Found problems: 85335
2007 AMC 8, 3
What is the sum of the two smallest prime factors of $250$?
$\textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ 12$
2017 Morocco TST-, 6
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
1962 All-Soviet Union Olympiad, 7
Let $a;b;c;d>0$ such that $abcd=1$. Prove that $a^2+b^2+c^2+d^2+a(b+c)+b(c+d)+c(d+a)\ge 10$
2012 Princeton University Math Competition, B2
A $6$-inch-wide rectangle is rotated $90$ degrees about one of its corners, sweeping out an area of $45\pi$ square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches.
2012 Princeton University Math Competition, A5
What is the smallest natural number $n$ greater than $2012$ such that the polynomial $f(x) =(x^6 + x^4)^n - x^{4n} - x^6$ is divisible by $g(x) = x^4 + x^2 + 1$?
1977 Canada National Olympiad, 3
$N$ is an integer whose representation in base $b$ is $777$. Find the smallest integer $b$ for which $N$ is the fourth power of an integer.
1998 AMC 12/AHSME, 10
A large square is divided into a small square surrounded by four congruent rectangles as shown. The perimeter of each of the congruent rectangles is 14. What is the area of the large square?
[asy]unitsize(3mm);
defaultpen(linewidth(.8pt));
draw((0,0)--(7,0)--(7,7)--(0,7)--cycle);
draw((1,0)--(1,6));
draw((7,1)--(1,1));
draw((6,7)--(6,1));
draw((0,6)--(6,6));[/asy]$ \textbf{(A)}\ \ 49 \qquad \textbf{(B)}\ \ 64 \qquad \textbf{(C)}\ \ 100 \qquad \textbf{(D)}\ \ 121 \qquad \textbf{(E)}\ \ 196$
2005 Germany Team Selection Test, 3
Let $a$, $b$, $c$, $d$ and $n$ be positive integers such that $7\cdot 4^n = a^2+b^2+c^2+d^2$. Prove that the numbers $a$, $b$, $c$, $d$ are all $\geq 2^{n-1}$.
2006 All-Russian Olympiad Regional Round, 8.5
The product $a_1 \cdot a_2 \cdot ... \cdot a_{100}$ is written on the board , where $a_1$, $a_2$, $ ... $, $a_{100}$, are natural numbers. Let's consider $99$ expressions, each of which is obtained by replacing one of the multiplication signs with an addition sign. It is known that the values of exactly $32$ of these expressions are even. What is the largest number of even numbers among $a_1$, $a_2$, $ ... $, $a_{100}$ could it be?
2015 India Regional MathematicaI Olympiad, 1
Let ABC be a triangle. Let B' and C' denote the reflection of B and C in the internal angle bisector of angle A. Show that the triangles ABC and AB'C' have the same incenter.
2003 AMC 10, 12
A point $ (x,y)$ is randomly picked from inside the rectangle with vertices $ (0,0)$, $ (4,0)$, $ (4,1)$, and $ (0,1)$. What is the probability that $ x<y$?
$ \textbf{(A)}\ \frac{1}{8} \qquad
\textbf{(B)}\ \frac{1}{4} \qquad
\textbf{(C)}\ \frac{3}{8} \qquad
\textbf{(D)}\ \frac{1}{2} \qquad
\textbf{(E)}\ \frac{3}{4}$
2020 New Zealand MO, 4
Determine all prime numbers $p$ such that $p^2 - 6$ and $p^2 + 6$ are both prime numbers.
2011 VJIMC, Problem 1
Let $n>k$ and let $A_1,\ldots,A_k$ be real $n\times n$ matrices of rank $n-1$. Prove that
$$A_1\cdots A_k\ne0.$$
2017 Polish MO Finals, 4
Prove that the set of positive integers $\mathbb Z^+$ can be represented as a sum of five pairwise distinct subsets with the following property: each $5$-tuple of numbers of form $(n,2n,3n,4n,5n)$, where $n\in\mathbb Z^+$, contains exactly one number from each of these five subsets.
2022 Stars of Mathematics, 4
Let $a{}$ be an even positive integer which is not a power of two. Prove that at least one of $2^{2^n}+1$ and $a^{2^n}+1$ is composite, for infinitely many positive integers $n$.
[i]Bojan Bašić[/i]
2016 CMIMC, 10
Let $\mathcal{P}$ be the unique parabola in the $xy$-plane which is tangent to the $x$-axis at $(5,0)$ and to the $y$-axis at $(0,12)$. We say a line $\ell$ is $\mathcal{P}$-friendly if the $x$-axis, $y$-axis, and $\mathcal{P}$ divide $\ell$ into three segments, each of which has equal length. If the sum of the slopes of all $\mathcal{P}$-friendly lines can be written in the form $-\tfrac mn$ for $m$ and $n$ positive relatively prime integers, find $m+n$.
1960 Putnam, A1
Let $n$ be a given positive integer. How many solutions are there in ordered positive integer pairs $(x,y)$ to the equation
$$\frac{xy}{x+y}=n?$$
2023 Princeton University Math Competition, A4 / B6
What is the smallest possible sum of six distinct positive integers for which the sum of any five of them is prime?
2006 Bulgaria Team Selection Test, 2
Find all couples of polynomials $(P,Q)$ with real coefficients, such that for infinitely many $x\in\mathbb R$ the condition \[ \frac{P(x)}{Q(x)}-\frac{P(x+1)}{Q(x+1)}=\frac{1}{x(x+2)}\]
Holds.
[i] Nikolai Nikolov, Oleg Mushkarov[/i]
2005 Cono Sur Olympiad, 2
We say that a number of 20 digits is [i]special[/i] if its impossible to represent it as an product of a number of 10 digits by a number of 11 digits. Find the maximum quantity of consecutive numbers that are specials.
2022 Dutch IMO TST, 2
Two circles $\Gamma_1$ and $\Gamma_2$are given with centres $O_1$ and $O_2$ and common exterior tangents $\ell_1$ and $\ell_2$. The line $\ell_1$ intersects $\Gamma_1$ in $A$ and $\Gamma_2$ in $B$. Let $X$ be a point on segment $O_1O_2$, not lying on $\Gamma_1$ or $\Gamma_2$. The segment $AX$ intersects $\Gamma_1$ in $Y \ne A$ and the segment $BX$ intersects $\Gamma_2$ in $Z \ne B$. Prove that the line through $Y$ tangent to $\Gamma_1$ and the line through $Z$ tangent to $\Gamma_2$ intersect each other on $\ell_2$.
1963 AMC 12/AHSME, 10
Point $P$ is taken interior to a square with side-length $a$ and such that is it equally distant from two consecutive vertices and from the side opposite these vertices. If $d$ represents the common distance, then $d$ equals:
$\textbf{(A)}\ \dfrac{3a}{5} \qquad
\textbf{(B)}\ \dfrac{5a}{8} \qquad
\textbf{(C)}\ \dfrac{3a}{8} \qquad
\textbf{(D)}\ \dfrac{a\sqrt{2}}{2} \qquad
\textbf{(E)}\ \dfrac{a}{2}$
2017 BMT Spring, 10
De ne $H_n =\sum^n_{k=1} \frac{1}{k}$ . Evaluate $\sum^{2017}_{n=1} {2017 \choose n} H_n(-1)^n$.
2010 Ukraine Team Selection Test, 2
Let $ABCD$ be a quadrilateral inscribled in a circle with the center $O, P$ be the point of intersection of the diagonals $AC$ and $BD$, $BC\nparallel AD$. Rays $AB$ and $DC$ intersect at the point $E$. The circle with center $I$ inscribed in the triangle $EBC$ touches $BC$ at point $T_1$. The $E$-excircle with center $J$ in the triangle $EAD$ touches the side $AD$ at the point T$_2$. Line $IT_1$ and $JT_2$ intersect at $Q$. Prove that the points $O, P$, and $Q$ lie on a straight line.
1975 Polish MO Finals, 6
On the interval $[0,1]$ are given functions $S(x) = 1 - x$ and $T(x) = x/2$. Does there exist a function of the form $f = g_1\circ g_2\circ ... \circ g_n$, where $n \in N$ and each $g_k$ is either $S(x)$ or $T(x)$, such that
$$f\left(\frac12\right)=\frac{1975}{2^{1975}} \, ?$$