Found problems: 85335
2013 South africa National Olympiad, 2
A is a two-digit number and B is a three-digit number such that A increased by B% equals B reduced by A%. Find all possible pairs (A, B).
2007 CHKMO, 4
Let a_1, a_2, a_3,... be a sequence of positive numbers. If there exists a positive number M such that for n = 1,2,3,...,
$a^{2}_{1}+a^{2}_{2}+...+a^{2}_{n}< Ma^{2}_{n+1}$
then prove that there exist a positive number M' such that for every n = 1,2,3,...,
$a_{1}+a_{2}+...+a_{n}< M'a_{n+1}$
2021 CCA Math Bonanza, L4.4
Let $ABC$ be a triangle with $AB=5$, $BC=7$, $CA=8$, and let $M$ be the midpoint of $BC$. Points $P$ and $Q$ are chosen on the circumcircle of $ABC$ such that $MPQ$ and $ABC$ are similar (with vertices in that order). The product of all different possible areas of $MPQ$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
[i]2021 CCA Math Bonanza Lightning Round #4.4[/i]
2001 IMC, 5
Let $A$ be an $n\times n$ complex matrix such that $A \ne \lambda I_{n}$ for all $\lambda \in \mathbb{C}$. Prove that $A$ is similar to a matrix having at most one non-zero entry on the maindiagonal.
2022 Indonesia TST, A
Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that
$$(a + b + c)(ab + bc + ca) + 3\ge 4(a + b + c).$$
2002 Moldova Team Selection Test, 1
Prove that for every positive integer n, there exists a polynomial p(x) with integer coefficients such that p(1), p(2),..., p(n-1), p(n) are distinct powers of 2.
2023 Baltic Way, 2
Let $a_1, a_2, \ldots, a_{2023}$ be positive reals such that $\sum_{i=1}^{2023}a_i^i=2023$. Show that $$\sum_{i=1}^{2023}a_i^{2024-i}>1+\frac{1}{2023}.$$
PEN A Problems, 115
Does there exist a $4$-digit integer (in decimal form) such that no replacement of three of its digits by any other three gives a multiple of $1992$?
2014 NIMO Problems, 4
Let $n$ be a positive integer. Determine the smallest possible value of $1-n+n^2-n^3+\dots+n^{1000}$.
[i]Proposed by Evan Chen[/i]
2014 Online Math Open Problems, 8
Let $a$ and $b$ be randomly selected three-digit integers and suppose $a > b$.
We say that $a$ is [i]clearly bigger[/i] than $b$ if each digit of $a$ is larger than the corresponding digit of $b$.
If the probability that $a$ is clearly bigger than $b$ is $\tfrac mn$, where $m$ and $n$ are relatively prime integers,
compute $m+n$.
[i]Proposed by Evan Chen[/i]
2019 China Girls Math Olympiad, 7
Let $DFGE$ be a cyclic quadrilateral. Line $DF$ intersects $EG$ at $C,$ and line $FE$ intersects $DG$ at $H.$ $J$ is the midpoint of $FG.$ The line $\ell$ is the reflection of the line $DE$ in $CH,$ and it intersects line $GF$ at $I.$
Prove that $C,J,H,I$ are concyclic.
2009 Today's Calculation Of Integral, 477
Suppose that $ P_1(x)\equal{}\frac{d}{dx}(x^2\minus{}1),\ P_2(x)\equal{}\frac{d^2}{dx^2}(x^2\minus{}1)^2,\ P_3(x)\equal{}\frac{d^3}{dx^3}(x^2\minus{}1)^3$.
Find all possible values for which $ \int_{\minus{}1}^1 P_k(x)P_l(x)\ dx\ (k\equal{}1,\ 2,\ 3,\ l\equal{}1,\ 2,\ 3)$ can be valued.
2011 Abels Math Contest (Norwegian MO), 4a
In a town there are $n$ avenues running from south to north. They are numbered $1$ through $n$ (from west to east). There are $n$ streets running from west to east – they are also numbered $1$ through $n$ (from south to north).
If you drive through the junction of the $k$th avenue and the $\ell$th street, you have to pay $k\ell$ kroner. How much do you at least have to pay for driving from the junction of the $1$st avenue and the $1$st street to the junction of the nth avenue and the $n$th street? (You also pay for the starting and finishing junctions.)
2013 Bulgaria National Olympiad, 2
Find all $f : \mathbb{R}\to \mathbb{R}$ , bounded in $(0,1)$ and satisfying:
$x^2 f(x) - y^2 f(y) = (x^2-y^2) f(x+y) -xy f(x-y)$
for all $x,y \in \mathbb{R}$
[i]Proposed by Nikolay Nikolov[/i]
1963 Dutch Mathematical Olympiad, 5
You want to color the side faces of a cube in such a way that each face is colored evenly. Six colors are available:
[i]red, white, blue, yellow, purple, orange[/i]. Two cube colors are called the same if one arises from the other by a rotation of the cube.
(a) How many different cube colorings are there, using six colors?
(b) How many different cube colorings are there, using exactly five colors?
1969 IMO Shortlist, 39
$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\{AB,AC,BC\}$ is the greatest possible.
2012 IMO Shortlist, N3
Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.
2013 Kazakhstan National Olympiad, 1
Find all triples of positive integer $(m,n,k)$ such that $ k^m|m^n-1$ and $ k^n|n^m-1$
2021 Francophone Mathematical Olympiad, 3
Let $ABCD$ be a square with incircle $\Gamma$. Let $M$ be the midpoint of the segment $[CD]$. Let $P \neq B$ be a point on the segment $[AB]$. Let $E \neq M$ be the point on $\Gamma$ such that $(DP)$ and $(EM)$ are parallel. The lines $(CP)$ and $(AD)$ meet each other at $F$. Prove that the line $(EF)$ is tangent to $\Gamma$
2011 Today's Calculation Of Integral, 694
Prove the following inequality:
\[\int_1^e \frac{(\ln x)^{2009}}{x^2}dx>\frac{1}{2010\cdot 2011\cdot2012}\]
created by kunny
2000 National Olympiad First Round, 10
$N$ is a $50-$digit number (in the decimal scale). All digits except the $26^{\text{th}}$ digit (from the left) are $1$. If $N$ is divisible by $13$, what is the $26^{\text{th}}$ digit?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 8
\qquad\textbf{(E)}\ \text{More information is needed}
$
2015 India IMO Training Camp, 1
Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ .
[i]Proposed by Serbia[/i]
1988 China Team Selection Test, 4
Let $k \in \mathbb{N},$ $S_k = \{(a, b) | a, b = 1, 2, \ldots, k \}.$ Any two elements $(a, b)$, $(c, d)$ $\in S_k$ are called "undistinguishing" in $S_k$ if $a - c \equiv 0$ or $\pm 1 \pmod{k}$ and $b - d \equiv 0$ or $\pm 1 \pmod{k}$; otherwise, we call them "distinguishing". For example, $(1, 1)$ and $(2, 5)$ are undistinguishing in $S_5$. Considering the subset $A$ of $S_k$ such that the elements of $A$ are pairwise distinguishing. Let $r_k$ be the maximum possible number of elements of $A$.
(i) Find $r_5$.
(ii) Find $r_7$.
(iii) Find $r_k$ for $k \in \mathbb{N}$.
1972 IMO Longlists, 38
Congruent rectangles with sides $m(cm)$ and $n(cm)$ are given ($m, n$ positive integers). Characterize the rectangles that can be constructed from these rectangles (in the fashion of a jigsaw puzzle). (The number of rectangles is unbounded.)
1990 Putnam, A3
Prove that any convex pentagon whose vertices (no three of which are collinear) have integer coordinates must have area greater than or equal to $ \dfrac {5}{2} $.