Found problems: 85335
1988 IMO Longlists, 42
Show that the solution set of the inequality
\[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4}
\]
is a union of disjoint intervals, the sum of whose length is 1988.
2011 Kurschak Competition, 2
Let $n$ be a positive integer. Denote by $a(n)$ the ways of expression $n=x_1+x_2+\dots$ where $x_1\leqslant x_2 \leqslant\dots$ are positive integers and $x_i+1$ is a power of $2$ for each $i$. Denote by $b(n)$ the ways of expression $n=y_1+y_2+\dots$ where $y_i$ is a positive integer and $2y_i\leqslant y_{i+1}$ for each $i$.
Prove that $a(n)=b(n)$.
2018 Tajikistan Team Selection Test, 9
Problem 9. The numbers 1,2,…,〖97〗^2 are written in the cells of a 97×97 board. In the center of each cell, there is a tower with the height equal to the number of that cell. Is it possible to see the top of any tower from the top of any other tower? (one point A can see the other point B, iff there is no other point on the segment AB).
1962 Miklós Schweitzer, 6
Let $ E$ be a bounded subset of the real line, and let $ \Omega$ be a system of (non degenerate) closed intervals such that for
each $ x \in E$ there exists an $ I \in \Omega$ with left endpoint $ x$. Show that for every $ \varepsilon > 0$ there exists a finite number of pairwise non overlapping intervals belonging to $ \Omega$ that cover $ E$ with the exception of a subset of outer measure less than $ \varepsilon$. [J. Czipszer]
2020 Canadian Mathematical Olympiad Qualification, 6
In convex pentagon $ABCDE, AC$ is parallel to $DE, AB$ is perpendicular to $AE$, and $BC$ is perpendicular to $CD$. If $H$ is the orthocentre of triangle $ABC$ and $M$ is the midpoint of segment $DE$, prove that $AD, CE$ and $HM$ are concurrent.
2000 AMC 12/AHSME, 6
Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
$ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 180\qquad \textbf{(E)}\ 231$
1991 Arnold's Trivium, 90
Calculate the sum of matrix commutators $[A, [B, C]] + [B, [C, A]] + [C, [A, B]]$, where $[A, B] = AB-BA$
1982 IMO Longlists, 38
Numbers $u_{n,k} \ (1\leq k \leq n)$ are defined as follows
\[u_{1,1}=1, \quad u_{n,k}=\binom{n}{k} - \sum_{d \mid n, d \mid k, d>1} u_{n/d, k/d}.\]
(the empty sum is defined to be equal to zero). Prove that $n \mid u_{n,k}$ for every natural number $n$ and for every $k \ (1 \leq k \leq n).$
2011 VJIMC, Problem 2
Let $(a_n)^\infty_{n=1}$ be an unbounded and strictly increasing sequence of positive reals such that the arithmetic mean of any four consecutive terms $a_n,a_{n+1},a_{n+2},a_{n+3}$ belongs to the same sequence. Prove that the sequence $\frac{a_{n+1}}{a_n}$ converges and find all possible values of its limit.
2017 ELMO Shortlist, 2
Let $ABC$ be a scalene triangle with $\angle A = 60^{\circ}$. Let $E$ and $F$ be the feet of the angle bisectors of $\angle ABC$ and $\angle ACB$, respectively, and let $I$ be the incenter of $\triangle ABC$. Let $P,Q$ be distinct points such that $\triangle PEF$ and $\triangle QEF$ are equilateral. If $O$ is the circumcenter of of $\triangle APQ$, show that $\overline{OI}\perp \overline{BC}$.
[i]Proposed by Vincent Huang
1997 AMC 12/AHSME, 4
If $ a$ is $ 50\%$ larger than $ c$, and $ b$ is $ 25\%$ larger than $ c$,then $ a$ is what percent larger than $ b$?
$ \textbf{(A)}\ 20\%\qquad \textbf{(B)}\ 25\%\qquad \textbf{(C)}\ 50\%\qquad \textbf{(D)}\ 100\%\qquad \textbf{(E)}\ 200\%$
2015 Kosovo Team Selection Test, 1
a)Prove that for every n,natural number exist natural numbers a and b such that
$(1-\sqrt{2})^n=a-b\sqrt{2}$ and $a^2-2b^2=(-1)^n$
b)Using first equation prove that for every n exist m such that
$(\sqrt{2}-1)^n=\sqrt{m}-\sqrt{m-1}$
1997 May Olympiad, 1
On a square board with $9$ squares (three by three), nine elements of the set $S=\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ must be placed, different from each other, so that each one is in a box and the following conditions are met:
$\bullet$ The sums of the numbers in the second and third rows are, respectively, double and triple the sum of the numbers in the first row.
$\bullet$ The sum of the numbers in the second and third columns are, respectively, double and triple the sum of the numbers in the first column.
Show all the possible ways to place elements of $S$ on the board, fulfilling the indicated conditions.
1987 Romania Team Selection Test, 9
Prove that for all real numbers $\alpha_1,\alpha_2,\ldots,\alpha_n$ we have \[ \sum_{i=1}^n \sum_{j=1}^n ij \cos {(\alpha_i - \alpha_j )} \geq 0. \]
[i]Octavian Stanasila[/i]
2017 Regional Competition For Advanced Students, 1
Let $x_1, x_2, \dots, x_n$ be non-negative real numbers such that
$$x_1^2+x_2^2 + \dots x_9^2 \ge 25.$$
Prove that one can choose three of these numbers such that their sum is at least $5$.
[i]Proposed by Karl Czakler[/i]
2014 VJIMC, Problem 4
Let $0<a<b$ and let $f:[a,b]\to\mathbb R$ be a continuous function with $\int^b_af(t)dt=0$. Show that
$$\int^b_a\int^b_af(x)f(y)\ln(x+y)dxdy\le0.$$
2022 Harvard-MIT Mathematics Tournament, 5
Let triangle $ABC$ be such that $AB = AC = 22$ and $BC = 11$. Point $D$ is chosen in the interior of the triangle such that $AD = 19$ and $\angle ABD + \angle ACD = 90^o$ . The value of $BD^2 + CD^2$ can be expressed as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.
2014 Junior Balkan Team Selection Tests - Romania, 1
Let $a, b, c, d$ be positive real numbers so that $abc+bcd+cda+dab = 4$.
Prove that $a^2 + b^2 + c^2 + d^2 \ge 4$
2019 JHMT, 5
Triangle $ABC$ has $AB = 8$, $BC = 12$, and $AC = 16$. Point $M$ is on $\overline{AC}$ so that $AM = MC$. Then, $\overline{BM}$ has length $x$. Find $x^2$
2023 Belarusian National Olympiad, 8.1
An unordered triple of numbers $(a,b,c)$ in one move you can change to either $(a,b,2a+2b-c)$, $(a,2a+2c-b,c)$ or $(2b+2c-a,b,c)$.
Can you from the triple $(3,5,14)$ get the triple $(3,13,6)$ in finite amount of moves?
2017 ASDAN Math Tournament, 2
Circles $A,B,C$ are externally tangent. Let $P$ be the tangent point between circles $A$ and $C$, and $Q$ be the tangent point between circles $B$ and $C$. Let $r_C$ be the radius of circle $C$. If the chord connecting $P$ and $Q$ has length $r_C\sqrt{2}$ and the radii of circles $A$ and $B$ are $4$ and $7$, respectively, what is the radius of circle $C$?
1974 IMO Longlists, 46
Outside an arbitrary triangle $ABC$, triangles $ADB$ and $BCE$ are constructed such that $\angle ADB=\angle BEC=90^{\circ}$ and $\angle DAB=\angle EBC=30^{\circ}$. On the segment $AC$ the point $F$ with $AF=3FC$ is chosen. Prove that $\angle DFE=90^{\circ}$ and $\angle FDE=30^{\circ}$.
2010 Iran MO (3rd Round), 1
suppose that $\mathcal F\subseteq X^{(k)}$ and $|X|=n$. we know that for every three distinct elements of $\mathcal F$ like $A,B,C$, at most one of $A\cap B$,$B\cap C$ and $C\cap A$ is $\phi$. for $k\le \frac{n}{2}$ prove that:
a) $|\mathcal F|\le max(1,4-\frac{n}{k})\times \dbinom{n-1}{k-1}$.(15 points)
b) find all cases of equality in a) for $k\le \frac{n}{3}$.(5 points)
PEN Q Problems, 6
Prove that for a prime $p$, $x^{p-1}+x^{p-2}+ \cdots +x+1$ is irreducible in $\mathbb{Q}[x]$.
2011 JBMO Shortlist, 3
$\boxed{\text{A3}}$If $a,b$ be positive real numbers, show that:$$ \displaystyle{\sqrt{\dfrac{a^2+ab+b^2}{3}}+\sqrt{ab}\leq a+b}$$