If Alice and Bob both use their best strategy to take as many candies as possible, how many candies can Alice take?
2023 Belarus Team Selection Test, 2.1
Find all positive integers $n>2$ such that
$$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$
2019 IMO Shortlist, C1
The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).
May Olympiad L2 - geometry, 1999.4
Let $ABC$ be an equilateral triangle. $M$ is the midpoint of segment $AB$ and $N$ is the midpoint of segment $BC$. Let $P$ be the point outside $ABC$ such that the triangle $ACP$ is isosceles and right in $P$. $PM$ and $AN$ are cut in $I$. Prove that $CI$ is the bisector of the angle $MCA$ .
2016 PUMaC Algebra Individual A, A5
Define a sequence $a_i$ as follows: $a_1 = 181$ and for $i \ge 2$, $a_i = a_{i-1}^2-1$ if $a_{i-1}$ is odd and $a_i = a_{i-1}/2$ if $a_{i-1}$ is even. Find the least $i$ such that $a_i = 0$.
2012 Today's Calculation Of Integral, 786
For each positive integer $n$, define $H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}.$
(1) Find $H_1(x),\ H_2(x),\ H_3(x)$.
(2) Express $\frac{d}{dx}H_n(x)$ interms of $H_n(x),\ H_{n+1}(x).$ Then prove that $H_n(x)$ is a polynpmial with degree $n$ by induction.
(3) Let $a$ be real number. For $n\geq 3$, express $S_n(a)=\int_0^a xH_n(x)e^{-x^2}dx$ in terms of $H_{n-1}(a),\ H_{n-2}(a),\ H_{n-2}(0)$.
(4) Find $\lim_{a\to\infty} S_6(a)$.
If necessary, you may use $\lim_{x\to\infty}x^ke^{-x^2}=0$ for a positive integer $k$.
2016 BMT Spring, 7
Let $ABC$ be a right triangle with $AB = BC = 2$. Construct point $D$ such that $\angle DAC = 30^o$ and $\angle DCA = 60^o$, and $\angle BCD > 90^o$. Compute the area of triangle $BCD$.
2021 Romanian Master of Mathematics Shortlist, A2
Let $n$ be a positive integer and let $x_1,\ldots,x_n,y_1,\ldots,y_n$ be integers satisfying the following
condition: the numbers $x_1,\ldots,x_n$ are pairwise distinct and for every positive integer $m$ there
exists a polynomial $P_m$ with integer coefficients such that $P_m(x_i) - y_i$, $i=1,\ldots,n$, are all divisible by $m$. Prove that there exists a polynomial $P$ with integer coefficients such that $P(x_i) = y_i$ for all $i=1,\ldots,n$.
2010 Slovenia National Olympiad, 3
Find all functions $f: [0, +\infty) \to [0, +\infty)$ satisfying the equation
\[(y+1)f(x+y) = f\left(xf(y)\right)\]
For all non-negative real numbers $x$ and $y.$
1998 Estonia National Olympiad, 3
A function $f$ satisfies the conditions $f (x) \ne 0$ and $f (x+2) = f (x-1) f (x+5)$ for all real x. Show that $f (x+18) = f (x)$ for any real $x$.
2016 Stars of Mathematics, 3
Let $ n $ be a natural number, and $ 2n $ nonnegative real numbers $ a_1,a_2,\ldots ,a_{2n} $ such that $ a_1a_2\cdots a_{2n}=1. $ Show that
$$ 2^{n+1} +\left( a_1^2+a_2^2 \right)\left( a_3^2+a_4^2 \right)\cdots\left( a_{2n-1}^2+a_{2n}^2 \right) \ge 3\left( a_1+a_2 \right)\left( a_3+a_4 \right)\cdots\left( a_{2n-1}+a_{2n} \right) , $$
and specify in which circumstances equality happens.
2018 HMNT, 1
Four standard six-sided dice are rolled. Find the probability that, for each pair of dice, the product of the two numbers rolled on those dice is a multiple of 4.
2012 Balkan MO Shortlist, A3
Determine the maximum possible number of distinct real roots of a polynomial $P(x)$ of degree $2012$ with real coefficients satisfying the condition
\begin{align*} P(a)^3 + P(b)^3 + P(c)^3 \geq 3 P(a) P(b) P(c) \end{align*}
for all real numbers $a,b,c \in \mathbb{R}$ with $a+b+c=0$
2022 Canadian Mathematical Olympiad Qualification, 2
Determine all pairs of integers $(m, n)$ such that $m^2 + n$ and $n^2 + m$ are both perfect squares.
2020 LIMIT Category 1, 15
In a $4\times 4$ chessboard, in how many ways can you place $3$ rooks and one bishop such that none of these pieces threaten another piece?
2020 Online Math Open Problems, 24
In graph theory, a [i]triangle[/i] is a set of three vertices, every two of which are connected by an edge. For an integer $n \geq 3$, if a graph on $n$ vertices does not contain two triangles that do not share any vertices, let $f(n)$ be the maximum number of triangles it can contain. Compute $f(3) + f(4) + \cdots + f(100).$
[i]Proposed by Edward Wan[/i]
2000 Switzerland Team Selection Test, 13
The incircle of a triangle $ABC$ touches the sides $AB,BC,CA$ at points $D,E,F$ respectively. Let $P$ be an internal point of triangle $ABC$ such that the incircle of triangle $ABP$ touches $AB$ at $D$ and the sides $AP$ and $BP$ at $Q$ and $R$. Prove that the points $E,F,R,Q$ lie on a circle.