Found problems: 85335
2022 Greece Team Selection Test, 4
In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country:
[i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]
2011-2012 SDML (High School), 4
What is the imaginary part of the complex number $\frac{-4+7i}{1+2i}$?
$\text{(A) }-\frac{1}{2}\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }\frac{7}{2}\qquad\text{(E) }-\frac{18}{5}$
2005 Belarusian National Olympiad, 5
For $0<a,b,c,d<\frac{\pi}{2}$ is true that $$\cos 2a+\cos 2b+ \cos 2c+ \cos 2d= 4 (\sin a \sin b \sin c \sin d -\cos a \cos b \cos c \cos d)$$
Find all possible values of $a+b+c+d$
ICMC 2, 2
In the symmetric group \(S_n\ (n \geq 3)\), let \(G_{a,b}\) be the subgroup generated by the 2-cycle \((a\ b)\) and the n-cycle \((1\ 2\ \cdots\ n)\). Find the index \(\left|S_n : G_{a,b}\right|\).
the 9th XMO, 3
A sequence $\{a_n\} $ satisfies $a_1$ is a positive integer and $a_{n+1}$ is the largest odd integer that divides $2^n-1+a_n$ for all $n\geqslant 1$. Given a positive integer $r$ which is greater than $1$. Is it possible that there exists infinitely many pairs of ordered positive integers $(m,n)$ for which $m>n$ and $a_m = ra_n$?
In other words, if you successfully find [b]an[/b] $a_1$ that yields infinitely many pairs of $(m,n)$ which work fine, you win and the answer is YES. Otherwise you have to proof NO for every possible $a_1$.
@below, XMO stands for Xueersi Mathematical Olympiad, where Xueersi (学而思) is a famous tutoring camp in China.
2018 PUMaC Number Theory B, 8
Find the smallest positive integer $G$ such that there exist distinct positive integers $a, b, c$ with the following properties:
$\: \bullet \: \gcd(a, b, c) = G$.
$\: \bullet \: \text{lcm}(a, b) = \text{lcm}(a, c) = \text{lcm}(b, c)$.
$\: \bullet \: \frac{1}{a} + \frac{1}{b}, \frac{1}{a} + \frac{1}{c},$ and $\frac{1}{b} + \frac{1}{c}$ are reciprocals of integers.
$\: \bullet \: \gcd(a, b) + \gcd(a, c) + \gcd(b, c) = 16G$.
2020 Princeton University Math Competition, 10
Let $N$ be the number of sequences of positive integers greater than $ 1$ where the product of all of the terms of the sequence is $12^{64}$. If $N$ can be expressed as $a(2^b)$ ), where $a$ is an odd positive integer, determine $b$.
2010 Stanford Mathematics Tournament, 1
Find the reflection of the point $(11, 16, 22)$ across the plane $3x+4y+5z=7$.
1969 IMO Shortlist, 45
Given $n>4$ points in the plane, no three collinear. Prove that there are at least $\frac{(n-3)(n-4)}{2}$ convex quadrilaterals with vertices amongst the $n$ points.
2001 Bosnia and Herzegovina Team Selection Test, 6
Prove that there exists infinitely many positive integers $n$ such that equation $(x+y+z)^3=n^2xyz$ has solution $(x,y,z)$ in set $\mathbb{N}^3$
2005 Today's Calculation Of Integral, 32
Evaluate
\[\int_0^1 e^{x+e^x+e^{e^x}+e^{e^{e^x}}}dx\]
2024 LMT Fall, B4
Let $S$, $K$, $I$, $B$, $D$, $Y$ be distinct integers from $0$ to $9,$ inclusive. Given that they follow this equation:
$$\begin{array}{rrrrr}
& S & K & I & B \\
- & I & D & I & D \\
\hline
& & & D & Y
\end{array}$$find the maximum value of $\overline{SKIBIDI}$.
2025 All-Russian Olympiad Regional Round, 10.5
The heights $BD$ and $CE$ of the acute-angled triangle $ABC$ intersect at point $H$, the heights of the triangle $ADE$ intersect at point $F$, point $M$ is the midpoint of side $BC$. Prove that $BH + CH \geqslant 2 FM$.
[i]A. Kuznetsov[/i]
2011 South africa National Olympiad, 6
In triangle $ABC$, the incircle touches $BC$ in $D$, $CA$ in $E$ and $AB$ in $F$. The bisector of $\angle BAC$ intersects $BC$ in $G$. The lines $BE$ and $CF$ intersect in $J$. The line through $J$ perpendicular to $EF$ intersects $BC$ in $K$. Prove that
$\frac{GK}{DK}=\frac{AE}{CE}+\frac{AF}{BF}$
1999 Rioplatense Mathematical Olympiad, Level 3, 2
Let $p_1, p_2, ..., p_k$ be $k$ different primes. We consider all positive integers that use only these primes (not necessarily all) in their prime factorization, and arrange those numbers in increasing order, forming an infinite sequence: $a_1 < a_2 < ... < a_n < ...$
Prove that, for every number $c$, there exists $n$ such that $a_{n+1} -a_n > c$.
2014 ISI Entrance Examination, 6
Define $\mathcal{A}=\{(x,y)|x=u+v,y=v, u^2+v^2\le 1\}$. Find the length of the longest segment that is contained in $\mathcal{A}$.
2016 PUMaC Combinatorics B, 8
Katie Ledecky and Michael Phelps each participate in $7$ swimming events in the Olympics (and there is no event that they both participate in). Ledecky receives $g_L$ gold, $s_L$ silver, and $b_L$ bronze medals, and Phelps receives $g_P$ gold, $s_P$ silver, and $b_P$ bronze medals. Ledecky notices that she performed objectively better than Phelps: for all positive real numbers $w_b<w_s<w_g$, we have
$$w_gg_l+w_ss_L+w_bb_L>w_gg_P+w_ss_P+w_bb_P.$$
Compute the number of possible $6$-tuples $(g_L,s_L,b_L,g_P,s_P,b_P).$
2024 All-Russian Olympiad, 7
There are $8$ different quadratic trinomials written on the board, among them there are no two that add up to a zero polynomial. It turns out that if we choose any two trinomials $g_1(x), g_2(X)$ from the board, then the remaining $6$ trinomials can be denoted as $g_3(x),g_4(x),\dots,g_8(x)$ so that all four polynomials $g_1(x)+g_2(x),g_3(x)+g_4(x),g_5(x)+g_6(x)$ and $g_7(x)+g_8(x)$ have a common root. Do all trinomials on the board necessarily have a common root?
[i]Proposed by S. Berlov[/i]
2023 May Olympiad, 5
There are $100$ boxes that were labeled with the numbers $00$, $01$, $02$,$…$, $99$ . The numbers $000$, $001$, $002$, $…$, $999$ were written on a thousand cards, one on each card. Placing a card in a box is permitted if the box number can be obtained by removing one of the digits from the card number. For example, it is allowed to place card $037$ in box $07$, but it is not allowed to place the card $156$ in box $65$.Can it happen that after placing all the cards in the boxes, there will be exactly $50$ empty boxes?
If the answer is yes, indicate how the cards are placed in the boxes; If the answer is no, explain why it is impossible
2009 Czech and Slovak Olympiad III A, 1
Knowing that the numbers $p, 3p+2, 5p+4, 7p+6, 9p+8$, and $11p+10$ are all primes, prove that $6p+11$ is a composite number.
2019 Jozsef Wildt International Math Competition, W. 27
Find all continuous functions $f : \mathbb{R} \to \mathbb{R}$ such that$$f(-x)+\int \limits_0^xtf(x-t)dt=x,\ \forall\ x\in \mathbb{R}$$
1997 Brazil Team Selection Test, Problem 3
Let $b$ be a positive integer such that $\gcd(b,6)=1$. Show that there are positive integers $x$ and $y$ such that $\frac1x+\frac1y=\frac3b$ if and only if $b$ is divisible by some prime number of form $6k-1$.
2016 District Olympiad, 2
For any natural number $ n, $ denote $ x_n $ as being the number of natural numbers of $ n $ digits that are divisible by $ 4 $ and formed only with the digits $ 0,1,2 $ or $ 6. $
[b]a)[/b] Calculate $ x_1,x_2,x_3,x_4. $
[b]b)[/b] Find the natural number $ m $ such that
$$ 1+\left\lfloor \frac{x_2}{x_1}\right\rfloor +\left\lfloor \frac{x_3}{x_2}\right\rfloor +\left\lfloor \frac{x_4}{x_3}\right\rfloor +\cdots +\left\lfloor \frac{x_{m+1}}{x_m}\right\rfloor =2016 , $$
where $ \lfloor\rfloor $ is the usual integer part.
2011 Gheorghe Vranceanu, 1
Let $ O $ be the circumcenter of $ ABC. $ The equalities
$$ |OA+2OB|=|OB+2OC|=|OC+2OA| $$
hold. Prove that $ ABC $ is equilateral.
2017 Hanoi Open Mathematics Competitions, 5
Let $a, b, c$ be two-digit, three-digit, and four-digit numbers, respectively. Assume that the sum of all digits of number $a+b$, and the sum of all digits of $b + c$ are all equal to $2$. The largest value of $a + b + c$ is
(A): $1099$ (B): $2099$ (C): $1199$ (D): $2199$ (E): None of the above.