Found problems: 85335
Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.
Let $ P (x) $ be a polynomial with real coefficients such that $ P (x)> 0 $ for all $ x \geq 0 $. Prove that there is a positive integer $ n $ such that $ (1 + x) ^ n P (x) $ polynomial with nonnegative coefficients.
Let $f : (a,b) \to \mathbb R$ be a function with the property that for all $x \in (a,b)$ there is a non-degenerated interval $[ a_x,b_x ]$ with $a < a_x \leq x \leq b_x < b$ such that $f$ is constant on $\left[ a_x,b_x \right]$.
(a) Prove that $\textrm{Im} \, f$ is finite or numerable.
(b) Find all continuous functions which have the property mentioned in the hypothesis.
Given $n > 1$ monic square trinomials $x^2 - a_1x + b_1$,$...$, $x^2-a_nx + b_n$, and all $2n$ numbers are $a_1$,$...$, $a_n$, $b_1$,$...$, $b_n$ are different. Can it happen that each of the numbers $a_1$,$...$, $a_n$, $b_1$,$...$, $b_n is the root of one of these trinomials?
Find the remainder of $\binom{3^n}{2^n}$ modulo $3^{n+1}$.
[i]Proposed by V. Rastorguev[/i]
Suppose that $ m$ and $ n$ are odd integers such that $ m^2\minus{}n^2\plus{}1$ divides $ n^2\minus{}1$. Prove that $ m^2\minus{}n^2\plus{}1$ is a perfect square.
In how many ways can we change the sign $ *$ with $ \plus{}$ or $ \minus{}$, such that the following equation is true?
\[ 1 *2*3*4*5*6*7*8*9*10\equal{}29\]
Find all surjective functions $f:\mathbb{N}\to\mathbb{N}$ such that for all positive integers $a$ and $b$, exactly one of the following equations is true:
\begin{align*}
f(a)&=f(b), <br /> \\
f(a+b)&=\min\{f(a),f(b)\}.
\end{align*}
[i]Remarks:[/i] $\mathbb{N}$ denotes the set of all positive integers. A function $f:X\to Y$ is said to be surjective if for every $y\in Y$ there exists $x\in X$ such that $f(x)=y$.
How many positive integers less than 2005 are relatively prime to 1001?
If $ \angle\text{CBD} $ is a right angle, then this protractor indicates that the measure of $ \angle\text{ABC} $ is approximately
[asy]
unitsize(36);
pair A,B,C,D;
A=3*dir(160); B=origin; C=3*dir(110); D=3*dir(20);
draw((1.5,0)..(0,1.5)..(-1.5,0));
draw((2.5,0)..(0,2.5)..(-2.5,0)--cycle);
draw(A--B); draw(C--B); draw(D--B);
label("O",(-2.5,0),W);
label("A",A,W);
label("B",B,S);
label("C",C,W);
label("D",D,E);
label("0",(-1.8,0),W);
label("20",(-1.7,.5),NW);
label("160",(1.6,.5),NE);
label("180",(1.7,0),E);[/asy]
$ \text{(A)}\ 20^\circ\qquad\text{(B)}\ 40^\circ\qquad\text{(C)}\ 50^\circ\qquad\text{(D)}\ 70^\circ\qquad\text{(E)}\ 120^\circ $
Let $a>1$ be a real number. Prove that for all $n\in\mathbb{N}*$ that :
$\frac{a^n-1}{n}\ge \sqrt{a}^{n+1}-\sqrt{a}^{n-1}$
Let $\{u_{n}\}_{n \ge 0}$ be a sequence of integers satisfying the recurrence relation $u_{n+2}=u_{n+1}^2 -u_{n}$ $(n \in \mathbb{N})$. Suppose that $u_{0}=39$ and $u_{1}=45$. Prove that $1986$ divides infinitely many terms of this sequence.
The sequence $a_1, a_2, a_3,...$ is defined by $$a_1 = 1\,\,\,, \,\,\,a_{n+1} =\frac{1}{16}(1 + 4a_n +\sqrt{1 + 24a_n}) \,\,\,(n \in N^* ).$$ Determine and prove a formula with which for every natural number $n$ the term $a_n$ can be computed directly without having to determine preceding terms of the sequence.
Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$
Let $p{}$ and $q{}$ be two coprime positive integers. A frog hops along the integer line so that on every hop it moves either $p{}$ units to the right or $q{}$ units to the left. Eventually, the frog returns to the initial point. Prove that for every positive integer $d{}$ with $d < p + q$ there are two numbers visited by the frog which differ just by $d{}$.
[i]Nikolay Belukhov[/i]
Fede chooses $50$ distinct integers from the set $\{1, 2, 3, \ldots, 100\}$ such that their sum equals $2900$. Determine the minimum number of even numbers that can be among the $50$ numbers chosen by Fede.
Point $M$ is the midpoint of the side $CD$ of the trapezoid $ABCD$, point $K$ is the foot of the perpendicular drawn from point $M$ to the side $AB$. Give that $3BK \le AK$. Prove that $BC + AD\ge 2BM$.
Let $ x_1, x_2,\ldots , x_n$ be positive real numbers such that $ x_1x_2\cdots x_n \equal{} 1$. Prove that
\[\sum_{i \equal{} 1}^n \frac {1}{n \minus{} 1 \plus{} x_i}\le 1.\]
A function $f$ satisfies $$f(x)+xf(1-x)=x$$ for all real numbers $x$. Determine the number $f (2)$. Find $f$ .
Compute the sum of the three smallest positive integers $n$ for which $$\frac{1+2+3+\cdots+(2024n-1)+2024n}{1+2+3+\cdots+(4n-1)+4n}$$ is an integer.
Find the smallest positive integer \( k \geq 2 \) for which there exists a polynomial \( f(x) \) of degree \( k \) with integer coefficients and a leading coefficient of \( 1 \) that satisfies the following condition:
(Condition) For any two integers \( m \) and \( n \), if \( f(m) - f(n) \) is a multiple of \( 31 \), then \( m - n \) is a multiple of \( 31 \).
For any positive integer $x$, let $S(x)$ be the sum of the digits of $x$, and let $T(x)$ be $|S(x+2)-S(x)|.$ For example, $T(199)=|S(201)-S(199)|=|3-19|=16.$ How many values $T(x)$ do not exceed 1999?
Let $N$ be an arbitrary positive integer. Prove that if, from among any $n$ consecutive integers larger than $N$, one may select $7$ of them, pairwise co-prime, then $n\geq 22$.
([i]Dan Schwarz[/i])
a) Four merchants want to travel from Athens to Rome by cart. On the same day, but different times they leave Athens and arrive on another day to Rome, but in reverse order. Every day, when the evening comes, each merchant enters the next inn on the way. When some merchants sleep in the same inn at night, then on the following day at dawn they leave in reverse order of arrival, because they can only park this way on the narrow streets next to the inns. They cannot overtake each other, their order only changes after a night spent together in the same inn. Eventually each merchant arrives in Rome while they sleep with every other merchant in the same inn exactly once. Is it possible, that the number of the inns they sleep in is even every night?
b) Is it possible if there are $8$ merchants instead of $4$ and every other condition is the same?
Consider a convex quadrilateral \(ABCD\) with \(AC > BD\). In the plane of this quadrilateral, points \(M\) and \(N\) are chosen such that triangles \(ABM\) and \(CDN\) are equilateral, and segments \(MD\) and \(NA\) intersect lines \(AB\) and \(CD\) respectively. Similarly, points \(P\) and \(Q\) are chosen such that triangles \(ADP\) and \(BCQ\) are equilateral, but here segments \(PB\) and \(QA\) do not intersect lines \(AD\) and \(BC\) respectively.
Prove that \(MN = AC + BD\) if and only if \(PQ = AC - BD\).
[i]Proposed by Zaza Meliqidze, Georgia [/i]