Found problems: 85335
2021-IMOC, N2
Show that for any two distinct odd primes $p, q$, there exists a positive integer $n$ such that $$\{d(n), d(n + 2) \} = \{p, q\}$$ where $d(n)$ is the smallest prime factor of $n$.
[i]Proposed By - ltf0501[/i]
2007 Greece JBMO TST, 4
Calculate the sum $$S=\sqrt{1+\frac{8\cdot 1^2-1}{1^2\cdot 3^2}}+\sqrt{1+\frac{8\cdot 2^2-1}{3^2\cdot 5^2}}+...+ \sqrt{1+\frac{8\cdot 1003^2-1}{2005^2\cdot 2007^2}}$$
1984 Tournament Of Towns, (057) O5
An infinite squared sheet is given, with squares of side length $1$. The “distance” between two squares is defined as the length of the shortest path from one of these squares to the other if moving between them like a chess rook (measured along the trajectory of the centre of the rook). Determine the minimum number of colours with which it is possible to colour the sheet (each square being given a single colour) in such a way that each pair of squares with distance between them equal to $6$ units is given different colours. Give an example of such a colouring and prove that using a smaller number of colours we cannot achieve this goal.
(AG Pechkovskiy, IV Itenberg)
Indonesia Regional MO OSP SMA - geometry, 2005.4
The lengths of the three sides $a, b, c$ with $a \le b \le c$, of a right triangle is an integer. Find all the sequences $(a, b, c)$ so that the values of perimeter and area of the triangle are the same.
2013 China Team Selection Test, 1
Let $p$ be a prime number and $a, k$ be positive integers such that $p^a<k<2p^a$. Prove that there exists a positive integer $n$ such that \[n<p^{2a}, C_n^k\equiv n\equiv k\pmod {p^a}.\]
2018 AMC 12/AHSME, 12
Let $S$ be a set of 6 integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$?
$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 7$
2024 JHMT HS, 3
Amelia has $27$ unit cubes. She selects one and paints one of its faces. She then randomly glues all $27$ cubes together to form a $3 \times 3 \times 3$ cube (with all possible arrangements of the unit cubes being equally likely). Compute the probability that the resulting cube appears unpainted.
1991 Arnold's Trivium, 28
Sketch the phase portrait and investigate its variation under variation of the small complex parameter $\epsilon$:
\[\dot{z}=\epsilon z-(1+i)z|z|^2+\overline{z}^4\]
2021 Argentina National Olympiad, 2
On each OMA lottery ticket there is a $9$-digit number that only uses the digits $1, 2$ and $3$ (not necessarily all three). Each ticket has one of the three colors red, blue or green. It is known that if two banknotes do not match in any of the $9$ figures, then they are of different colors. Bill $122222222$ is red, $222222222$ is green, what color is bill $123123123$?
2013 China Team Selection Test, 2
Find the greatest positive integer $m$ with the following property:
For every permutation $a_1, a_2, \cdots, a_n,\cdots$ of the set of positive integers, there exists positive integers $i_1<i_2<\cdots <i_m$ such that $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$ is an arithmetic progression with an odd common difference.
2016 India IMO Training Camp, 1
Suppose $\alpha, \beta$ are two positive rational numbers. Assume for some positive integers $m,n$, it is known that $\alpha^{\frac 1n}+\beta^{\frac 1m}$ is a rational number. Prove that each of $\alpha^{\frac 1n}$ and $\beta^{\frac 1m}$ is a rational number.
1962 Miklós Schweitzer, 5
Let $ f$ be a finite real function of one variable. Let $ \overline{D}f$ and $ \underline{D}f$ be its upper and lower derivatives, respectively, that is, \[ \overline{D}f\equal{}\limsup_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}\] ,
\[ \underline{D}f\equal{}\liminf_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}.\] Show that $ \overline{D}f$ and $ \underline{D}f$ are Borel-measurable functions. [A. Csaszar]
2016 Turkey Team Selection Test, 3
Let $a,b,c$ be non-negative real numbers such that $a^2+b^2+c^2 \le 3$ then prove that;
$$(a+b+c)(a+b+c-abc)\ge2(a^2b+b^2c+c^2a)$$
2000 Belarus Team Selection Test, 1.4
A closed pentagonal line is inscribed in a sphere of the diameter $1$, and has all edges of length $\ell$.
Prove that $\ell \le \sin \frac{2\pi}{5}$
.
1995 AMC 12/AHSME, 3
The total in-store price for an appliance is $\$99.99$. A television commercial advertises the same product for three easy payments of $\$29.98$ and a one-time shipping and handling charge of $\$9.98$. How much is saved by buying the appliance from the television advertiser?
$\textbf{(A)}\ \text{6 cents} \qquad
\textbf{(B)}\ \text{7 cents} \qquad
\textbf{(C)}\ \text{8 cents} \qquad
\textbf{(D)}\ \text{9 cents} \qquad
\textbf{(E)}\ \text{10 cents}$
2005 CentroAmerican, 2
Show that the equation $a^{2}b^{2}+b^{2}c^{2}+3b^{2}-c^{2}-a^{2}=2005$ has no integer solutions.
[i]Arnoldo Aguilar, El Salvador[/i]
2010 Kosovo National Mathematical Olympiad, 2
The set $S\subseteq \mathbb{R}$ is given with the properties:
$(a) \mathbb{Z}\subset S$,
$(b) (\sqrt 2 +\sqrt 3)\in S$,
$(c)$ If $x,y\in S$ then $x+y\in S$, and
$(d)$ If $x,y\in S$ then $x\cdot y\in S$.
Prove that $(\sqrt 2+\sqrt 3)^{-1}\in S$.
2013 BMT Spring, 14
Triangle $ABC$ has incircle $O$ that is tangent to $AC$ at $D$. Let $M$ be the midpoint of $AC$. $E$ lies on $BC$ so that line $AE$ is perpendicular to $BO$ extended. If $AC = 2013$, $AB = 2014$, $DM = 249$, find $CE$.
2003 AMC 12-AHSME, 17
Square $ ABCD$ has sides of length $ 4$, and $ M$ is the midpoint of $ \overline{CD}$. A circle with radius $ 2$ and center $ M$ intersects a circle with raidus $ 4$ and center $ A$ at points $ P$ and $ D$. What is the distance from $ P$ to $ \overline{AD}$?
[asy]unitsize(8mm);
defaultpen(linewidth(.8pt));
dotfactor=4;
draw(Circle((2,0),2));
draw(Circle((0,4),4));
clip(scale(4)*unitsquare);
draw(scale(4)*unitsquare);
filldraw(Circle((2,0),0.07));
filldraw(Circle((3.2,1.6),0.07));
label("$A$",(0,4),NW);
label("$B$",(4,4),NE);
label("$C$",(4,0),SE);
label("$D$",(0,0),SW);
label("$M$",(2,0),S);
label("$P$",(3.2,1.6),N);[/asy]$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac {16}{5} \qquad \textbf{(C)}\ \frac {13}{4} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {7}{2}$
2024 CMIMC Integration Bee, 1
\[\int_1^e \frac{\log(x^{2024})}{x} \mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2011 Bogdan Stan, 3
Solve in $ \mathbb{R} $ the equation $ 4^{x^2-x}=\log_2 x+\sqrt{x-1} +14. $
[i]Marin Tolosi[/i]
1982 IMO Longlists, 7
Find all solutions $(x, y) \in \mathbb Z^2$ of the equation
\[x^3 - y^3 = 2xy + 8.\]
2023 AMC 10, 11
Suzanne went to the bank and withdrew \$$800$. The teller gave her this amount using \$$20$ bills, \$$50$ bills, and \$$100$ bills, with at least one of each denomination. How many different collections of bills could Suzanne have received?
$\textbf{(A) }45\qquad\textbf{(B) }21\qquad\textbf{(C) }36\qquad\textbf{(D) }28\qquad\textbf{(E) }32$
1974 IMO Longlists, 42
In a certain language words are formed using an alphabet of three letters. Some words of two or more letters are not allowed, and any two such distinct words are of different lengths. Prove that one can form a word of arbitrary length that does not contain any non-allowed word.
1974 Polish MO Finals, 6
Several diagonals in a convex $n$-gon are drawn so as to divide the $n$-gon into triangles and:
(i) the number of diagonals drawn at each vertex is even;
(ii) no two of the diagonals have a common interior point.
Prove that $n$ is divisible by $3$.