Found problems: 85335
2010 ISI B.Math Entrance Exam, 8
Let $f$ be a real-valued differentiable function on the real line $\mathbb{R}$ such that
$\lim_{x\to 0} \frac{f(x)}{x^2}$ exists, and is finite . Prove that $f'(0)=0$.
1992 Irish Math Olympiad, 4
A convex pentagon has the property that each of its diagonals cuts off a triangle of unit area. Find the area of the pentagon.
1990 Bulgaria National Olympiad, Problem 3
Let $n=p_1p_2\cdots p_s$, where $p_1,\ldots,p_s$ are distinct odd prime numbers.
(a) Prove that the expression
$$F_n(x)=\prod\left(x^{\frac n{p_{i_1}\cdots p_{i_k}}}-1\right)^{(-1)^k},$$where the product goes over all subsets $\{p_{i_1},\ldots,p_{i_k}\}$ or $\{p_1,\ldots,p_s\}$ (including itself and the empty set), can be written as a polynomial in $x$ with integer coefficients.
(b) Prove that if $p$ is a prime divisor of $F_n(2)$, then either $p\mid n$ or $n\mid p-1$.
2023 Romanian Master of Mathematics, 2
Fix an integer $n \geq 3$. Let $\mathcal{S}$ be a set of $n$ points in the plane, no three of which are collinear. Given different points $A,B,C$ in $\mathcal{S}$, the triangle $ABC$ is [i]nice[/i] for $AB$ if $[ABC] \leq [ABX]$ for all $X$ in $\mathcal{S}$ different from $A$ and $B$. (Note that for a segment $AB$ there could be several nice triangles). A triangle is [i] beautiful [/i] if its vertices are all in $\mathcal{S}$ and is nice for at least two of its sides.
Prove that there are at least $\frac{1}{2}(n-1)$ beautiful triangles.
2019 Belarus Team Selection Test, 5.1
A function $f:\mathbb N\to\mathbb N$, where $\mathbb N$ is the set of positive integers, satisfies the following condition: for any positive integers $m$ and $n$ ($m>n$) the number $f(m)-f(n)$ is divisible by $m-n$.
Is the function $f$ necessarily a polynomial? (In other words, is it true that for any such function there exists a polynomial $p(x)$ with real coefficients such that $f(n)=p(n)$ for all positive integers $n$?)
[i](Folklore)[/i]
2003 USA Team Selection Test, 5
Let $A, B, C$ be real numbers in the interval $\left(0,\frac{\pi}{2}\right)$. Let \begin{align*} X &= \frac{\sin A\sin (A-B)\sin (A-C)}{\sin (B+C)} \\ Y &= \frac{\sin B\sin(B-C)\sin (B-A)}{\sin (C+A)} \\ Z &= \frac{\sin C\sin (C-A)\sin (C-B)}{\sin (A+B)} . \end{align*} Prove that $X+Y+Z \geq 0$.
2009 VJIMC, Problem 4
Let $k,m,n$ be positive integers such that $1\le m\le n$ and denote $S=\{1,2,\ldots,n\}$. Suppose that $A_1,A_2,\ldots,A_k$ are $m$-element subsets of $S$ with the following property: for every $i=1,2,\ldots,k$ there exists a partition $S=S_{1,i}\cup S_{2,i}\cup\ldots\cup S_{m,i}$ (into pairwise disjoint subsets) such that
(i) $A_i$ has precisely one element in common with each member of the above partition.
(ii) Every $A_j,j\ne i$ is disjoint from at least one member of the above partition.
Show that $k\le\binom{n-1}{m-1}$.
2017 Harvard-MIT Mathematics Tournament, 5
[b]E[/b]ach of the integers $1,2,...,729$ is written in its base-$3$ representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: $12101112202122...$ How many times in this string does the substring $012$ appear?
2020 Moldova Team Selection Test, 1
All members of geometrical progression $(b_n)_{n\geq1}$ are members of some arithmetical progression. It is known that $b_1$ is an integer. Prove that all members of this geometrical progression are integers. (progression is infinite)
2014 AMC 12/AHSME, 16
The product $(8)(888\ldots 8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$?
$\textbf{(A) }901\qquad
\textbf{(B) }911\qquad
\textbf{(C) }919\qquad
\textbf{(D) }991\qquad
\textbf{(E) }999\qquad$
2011 Pre-Preparation Course Examination, 5
suppose that $v(x)=\sum_{p\le x,p\in \mathbb P}log(p)$ (here $\mathbb P$ denotes the set of all positive prime numbers). prove that the two statements below are equivalent:
[b]a)[/b] $v(x) \sim x$ when $x \longrightarrow \infty$
[b]b)[/b] $\pi (x) \sim \frac{x}{ln(x)}$ when $x \longrightarrow \infty$. (here $\pi (x)$ is number of the prime numbers less than or equal to $x$).
1998 Taiwan National Olympiad, 5
For a positive integer $n$, let $\omega(n)$ denote the number of positive prime divisors of $n$. Find the smallest positive tinteger $k$ such that $2^{\omega(n)}\leq k\sqrt[4]{n}\forall n\in\mathbb{N}$.
2019 CMIMC, 4
Define a search algorithm called $\texttt{powSearch}$. Throughout, assume $A$ is a 1-indexed sorted array of distinct integers. To search for an integer $b$ in this array, we search the indices $2^0,2^1,\ldots$ until we either reach the end of the array or $A[2^k] > b$. If at any point we get $A[2^k] = b$ we stop and return $2^k$. Once we have $A[2^k] > b > A[2^{k-1}]$, we throw away the first $2^{k-1}$ elements of $A$, and recursively search in the same fashion. For example, for an integer which is at position $3$ we will search the locations $1, 2, 4, 3$.
Define $g(x)$ to be a function which returns how many (not necessarily distinct) indices we look at when calling $\texttt{powSearch}$ with an integer $b$ at position $x$ in $A$. For example, $g(3) = 4$. If $A$ has length $64$, find
\[g(1) + g(2) + \ldots + g(64).\]
2022 Oral Moscow Geometry Olympiad, 6
In an acute non-isosceles triangle $ABC$, the inscribed circle touches side $BC$ at point $T, Q$ is the midpoint of altitude $AK$, $P$ is the orthocenter of the triangle formed by the bisectors of angles $B$ and $C$ and line $AK$. Prove that the points $P, Q$ and $T$ lie on the same line.
(D. Prokopenko)
1957 Czech and Slovak Olympiad III A, 1
Find all real numbers $p$ such that the equation $$\sqrt{x^2-5p^2}=px-1$$ has a root $x=3$. Then, solve the equation for the determined values of $p$.
2006 Oral Moscow Geometry Olympiad, 4
The quadrangle $ABCD$ is inscribed in a circle, the center $O$ of which lies inside it. The tangents to the circle at points $A$ and $C$ and a straight line, symmetric to $BD$ wrt point $O$, intersect at one point. Prove that the products of the distances from $O$ to opposite sides of the quadrilateral are equal.
(A. Zaslavsky)
1971 Kurschak Competition, 1
A straight line cuts the side $AB$ of the triangle $ABC$ at $C_1$, the side $AC$ at $B_1$ and the line $BC$ at $A_1$. $C_2$ is the reflection of $C_1$ in the midpoint of $AB$, and $B_2$ is the reflection of $B_1$ in the midpoint of $AC$. The lines $B_2C_2$ and $BC$ intersect at $A_2$. Prove that $$\frac{sen \, \, B_1A_1C}{sen\, \, C_2A_2B} = \frac{B_2C_2}{B_1C_1}$$
[img]https://cdn.artofproblemsolving.com/attachments/3/8/774da81495df0a0f7f2f660ae9f516cf70df06.png[/img]
2019 AMC 12/AHSME, 5
Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of $n$?
$\textbf{(A) } 18 \qquad \textbf{(B) } 21 \qquad \textbf{(C) } 24\qquad \textbf{(D) } 25 \qquad \textbf{(E) } 28$
2024 Princeton University Math Competition, A3 / B5
$T_1$ consists of a single branch from 0 to 1 in the complex plane. This branch then splits into two branches at the endpoint which each form a $135^\circ$ angle with the branch in $T_1$, and each branch has length $\frac{1}{\sqrt{2}}$. This process is repeated so that from each terminal branch in $T_n$, we form two more branches at angles $135^\circ$ with $\frac{1}{\sqrt{2}}$ the length. Let $L_n$ be the collection of the $2^{n-1}$ endpoints of the tree $T_n$. If multiple terminal branches end at the same point, then that point is counted multiple times in $L_n$. Shown below is $T_k$ for $k=1, 2, 3$ with dots at the points in $L_k$. Find the sum of $\ell^2$ over all points $\ell$ in $L_{10}$.
2013 Miklós Schweitzer, 9
Prove that there is a function ${f: (0,\infty) \rightarrow (0,\infty)}$ which is nowhere continuous and for all ${x,y \in (0,\infty)}$ and any rational ${\alpha}$ we have
\[ \displaystyle f\left( \left(\frac{x^\alpha+y^\alpha}{2}\right)^{\frac{1}{\alpha}}\right)\leq \left(\frac{f(x)^\alpha +f(y)^\alpha }{2}\right)^{\frac{1}{\alpha}}. \]
Is there such a function if instead the above relation holds for every ${x,y \in (0,\infty)}$ and for every irrational ${\alpha}?$
[i]Proposed by Maksa Gyula and Zsolt Páles[/i]
2019 Online Math Open Problems, 12
Let $F(n)$ denote the smallest positive integer greater than $n$ whose sum of digits is equal to the sum of the digits of $n$. For example, $F(2019) = 2028$. Compute $F(1) + F(2) + \dots + F(1000).$
[i]Proposed by Sean Li[/i]
1999 Singapore Team Selection Test, 1
Find all integers $m$ for which the equation $$x^3 - mx^2 + mx - (m^2 + 1) = 0$$ has an integer solution.
2007 Belarusian National Olympiad, 1
Find all polynomials with degree $\leq n$ and nonnegative coefficients, such that $P(x)P(\frac{1}{x}) \leq P(1)^2$ for every positive $x$
2025 Romania National Olympiad, 2
Let $f \colon [0,1] \to \mathbb{R} $ be a differentiable function such that its derivative is an integrable function on $[0,1]$, and $f(1)=0$. Prove that \[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]
2024 Korea - Final Round, P5
A positive integer $n (\ge 4)$ is given. Let $a_1, a_2, \cdots ,a_n$ be $n$ pairwise distinct positive integers where $a_i \le n$ for all $1 \le i \le n$. Determine the maximum value of
$$\sum_{i=1}^{n}{|a_i - a_{i+1} + a_{i+2} - a_{i+3}|}$$
where all indices are modulo $n$