Found problems: 85335
2011 Ukraine Team Selection Test, 4
Suppose an ordered set of $ ({{a} _{1}}, \ {{a} _{2}},\ \ldots,\ {{a} _{n}}) $ real numbers, $n \ge 3 $. It is possible to replace the number $ {{a} _ {i}} $, $ i = \overline {2, \ n-1} $ by the number $ a_ {i} ^ {*} $ that $ {{a} _ {i}} + a_ {i} ^ {*} = {{a} _ {i-1}} + {{a} _ {i + 1}} $. Let $ ({{b} _ {1}},\ {{b} _ {2}}, \ \ldots, \ {{b} _ {n}}) $ be the set with the largest sum of numbers that can be obtained from this, and $ ({{c} _ {1}},\ {{c} _ {2}}, \ \ldots, \ {{c} _ {n}}) $ is a similar set with the least amount.
For the odd $n \ge 3 $ and set $ (1,\ 3, \ \ldots, \ n, \ 2, \ 4, \ \ldots,\ n-1) $ find the values of the expressions $ {{b} _ {1}} + {{b} _ {2}} + \ldots + {{b} _ {n}} $ and $ {{c} _ {1}} + {{c} _ {2}} + \ldots + {{c} _ {n}} $.
2016 Putnam, A2
Given a positive integer $n,$ let $M(n)$ be the largest integer $m$ such that
\[\binom{m}{n-1}>\binom{m-1}{n}.\]
Evaluate
\[\lim_{n\to\infty}\frac{M(n)}{n}.\]
2018 AMC 12/AHSME, 17
Let $p$ and $q$ be positive integers such that \[\frac{5}{9} < \frac{p}{q} < \frac{4}{7}\] and $q$ is as small as possible. What is $q-p$?
$\textbf{(A) } 7 \qquad \textbf{(B) } 11 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19 $
1999 Bosnia and Herzegovina Team Selection Test, 5
For any nonempty set $S$, we define $\sigma(S)$ and $\pi(S)$ as sum and product of all elements from set $S$, respectively. Prove that
$a)$ $\sum \limits_{} \frac{1}{\pi(S)} =n$
$b)$ $\sum \limits_{} \frac{\sigma(S)}{\pi(S)} =(n^2+2n)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)(n+1)$
where $\sum$ denotes sum by all nonempty subsets $S$ of set $\{1,2,...,n\}$
2023 Romania National Olympiad, 1
Determine twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which verify relation
\[
\left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
\]
2009 Indonesia TST, 2
Find the formula to express the number of $ n\minus{}$series of letters which contain an even number of vocals (A,I,U,E,O).
2013 Moldova Team Selection Test, 4
Prove that for any positive real numbers $a_i,b_i,c_i$ with $i=1,2,3$,
$(a_1^3+b_1^3+c_1^3+1)(a_2^3+b_2^3+c_2^3+1)(a_3^3+b_3^3+c_3^3+1)\geq \frac{3}{4} (a_1+b_1+c_1)(a_2+b_2+c_2)(a_3+b_3+c_3)$
2013 NIMO Problems, 7
Tyler has two calculators, both of which initially display zero. The first calculators has only two buttons, $[+1]$ and $[\times 2]$. The second has only the buttons $[+1]$ and $[\times 4]$. Both calculators update their displays immediately after each keystroke.
A positive integer $n$ is called [i]ambivalent[/i] if the minimum number of keystrokes needed to display $n$ on the first calculator equals the minimum number of keystrokes needed to display $n$ on the second calculator. Find the sum of all ambivalent integers between $256$ and $1024$ inclusive.
[i]Proposed by Joshua Xiong[/i]
2008 Irish Math Olympiad, 2
Circles $ S$ and $ T$ intersect at $ P$ and $ Q$, with $ S$ passing through the centre of $ T$. Distinct points $ A$ and $ B$ lie on $ S$, inside $ T$, and are equidistant from the centre of $ T$. The line $ PA$ meets $ T$ again at $ D$. Prove that $ |AD| \equal{} |PB|$.
Geometry Mathley 2011-12, 13.3
Let $ABCD$ be a quadrilateral inscribed in circle $(O)$. Let $M,N$ be the midpoints of $AD,BC$. A line through the intersection $P$ of the two diagonals $AC,BD$ meets $AD,BC$ at $S, T$ respectively. Let $BS$ meet $AT$ at $Q$. Prove that three lines $AD,BC,PQ$ are concurrent if and only if $M, S, T,N$ are on the same circle.
Đỗ Thanh Sơn
2002 AMC 12/AHSME, 13
Two different positive numbers $ a$ and $ b$ each differ from their reciprocals by 1. What is $ a \plus{} b$?
\[ \textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } \sqrt {5} \qquad \textbf{(D) } \sqrt {6} \qquad \textbf{(E) } 3
\]
STEMS 2024 Math Cat B, P6
All the rationals are coloured with $n$ colours so that, if rationals $a$ and $b$ are colored with different colours then $\frac{a+b}2$ is coloured with a colour different from both $a$ and $b$. Prove that every rational is coloured with the same colour.
1993 Tournament Of Towns, (396) 4
A convex $1993$-gon is divided into convex $7$-gons. Prove that there are $3$ neighbouring sides of the $1993$-gon belonging to one such $7$-gon. (A vertex of a $7$-gon may not be positioned on the interior of a side of the $1993$-gon, and two $7$-gons either have no common points, exactly one common vertex or a complete common side.)
(A Kanel-Belov)
2023 Girls in Mathematics Tournament, 2
Let $a,b,c$ real numbers such that $a^n+b^n= c^n$ for three positive integers consecutive of $n$. Prove that $abc= 0$
1995 AMC 12/AHSME, 19
Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ such that $\overline{DE} \perp \overline{BC}$. The ratio of the area of $\triangle DEF$ to the area of $\triangle ABC$ is
[asy]
size(180);
pathpen = linewidth(0.7); pointpen = black; pointfontpen = fontsize(10);
pair B = (0,0), C = (1,0), A = dir(60), D = C*2/3, E = (2*A+C)/3, F = (2*B+A)/3;
D(D("A",A,N)--D("B",B,SW)--D("C",C,SE)--cycle); D(D("D",D)--D("E",E,NE)--D("F",F,NW)--cycle); D(rightanglemark(C,D,E,1.5));[/asy]
$\textbf{(A)}\ \dfrac{1}{6}\qquad
\textbf{(B)}\ \dfrac{1}{4} \qquad
\textbf{(C)}\ \dfrac{1}{3} \qquad
\textbf{(D)}\ \dfrac{2}{5} \qquad
\textbf{(E)}\ \dfrac{1}{2}$
PEN J Problems, 18
Prove that for any $\delta$ greater than 1 and any positive number $\epsilon$, there is an $n$ such that $\left \vert \frac{\sigma (n)}{n} -\delta \right \vert < \epsilon$.
1967 German National Olympiad, 3
Prove the following theorem:
If $n > 2$ is a natural number, $a_1, ..., a_n$ are positive real numbers and becomes $\sum_{i=1}^n a_i = s$, then the following holds
$$\sum_{i=1}^n \frac{a_i}{s - a_i} \ge \frac{n}{n - 1}$$
2023 Stanford Mathematics Tournament, 3
How many trailing zeros does the value
\[300\cdot305\cdot310\dots1090\cdot1095\cdot1100\]
end with?
1980 IMO Longlists, 16
Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)
2021 CMIMC Integration Bee, 7
$$\int_0^\infty \frac{1}{(x^2+4)^{5/2}}\,dx$$
[i]Proposed by Connor Gordon[/i]
2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 1
Introduce a standard scalar product in $\mathbb{R}^4.$ Let $V$ be a partial vector space in $\mathbb{R}^4$ produced by $\left(
\begin{array}{c}
1 \\
-1 \\
-1 \\
1
\end{array}
\right),\left(
\begin{array}{c}
1 \\-1 \\
1 \\
-1
\end{array}
\right).$
Find a pair of base of orthogonal complement $W$ for $V$ in $\mathbb{R}^4.$
2008 Finnish National High School Mathematics Competition, 5
The closed line segment $I$ is covered by finitely many closed line segments.
Show that one can choose a subfamily $S$ of the family of line segments having the properties:
(1) the chosen line segments are disjoint,
(2) the sum of the lengths of the line segments of S is more than half of the length of $I.$
Show that the claim does not hold any more if the line segment $I$ is replaced by a circle and other occurences of the compound word ''line segment" by the word ''circular arc".
2021 MMATHS, 2
Define the [i]digital reduction[/i] of a two-digit positive integer $\underline{AB}$ to be the quantity $\underline{AB} - A - B$. Find the greatest common divisor of the digital reductions of all the two-digit positive integers. (For example, the digital reduction of $62$ is $62 - 6 - 2 = 54.$)
[i]Proposed by Andrew Wu[/i]
2022 IFYM, Sozopol, 2
Let $ABC$ be a triangle with $\angle BAC=40^\circ $, $O$ be the center of its circumscribed circle and $G$ is its centroid. Point $D$ of line $BC$ is such that $CD=AC$ and $C$ is between $B$ and $D$. If $AD\parallel OG$, find $\angle ACB$.
2021 Federal Competition For Advanced Students, P1, 1
Let $a,b,c\geq 0$ and $a+b+c=1.$ Prove that$$\frac{a}{2a+1}+\frac{b}{3b+1}+\frac{c}{6c+1}\leq \frac{1}{2}.$$
[size=50](Marian Dinca)[/size]