Found problems: 85335
2021 MOAA, 6
Find the sum of all two-digit prime numbers whose digits are also both prime numbers.
[i]Proposed by Nathan Xiong[/i]
2007 Hanoi Open Mathematics Competitions, 15
Let $p = \overline{abcd}$ be a $4$-digit prime number. Prove that the equation $ax^3+bx^2+cx+d=0$ has no rational roots.
2018 CCA Math Bonanza, T8
A rectangular prism with positive integer side lengths formed by stacking unit cubes is called [i]bipartisan[/i] if the same number of unit cubes can be seen on the surface as those which cannot be seen on the surface. How many non-congruent bipartisan rectangular prisms are there?
[i]2018 CCA Math Bonanza Team Round #8[/i]
2010 Iran MO (3rd Round), 2
$R$ is a ring such that $xy=yx$ for every $x,y\in R$ and if $ab=0$ then $a=0$ or $b=0$. if for every Ideal $I\subset R$ there exist $x_1,x_2,..,x_n$ in $R$ ($n$ is not constant) such that $I=(x_1,x_2,...,x_n)$, prove that every element in $R$ that is not $0$ and it's not a unit, is the product of finite irreducible elements.($\frac{100}{6}$ points)
2011 China Team Selection Test, 1
Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$. Let $P$ be an arbitrary point in the interior of $\triangle ABC$, and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$, respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$, let $Y$ be the point such that $E$ is the midpoint of $B'Y$, and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$. Prove that triangle $XYZ$ is similar to triangle $ABC$.
2014 MMATHS, 3
Let $f : R^+ \to R^+$ be a function satisfying $$f(\sqrt{x_1x_2}) =\sqrt{f(x_1)f(x_2)}$$ for all positive real numbers $x_1, x_2$. Show that $$f( \sqrt[n]{x_1x_2... x_n}) = \sqrt[n]{f(x_1)f(x_2) ... f(x_n)}$$ for all positive integers $n$ and positive real numbers $x_1, x_2,..., x_n$.
1908 Eotvos Mathematical Competition, 2
Let $n$ be an integer greater than $2$. Prove that the $n$th power of the length of the hypotenuse of a right triangle is greater than the sum of the $n$th powers of the lengths of the legs.
2024 AMC 8 -, 10
In January 1980 the Moana Loa Observation recorded carbon dioxide levels of 338 ppm (parts per million). Over the years the average carbon dioxide reading has increased by about 1.515 ppm each year. What is the expected carbon dioxide level in ppm in January 2030? Round your answer to the nearest integer.
$\textbf{(A) } 399\qquad\textbf{(B) } 414\qquad\textbf{(C) } 420\qquad\textbf{(D) } 444\qquad\textbf{(E) } 459$
2002 SNSB Admission, 3
Classify up to homeomorphism the topological spaces of the support of functions that are real quadratic polynoms of three variables and and irreducible over the set of real numbers.
2012 Princeton University Math Competition, B4
For a set $S$ of integers, define $\max (S)$ to be the maximal element of $S$. How many non-empty subsets $S \subseteq \{1, 2, 3, ... , 10\}$ satisfy $\max (S) \le |S| + 2$?
2019 Belarus Team Selection Test, 2.1
Given a quadratic trinomial $p(x)$ with integer coefficients such that $p(x)$ is not divisible by $3$ for all integers $x$.
Prove that there exist polynomials $f(x)$ and $h(x)$ with integer coefficients such that
$$
p(x)\cdot f(x)+3h(x)=x^6+x^4+x^2+1.
$$
[i](I. Gorodnin)[/i]
2007 AMC 8, 24
A bag contains four pieces of paper, each labeled with one of the digits $1$, $2$, $3$ or $4$, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multiple of $3$?
$\textbf{(A)}\ \frac{1}{4} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{1}{2} \qquad
\textbf{(D)}\ \frac{2}{3} \qquad
\textbf{(E)}\ \frac{3}{4}$
Russian TST 2014, P2
The polygon $M{}$ is bicentric. The polygon $P{}$ has vertices at the points of contact of the sides of $M{}$ with the inscribed circle. The polygon $Q{}$ is formed by the external bisectors of the angles of $M{}.$ Prove that $P{}$ and $Q{}$ are homothetic.
2016 Romania Team Selection Tests, 3
Prove that:
[b](a)[/b] If $(a_n)_{n\geq 1}$ is a strictly increasing sequence of positive integers such that $\frac{a_{2n-1}+a_{2n}}{a_n}$ is a constant as $n$ runs through all positive integers, then this constant is an integer greater than or equal to $4$; and
[b](b)[/b] Given an integer $N\geq 4$, there exists a strictly increasing sequene $(a_n)_{n\geq 1}$ of positive integers such that $\frac{a_{2n-1}+a_{2n}}{a_n}=N$ for all indices $n$.
2020 ASDAN Math Tournament, 2
Sam's cup has a $400$ mL mixture of coffee and milk tea. He pours $200$ mL into Ben's empty cup. Ben then adds 100mL of co ee to his cup and stirs well. Finally, Ben pours $200$ mL out of his cup back into Sam's cup. If the mixture in Sam's cup is now $50\%$ milk tea, then how many milliliters of milk tea were in it originally?
2019 India IMO Training Camp, P2
Determine all positive integers $m$ satisfying the condition that there exists a unique positive integer $n$ such that there exists a rectangle which can be decomposed into $n$ congruent squares and can also be decomposed into $m+n$ congruent squares.
1961 IMO Shortlist, 3
Solve the equation $\cos^n{x}-\sin^n{x}=1$ where $n$ is a natural number.
2011 Postal Coaching, 6
Prove that there exist integers $a, b, c$ all greater than $2011$ such that
\[(a+\sqrt{b})^c=\ldots 2010 \cdot 2011\ldots\]
[Decimal point separates an integer ending in $2010$ and a decimal part beginning with $2011$.]
2024 Australian Mathematical Olympiad, P4
Consider a $2024 \times 2024$ grid of unit squares. Two distinct unit squares are adjacent if they share a common side. Each unit square is to be coloured either black or white. Such a colouring is called $\textit{evenish}$ if every unit square in the grid is adjacent to an even number of black unit squares. Determine the number of $\textit{evenish}$ colourings.
2024 District Olympiad, P1
Determine the integers $n\geqslant 2$ for which the equation $x^2-\hat{3}\cdot x+\hat{5}=\hat{0}$ has a unique solution in $(\mathbb{Z}_n,+,\cdot).$
2021 AMC 12/AHSME Spring, 1
What is the value of $$2^{1+2+3}-(2^1+2^2+2^3)?$$
$\textbf{(A) }0 \qquad \textbf{(B) }50 \qquad \textbf{(C) }52 \qquad \textbf{(D) }54 \qquad \textbf{(E) }57$
Proposed by [b]djmathman[/b]
2023 Hong Kong Team Selection Test, Problem 4
Let $x$, $y$, $z$ be real numbers such that $x+y+z \ne 0$. Find the minimum value of
$\frac{|x|+|x+4y|+|y+7z|+2|z|}{|x+y+z|}$
1975 AMC 12/AHSME, 23
In the adjoining figure $AB$ and $BC$ are adjacent sides of square $ABCD$; $M$ is the midpoint of $AB$; $N$ is the midpoint of $BC$; and $AN$ and $CM$ intersect at $O$. The ratio of the area of $AOCD$ to the area of $ABCD$ is
[asy]
draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)--(2,1)--(2,2)--(1,0));
label("A", (0,0), S);
label("B", (2,0), S);
label("C", (2,2), N);
label("D", (0,2), N);
label("M", (1,0), S);
label("N", (2,1), E);
label("O", (1.2, .8));
[/asy]
$ \textbf{(A)}\ \frac{5}{6} \qquad\textbf{(B)}\ \frac{3}{4} \qquad\textbf{(C)}\ \frac{2}{3} \qquad\textbf{(D)}\ \frac{\sqrt{3}}{2} \qquad\textbf{(E)}\ \frac{(\sqrt{3}-1)}{2} $
2014 ISI Entrance Examination, 2
Let us consider a triangle $\Delta{PQR}$ in the co-ordinate plane. Show for every function $f: \mathbb{R}^2\to \mathbb{R}\;,f(X)=ax+by+c$ where $X\equiv (x,y) \text{ and } a,b,c\in\mathbb{R}$ and every point $A$ on $\Delta PQR$ or inside the triangle we have the inequality:
\begin{align*} & f(A)\le \text{max}\{f(P),f(Q),f(R)\} \end{align*}
2000 Harvard-MIT Mathematics Tournament, 7
Suppose you are given a fair coin and a sheet of paper with the polynomial $x^m$ written on it. Now for each toss of the coin, if heads show up, you must erase the polynomial $x^r$ (where $r$ is going to change with time - initially it is $m$) written on the paper and replace it with $x^{r-1}$. If tails show up, replace it with $x^{r+1}$. What is the expected value of the polynomial I get after $m$ such tosses? (Note: this is a different concept from the most probable value)