Found problems: 85335
Bill’s age is one third larger than Tracy’s age. In $30$ years Bill’s age will be one eighth larger than Tracy’s age. How many years old is Bill?
Baron Munchausen claims that he got a map of a country that consists of five cities. Each two cities are connected by a direct road. Each road intersects no more than one another road (and no more than once). On the map, the roads are colored in yellow or red, and while circling any city (along its border) one can notice that the colors of crossed roads alternate. Can Baron's claim be true?
Suppose the function $f:[1,+\infty)\to[1,+\infty)$ satisfies the following two conditions:
(i) $f(f(x))=x^2$ for any $x\geq 1$;
(ii) $f(x)\leq x^2+2021x$ for any $x\geq 1$.
1. Prove that $x<f(x)<x^2$ for any $x\geq 1$.
2. Prove that there exists a function $f$ satisfies the above two conditions and the following one:
(iii) There are no real constants $c$ and $A$, such that $0<c<1$, and $\frac{f(x)}{x^2}<c$ for any $x>A$.
Determine the maximum value of $x^2 y^2 z^2 w$ for $\{x,y,z,w\}\in\mathbb{R}^{+} \cup\{0\}$ and $2x+xy+z+yzw=1$.
Let $ANC$, $CLB$ and $BKA$ be triangles erected on the outside of the triangle $ABC$ such that $\angle NAC = \angle KBA = \angle LCB$ and $\angle NCA = \angle KAB = \angle LBC$. Let $D$, $E$, $G$ and $H$ be the midpoints of $AB$, $LK$, $CA$ and $NA$ respectively. Prove that $DEGH$ is a parallelogram.
The numbers $1, 2, 3, ... , 64$ are written in the cells of an $8 \times 8$ board (in some order, one per cell). Show that at least four $2 \times 2$ squares have sum greater than $100$.
Find the least positive integer $m$ such that $105| 9^{(p^2)} -29^p +m$ for all prime numbers $p > 3$.
Let $n$ be square with 4 digits, such that all its digits are less than 6. If we add 1 to each digit the resulting number is another square. Find $n$
Let $V_n=\sqrt{F_n^2+F_{n+2}^2}$, where $F_n$ is the Fibonacci sequence
($F_1=F_2=1,F_{n+2}=F_{n+1}+F_{n}$)
Show that $V_n,V_{n+1},V_{n+2}$ are the sides of a triangle with area $1/2$
Suppose, medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that:
(a) The medians of the triangle correspond to the sides of a right-angled triangle.
(b) If $a,b,c$ are the side-lengths of the triangle, then, the following inequality holds:\[5(a^2+b^2-c^2)\geq 8ab\]
Prove that for any odd prime number $ p,$ the number of positive integer $ n$ satisfying $ p|n! \plus{} 1$ is less than or equal to $ cp^\frac{2}{3}.$ where $ c$ is a constant independent of $ p.$
$P$ is a point on arc $\overarc{BC}$ of the circumcircle of $\triangle ABC$ not containing $A$, $K$ lies on segment $AP$ such that $BK$ bisects $\angle ABC$. The circumcircle of $\triangle KPC$ meets $AC,BD$ at $D,E$ respectively. $PE$ meets $AB$ at $F$. Prove that $\angle ABC=2\angle FCB$.
Given a horizontal strip on the plane (its sides are parallel lines) and $n$ lines intersecting the strip. Every two of them intersect inside the strip, and not a triple has a common point. Consider all the paths along the segments of those lines, starting on the lower side of the strip and ending on the upper side with the properties: moving along such a path we are constantly rising up, and, having reached the intersection, we are obliged to turn to another line. Prove that:
a) there are not less than $n/2$ such a paths without common points;
b) there is a path consisting of not less than of $n$ segments;
c) there is a path that goes along not more than along $n/2+1$ lines;
d) there is a path that goes along all the $n$ lines.
Decide whether there is a member in the arithmetic sequence $\{a_n\}_{n=1}^\infty$ whose first member is $a_1=1998$ and the common difference $d=131$ which is a palindrome (palindrome is a number such that its decimal expansion is symmetric, e.g., $7$, $33$, $433334$, $2135312$ and so on).
If $(a,~b,~c)$ is a triple of real numbers, de fine
[list]
[*] $g(a,~b,~c)=(a+b,~b+c,~a+c)$, and
[*] $g^n(a,~b,~c)=g(g^{n-1}(a,~b,~c))$ for $n\ge 2$[/list]
Suppose that there exists a positive integer $n$ so that $g^n(a,~b,~c)=(a,~b,~c)$ for some $(a,~b,~c)\neq (0,~0,~0)$. Prove that $g^6(a,~b,~c)=(a,~b,~c)$
Let $O$ and $\Omega$ denote the circumcenter and circumcircle, respectively, of scalene triangle $\triangle ABC$. Furthermore, let $M$ be the midpoint of side $BC$. The tangent to $\Omega$ at $A$ intersects $BC$ and $OM$ at points $X$ and $Y$, respectively. If the circumcircle of triangle $\triangle OXY$ intersects $\Omega$ at two distinct points $P$ and $Q$, prove that $PQ$ bisects $\overline{AM}$.
[i]Proposed by Andrew Wen[/i]
$ABC$ is a triangle with $AB = 33$, $AC = 21$ and $BC = m$, an integer. There are points $D$, $E$ on the sides $AB$, $AC$ respectively such that $AD = DE = EC = n$, an integer. Find $m$.
Let $\ell$ be a straight line and $P \notin \ell$ be a point in the plane. On $\ell$ are, in this arrangement, points $A_1, A_2,...$ such that the radii of the incircles of all triangles $P A_iA_{i + 1}$ are equal. Let $k \in N$. Show that the radius of the incircle of the triangle $P A_j A_{j + k}$ does not depend on the choice of $j \in N$ .
The polynomial $ f(x)\equal{}ax^2\plus{}bx\plus{}c$ has real coefficients and satisfies $ \left|f(x)\right|\le 1$ for all $ x\in [0, 1]$. Find the maximal value of $ |a|\plus{}|b|\plus{}|c|$.
In terms of the real numbers $a$ and $b$ determine the minimum value of $$ \sqrt{(x+a)^2+1}+\sqrt{(x+1-a)^2+1}+\sqrt{(x+b)^2+1}+\sqrt{(x+1-b)^2+1}$$ as well as all values of $x$ which attain it.
Is there an increasing sequence of integers $ 0 = {{a} _{0}} <{{a} _{1}} <{{a} _{2}} <\ldots $ for which the following two conditions are satisfied simultaneously:
1) any natural number can be given as $ {{a} _{i}} + {{a} _{j}} $ for some (possibly equal) $ i \ge 0 $, $ j \ge 0$ ,
2) $ {{a} _ {n}}> \tfrac {{{n} ^ {2}}} {16} $ for all natural $ n $?
Let $ABC$ be an acute-angled triangle such that $AB<AC$. Let $D$ be the point of intersection of the perpendicular bisector of the side $BC$ with the side $AC$. Let $P$ be a point on the shorter arc $AC$ of the circumcircle of the triangle $ABC$ such that $DP \parallel BC$. Finally, let $M$ be the midpoint of the side $AB$. Prove that $\angle APD=\angle MPB$.
[i]Proposed by Dominik Burek, Poland[/i]
In acute triangle $ABC$, $\angle A = 20^o$. Prove that the triangle is isosceles if and only if $$\sqrt[3]{a^3 + b^3 + c^3 -3abc} = \min\{b, c\}$$, where $a,b, c$ are the side lengths of triangle $ABC$.
Find the largest integer $n$ for which $2^n$ divides \[ \binom 21 \binom 42 \binom 63 \dots \binom {128}{64}. \][i]Proposed by Evan Chen[/i]
On two sheets of paper are written more than one positive integers. On the first paper $n$ numbers are written and on the second paper $m$ numbers are written. If one number is written on any of the papers then on the first paper is written also the sum of that number and $13$, and on the second paper the difference of that number and $23$. Calculate the value of $\frac{m}{n}$.
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