Found problems: 85335
In a set of 17 externally identical coins, two are counterfeit, differing from the rest in weight. It is known that the total weight of two counterfeit coins is twice the weight of a real one.s it always possible to determine the couple of counterfeit coins, having made $5$ weighings on a cup scale without weights? (It is not necessary to determine which of the fakes is heavier.)
Let $p(z)=a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a complex polynomial. Suppose that $1=c_0\ge c_1\ge \cdots \ge c_n\ge 0$ is a sequence of real numbers which form a convex sequence. (That is $2c_k\le c_{k-1}+c_{k+1}$ for every $k=1,2,\cdots ,n-1$ ) and consider the polynomial
\[ q(z)=c_0a_0+c_1a_1z+c_2a_2z^2+\cdots +c_na_nz^n \]
Prove that :
\[ \max_{|z|\le 1}q(z)\le \max_{|z|\le 1}p(z) \]
The number $2024$ is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?
$\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24$
Given the two hypotheses: $\text{I}$ Some Mems are not Ens and $\text{II}$ No Ens are Veens. If "some" means "at least one," we can conclude that:
$\textbf{(A)}\ \text{Some Mems are not Veens}\qquad
\textbf{(B)}\ \text{Some Vees are not Mems}\\
\textbf{(C)}\ \text{No Mem is a Vee}\qquad
\textbf{(D)}\ \text{Some Mems are Vees}\\
\textbf{(E)}\ \text{Neither} \; \textbf{(A)} \; \text{nor} \; \textbf{(B)} \; \text{nor} \; \textbf{(C)} \; \text{nor} \; \textbf{(D)} \; \text{is deducible from the given statements}$
The positive integers $v, w, x, y$, and $z$ satisfy the equation \[v + \frac{1}{w + \frac{1}{x + \frac{1}{y + \frac{1}{z}}}} = \frac{222}{155}.\] Compute $10^4 v + 10^3 w + 10^2 x + 10 y + z$.
Compute $\left|\sum_{i=1}^{2022} \sum_{j=1}^{2022} \cos\left(\frac{ij\pi}{2023}\right)\right|.$
If $x,y,z$ are positive integers and $z(xz+1)^2=(5z+2y)(2z+y)$, prove that $z$ is an odd perfect square.
For a polynomial $ P$ of degree 2000 with distinct real coefficients let $ M(P)$ be the set of all polynomials that can be produced from $ P$ by permutation of its coefficients. A polynomial $ P$ will be called [b]$ n$-independent[/b] if $ P(n) \equal{} 0$ and we can get from any $ Q \in M(P)$ a polynomial $ Q_1$ such that $ Q_1(n) \equal{} 0$ by interchanging at most one pair of coefficients of $ Q.$ Find all integers $ n$ for which $ n$-independent polynomials exist.
A positive integer $n$ is called [i]egyptian[/i] if there exists a strictly increasing sequence $0<a_1<a_2<\dots<a_k=n$ of integers with last term $n$ such that
\[\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_k}=1.\]
(a) Determine if $n=72$ is egyptian.
(b) Determine if $n=71$ is egyptian.
(c) Determine if $n=72^{71}$ is egyptian.
Find the largest positive real number $p$ (if it exists) such that the inequality
\[x^2_1+ x_2^2+ \cdots + x^2_n\ge p(x_1x_2 + x_2x_3 + \cdots + x_{n-1}x_n)\]
is satisfied for all real numbers $x_i$, and $(a) n = 2; (b) n = 5.$
Find the largest positive real number $p$ (if it exists) such that the inequality holds for all real numbers $x_i$ and all natural numbers $n, n \ge 2.$
Given a prime $p \geq 5$ , show that there exist at least two distinct primes $q$ and $r$ in the range $2, 3, \ldots p-2$ such that $q^{p-1} \not\equiv 1 \pmod{p^2}$ and $r^{p-1} \not\equiv 1 \pmod{p^2}$.
In an election, there are a total of $12$ candidates. An election committee has $6$ members voting. It is known that at most two candidates voted by any two committee members are the same. Find the maximum number of committee members.
An equilateral triangle $T$ and a circle $C$ are on the same plane. Suppose each side length of $T$ is $6\sqrt3$ and the radius of $C$ is $2.$ The distance between the centers of $T$ and $C$ is $15.$ For every two points $X$ on $T$ and $Y$ on $C,$ let $M(X, Y)$ be the midpoint of segment $\overline{XY}.$ The points $M(X, Y)$ as $X$ varies on $T$ and $Y$ varies on $C$ create a region whose area is $A.$ Find $A.$
\[\mathrm a. ~\pi + 14\sqrt3 \qquad \mathrm b. ~3\pi + 10\sqrt3 \qquad \mathrm c. ~4\pi+9\sqrt3 \qquad\mathrm d. ~\pi + 15\sqrt3 \qquad\mathrm e. ~4\pi+6\sqrt3\]
Circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively lie on a plane such that that the circle $C_2$ passes through $O_1$. The ratio of radius of circle $C_1$ to $O_1O_2$ is $\sqrt{2+\sqrt3}$.
a) Prove that the circles $C_1$ and $C_2$ intersect at two distinct points.
b) Let $A,B$ be these points of intersection. What proportion of the area of circle is $C_1$ is the area of the sector $AO_1B$ ?
Four balls of radius $1$ are placed in space so that each of them touches the other three. What is the radius of the smallest sphere containing all of them?
Natural numbers $a$ and $b$ are given such that the number
$$ P = \frac{[a, b]}{a + 1} + \frac{[a, b]}{b + 1} $$
Is a prime. Prove that $4P + 5$ is the square of a natural number.
For how many positive integers $n\le100$ is it true that $10n$ has exactly three times as many positive divisors as $n$ has?
In triangle $ABC$, the points $M$ and $N$ are the midpoints of the sides $AB$ and $AC$, respectively. The bisectors of interior angles $\angle ABC$ and $\angle BCA$ intersect the line $MN$ at points $P$ and $Q$, respectively. Let $p$ be the tangent to the circumscribed circle of the triangle $AMP$ passing through point $P$, and $q$ be the tangent to the circumscribed circle of the triangle $ANQ$ passing through point $Q$. Prove that the lines $p$ and $q$ intersect on line $BC$.
A town has a road network that consists entirely of one-way streets that are used for bus routes. Along these routes, bus stops have been set up. If the one-way signs permit travel from bus stop $X$ to bus stop $Y \neq X$, then we shall say [i]$Y$ can be reached from $X$[/i]. We shall use the phrase [i]$Y$ comes after $X$[/i] when we wish to express that every bus stop from which the bus stop $X$ can be reached is a bus stop from which the bus stop $Y$ can be reached, and every bus stop that can be reached from $Y$ can also be reached from $X$. A visitor to this town discovers that if $X$ and $Y$ are any two different bus stops, then the two sentences [i]“$Y$ can be reached from $X$”[/i] and [i]“$Y$ comes after $X$”[/i] have exactly the same meaning in this town. Let $A$ and $B$ be two bus stops. Show that of the following two statements, exactly one is true:
[b] (i)[/b] $B$ can be reached from $A;$
[b] (ii) [/b] $A$ can be reached from $B.$
The bisectors of trapezoid's angles form a quadrilateral with perpendicular diagonals. Prove that this trapezoid is isosceles.
Determine all sequences of integers $a_1, a_2,. . .,$ such that:
(i) $1 \le a_i \le n$ for all $1 \le i \le n$.
(ii) $| a_i - a_j| = | i - j |$ for any $1 \le i, j \le n$
Two graphs $G_1$ and $G_2$ of quadratic polynomials intersect at points $A$ and $B$. Let $O$ be the vertex of $G_1$. Lines $OA$ and $OB$ intersect $G_2$ again at points $C$ and $D$. Prove that $CD$ is parallel to the $x$-axis.
For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:
(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$;
(b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$.
Determine $N(n)$ for all $n\ge 2$.
Sita and Geeta are two sisters. If Sita's age is written after Geeta's age a four digit perfect square (number) is obtained. If the same exercise is repeated after 13 years another four digit perfect square (number) will be obtained. What is the sum of the present ages of Sita and Geeta?
What is $ (20\minus{}(2010\minus{}201)) \plus{} (2010\minus{}(201\minus{}20))$?
$ \textbf{(A)}\ \minus{}4020\qquad \textbf{(B)}\ 0\qquad \textbf{(C)}\ 40\qquad \textbf{(D)}\ 401\qquad \textbf{(E)}\ 4020$