Found problems: 85335
2003 France Team Selection Test, 3
Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.
2024 Iran Team Selection Test, 9
Prove that for any natural numbers $a , b , c$ that $b>a>1$ and $gcd(c,ab)=1$ , there exist a natural number $n$ such that :
$$c | \binom{b^n}{a^n}$$
[i]Proposed by Navid Safaei[/i]
2019 HMNT, 2
Sandy likes to eat waffles for breakfast. To make them, she centers a circle of wafflebatter of radius $3$ cm at the origin of the coordinate plane and her waffle iron imprints non-overlapping unit-square holes centered at each lattice point. How many of these holes are contained entirely within the area of the waffle?
2019 LIMIT Category A, Problem 11
$\angle A,\angle B,\angle C$ are angles of a triangle such that $\sin^2A+\sin^2B=\sin^2C$, then $\angle C$ in degrees is equal to
$\textbf{(A)}~30$
$\textbf{(B)}~90$
$\textbf{(C)}~45$
$\textbf{(D)}~\text{none of the above}$
2023 Harvard-MIT Mathematics Tournament, 4
Let $ABCD$ be a square, and let $M$ be the midpoint of side $BC$. Points $P$ and $Q$ lie on segment $AM$ such that $\angle BPD=\angle BQD=135^\circ$. Given that $AP<AQ$, compute $\tfrac{AQ}{AP}$.
1972 AMC 12/AHSME, 12
The number of cubic feet in the volume of a cube is the same as the number of square inches in its surface area. The length of the edge expressed as a number of feet is
$\textbf{(A) }6\qquad\textbf{(B) }864\qquad\textbf{(C) }1728\qquad\textbf{(D) }6\times 1728\qquad \textbf{(E) }2304$
2015 USA TSTST, 4
Let $x$, $y$, and $z$ be real numbers (not necessarily positive) such that $x^4+y^4+z^4+xyz=4$.
Show that $x\le2$ and $\sqrt{2-x}\ge\frac{y+z}{2}$.
[i]Proposed by Alyazeed Basyoni[/i]
2017 JBMO Shortlist, A2
Let $a$ and $b$ be positive real numbers such that $3a^2 + 2b^2 = 3a + 2b$. Find the minimum value of $A =\sqrt{\frac{a}{b(3a+2)}} + \sqrt{\frac{b}{a(2b+3)}} $
2021 Romania National Olympiad, 1
Find all continuous functions $f:\left[0,1\right]\rightarrow[0,\infty)$ such that:
$\int_{0}^{1}f\left(x\right)dx\cdotp\int_{0}^{1}f^{2}\left(x\right)dx\cdotp...\cdotp\int_{0}^{1}f^{2020}\left(x\right)dx=\left(\int_{0}^{1}f^{2021}\left(x\right)dx\right)^{1010}$
1997 Polish MO Finals, 3
Given any $n$ points on a unit circle show that at most $\frac{n^2}{3}$ of the segments joining two points have length $> \sqrt{2}$.
2010 Contests, 3
Let $n$ be a positive integer. $n$ people take part in a certain party. For any pair of the participants, either the two are acquainted with each other or they are not. What is the maximum possible number of the pairs for which the two are not acquainted but have a common acquaintance among the participants?
2001 Cuba MO, 4
The tangents at four different points of an arc of a circle less than $180^o$ intersect forming a convex quadrilateral $ABCD$. Prove that two of the vertices belong to an ellipse whose foci to the other two vertices.
2010 Today's Calculation Of Integral, 612
For $f(x)=\frac{1}{x}\ (x>0)$, prove the following inequality.
\[f\left(t+\frac 12 \right)\leq \int_t^{t+1} f(x)\ dx\leq \frac 16\left\{f(t)+4f\left(t+\frac 12\right)+f(t+1)\right\}\]
2007 Grigore Moisil Intercounty, 3
Find the natural numbers $ a $ that have the property that there exists a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ such that $ f(f(n))=a+n, $ for any natural number $ n, $ and the function $ g:\mathbb{N}\longrightarrow\mathbb{N} $ defined as $ g(n)=f(n)-n $ is injective.
2023 Romanian Master of Mathematics Shortlist, G3
A point $P$ is chosen inside a triangle $ABC$ with circumcircle $\Omega$. Let $\Gamma$ be the circle passing
through the circumcenters of the triangles $APB$, $BPC$, and $CPA$. Let $\Omega$ and $\Gamma$ intersect at
points $X$ and $Y$. Let $Q$ be the reflection of $P$ in the line $XY$ . Prove that $\angle BAP = \angle CAQ$.
2021 AMC 10 Spring, 2
Portia’s high school has $3$ times as many students as Lara’s high school. The two high schools have a total of
$2600$ students. How many students does Portia’s high school have?
$\textbf{(A) }600 \qquad \textbf{(B) }650 \qquad \textbf{(C) }1950 \qquad \textbf{(D) }2000 \qquad \textbf{(E) }2050$
1977 IMO Longlists, 42
The sequence $a_{n,k} \ , k = 1, 2, 3,\ldots, 2^n \ , n = 0, 1, 2,\ldots,$ is defined by the following recurrence formula:
\[a_1 = 2,\qquad a_{n,k} = 2a_{n-1,k}^3, \qquad , a_{n,k+2^{n-1}} =\frac 12 a_{n-1,k}^3\]\[\text{for} \quad k = 1, 2, 3,\ldots, 2^{n-1} \ , n = 0, 1, 2,\ldots\]
Prove that the numbers $a_{n,k}$ are all different.
2006 Oral Moscow Geometry Olympiad, 1
An arbitrary triangle $ABC$ is given. Construct a line that divides it into two polygons, which have equal radii of the circumscribed circles.
(L. Blinkov)
1992 AMC 8, 4
During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single?
$\text{(A)}\ 28\% \qquad \text{(B)}\ 35\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 75\% \qquad \text{(E)}\ 80\% $
2002 Tuymaada Olympiad, 4
A rectangular table with 2001 rows and 2002 columns is partitioned into $1\times 2$ rectangles. It is known that any other partition of the table into $1\times 2$ rectangles contains a rectangle belonging to the original partition.
Prove that the original partition contains two successive columns covered by 2001 horizontal rectangles.
[i]Proposed by S. Volchenkov[/i]
2012 Today's Calculation Of Integral, 843
Let $f(x)$ be a continuous function such that $\int_0^1 f(x)\ dx=1.$ Find $f(x)$ for which $\int_0^1 (x^2+x+1)f(x)^2dx$ is minimized.
2022 Estonia Team Selection Test, 3
Determine all tuples of integers $(a,b,c)$ such that:
$$(a-b)^3(a+b)^2 = c^2 + 2(a-b) + 1$$
1983 Bundeswettbewerb Mathematik, 4
Let $g$ be a straight line and $n$ a given positive integer. Prove that there are always n different points on g to choose as well as a point not lying on g in such a way that the distance between each two of these $n + 1$ points is an integer.
Novosibirsk Oral Geo Oly VII, 2021.5
In an acute-angled triangle $ABC$ on the side $AC$, point $P$ is chosen in such a way that $2AP = BC$. Points $X$ and $Y$ are symmetric to $P$ with respect to vertices $A$ and $C$, respectively. It turned out that $BX = BY$. Find $\angle BCA$.
2016 Iranian Geometry Olympiad, 2
Let two circles $C_1$ and $C_2$ intersect in points $A$ and $B$. The tangent to $C_1$ at $A$ intersects $C_2$ in $P$ and the line $PB$ intersects $C_1$ for the second time in $Q$ (suppose that $Q$ is outside $C_2$). The tangent to $C_2$ from $Q$ intersects $C_1$ and $C_2$ in $C$ and $D$, respectively. (The points $A$ and $D$ lie on different sides of the line $PQ$.) Show that $AD$ is the bisector of $\angle CAP$.
[i]Proposed by Iman Maghsoudi[/i]