Found problems: 85335
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:$$ \displaystyle{\sqrt{\dfrac{a^2+ab+b^2}{3}}+\sqrt{ab}\leq a+b}$$
2005 Vietnam Team Selection Test, 3
Find all functions $f: \mathbb{Z} \mapsto \mathbb{Z}$ satisfying the condition: $f(x^3 +y^3 +z^3 )=f(x)^3+f(y)^3+f(z)^3.$
2024 Harvard-MIT Mathematics Tournament, 19
let $A_1A_2\ldots A_{19}$ be a regular nonadecagon. Lines $A_1A_5$ and $A_3A_4$ meet at $X.$ Compute $\angle A_7 X A_5.$
2003 Mediterranean Mathematics Olympiad, 1
Prove that the equation $x^2 + y^2 + z^2 = x + y + z + 1$ has no rational solutions.
2006 Bulgaria Team Selection Test, 2
[b] Problem 5. [/b]Denote with $d(a,b)$ the numbers of the divisors of natural $a$, which are greater or equal to $b$. Find all natural $n$, for which
$d(3n+1,1)+d(3n+2,2)+\ldots+d(4n,n)=2006.$
[i]Ivan Landgev[/i]
2012 Lusophon Mathematical Olympiad, 5
5)Players $A$ and $B$ play the following game: a player writes, in a board, a positive integer $n$, after this they delete a number in the board and write a new number where can be:
i)The last number $p$, where the new number will be $p - 2^k$ where $k$ is greatest number such that $p\ge 2^k$
ii) The last number $p$, where the new number will be $\frac{p}{2}$ if $p$ is even.
The players play alternately, a player win(s) if the new number is equal to $0$ and player $A$ starts.
a)Which player has the winning strategy with $n = 40$??
b)Which player has the winning strategy with $n = 2012$??
1947 Putnam, B4
Given $P(z)= z^2 +az +b,$ where $a,b \in \mathbb{C}.$ Suppose that $|P(z)|=1$ for every complex number $z$ with $|z|=1.$ Prove that $a=b=0.$