Found problems: 85335
2003 India Regional Mathematical Olympiad, 3
Let $a,b,c$ be three positive real numbers such that $a + b +c =1$ . prove that among the three numbers $a-ab, b - bc, c-ca$ there is one which is at most $\frac{1}{4}$ and there is one which is at least $\frac{2}{9}$.
1951 Moscow Mathematical Olympiad, 194
One side of a convex polygon is equal to $a$, the sum of exterior angles at the vertices not adjacent to this side are equal to $120^o$. Among such polygons, find the polygon of the largest area.
2021 Turkey MO (2nd round), 2
If a polynomial with real coefficients of degree $d$ has at least $d$ coefficients equal to $1$ and has $d$ real roots, what is the maximum possible value of $d$?
(Note: The roots of the polynomial do not have to be different from each other.)
PEN D Problems, 21
Determine the last three digits of \[2003^{2002^{2001}}.\]
LMT Speed Rounds, 2016.9
An acute triangle has area $84$ and perimeter $42$, with each side being at least $10$ units long. Let $S$ be the set of points that are within $5$ units of some vertex of the triangle. What fraction of the area of $S$ lies outside the triangle?
[i]Proposed by Nathan Ramesh
1995 IberoAmerican, 3
Let $ r$ and $ s$ two orthogonal lines that does not lay on the same plane. Let $ AB$ be their common perpendicular, where $ A\in{}r$ and $ B\in{}s$(*).Consider the sphere of diameter $ AB$. The points $ M\in{r}$ and $ N\in{s}$ varies with the condition that $ MN$ is tangent to the sphere on the point $ T$. Find the locus of $ T$.
Note: The plane that contains $ B$ and $ r$ is perpendicular to $ s$.
2023 IMC, 7
Let $V$ be the set of all continuous functions $f\colon [0,1]\to \mathbb{R}$, differentiable on $(0,1)$, with the property that $f(0)=0$ and $f(1)=1$. Determine all $\alpha \in \mathbb{R}$ such that for every $f\in V$, there exists some $\xi \in (0,1)$ such that
\[f(\xi)+\alpha = f'(\xi)\]
1989 Bulgaria National Olympiad, Problem 6
Let $x,y,z$ be pairwise coprime positive integers and $p\ge5$ and $q$ be prime numbers which satisfy the following conditions:
(i) $6p$ does not divide $q-1$;
(ii) $q$ divides $x^2+xy+y^2$;
(iii) $q$ does not divide $x+y-z$.
Prove that $x^p+y^p\ne z^p$.
2024 Princeton University Math Competition, 15
There are $10$ teams, named $T_1$ through $T_{10},$ participating in a draft in which there are $20$ players available, named $P_1$ through $P_{20}.$ Suppose each team independent of the others has uniform random preference on the $20$ players. Team $T_1$ will draft their favorite player, and then each subsequent team $T_2, \ldots , T_{10}$ draft their favorite player among the ones not already drafted. Each team drafts exactly one player. Given that $P_1$ is among the $10$ favorite players for each team, the probability that $P_1$ is drafted can be written as $\tfrac{m}{n}$ where $m$ and $n$ are coprime positive integers. Find $m + n.$
2018 AMC 8, 1
An amusement park has a collection of scale models, with ratio $1 : 20$, of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its replica to the nearest whole number?
$\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }18\qquad\textbf{(E) }20$
2011 AMC 8, 21
Students guess that Norb's age is $24, 28, 30, 32, 36, 38, 41, 44, 47$, and $49$. Norb says, "At least half of you guessed too low, two of you are off by one, and my age is a prime number." How old is Norb?
$ \textbf{(A)}29\qquad\textbf{(B)}31\qquad\textbf{(C)}37\qquad\textbf{(D)}43\qquad\textbf{(E)}48 $
2023 HMNT, 26
Compute the smallest multiple of $63$ with an odd number of ones in its base two representation.
2000 AIME Problems, 11
The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107).$ The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum o f the absolute values of all possible slopes for $\overline{AB}$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
1994 Vietnam National Olympiad, 2
$S$ is a sphere center $O. G$ and $G'$ are two perpendicular great circles on $S$. Take $A, B, C$ on $G$ and $D$ on $G'$ such that the altitudes of the tetrahedron $ABCD$ intersect at a point. Find the locus of the intersection.
Today's calculation of integrals, 869
Let $I_n=\frac{1}{n+1}\int_0^{\pi} x(\sin nx+n\pi\cos nx)dx\ \ (n=1,\ 2,\ \cdots).$
Answer the questions below.
(1) Find $I_n.$
(2) Find $\sum_{n=1}^{\infty} I_n.$
1992 Tournament Of Towns, (329) 6
A circle is divided into $n$ sectors. Pawns stand on some of the sectors; the total number of pawns equals $n + 1$. This configuration is changed as follows. Any two of the pawns standing on the same sector move simultaneously to the neighbouring sectors in different directions. Prove that after several such transformations a configuration in which no less than half of the sectors are occupied by pawns, will inevitably appear.
(D. Fomin, St Petersburg)
2008 Junior Balkan Team Selection Tests - Moldova, 8
Archipelago consists of $ n$ islands : $ I_1,I_2,...,I_n$ and $ a_1,a_2,...,a_n$ - number of the roads on each island. $ a_1 \equal{} 55$, $ a_k \equal{} a_{k \minus{} 1} \plus{} (k \minus{} 1)$, ($ k \equal{} 2,3,...,n$)
a) Does there exist an island with 2008 roads?
b) Calculate $ a_1 \plus{} a_2 \plus{} ... \plus{} a_n.$
2019 Online Math Open Problems, 4
Compute $\left\lceil\displaystyle\sum_{k=2018}^{\infty}\frac{2019!-2018!}{k!}\right\rceil$. (The notation $\left\lceil x\right\rceil$ denotes the least integer $n$ such that $n\geq x$.)
[i]Proposed by Tristan Shin[/i]
2006 IMO Shortlist, 1
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula
\[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large.
[i]Proposed by Harmel Nestra, Estionia[/i]
2008 Hanoi Open Mathematics Competitions, 8
Consider a convex quadrilateral $ABCD$. Let $O$ be the intersection of $AC$ and $BD$; $M, N$ be the centroid of $\Delta AOB$ and $\Delta COD$ and $P, Q$ be orthocenter of $\Delta BOC$ and $\Delta DOA$, respectively.
Prove that $MN\bot PQ$.
2013 AMC 8, 1
Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way?
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
2017 Princeton University Math Competition, A7
Let $ACDB$ be a cyclic quadrilateral with circumcenter $\omega$. Let $AC=5$, $CD=6$, and $DB=7$. Suppose that there exists a unique point $P$ on $\omega$ such that $\overline{PC}$ intersects $\overline{AB}$ at a point $P_1$ and $\overline{PD}$ intersects $\overline{AB}$ at a point $P_2$, such that $AP_1=3$ and $P_2B=4$. Let $Q$ be the unique point on $\omega$ such that $\overline{QC}$ intersects $\overline{AB}$ at a point $Q_1$, $\overline{QD}$ intersects $\overline{AB}$ at a point $Q_2$, $Q_1$ is closer to $B$ than $P_1$ is to $B$, and $P_2Q_2=2$. The length of $P_1Q_1$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2021 Oral Moscow Geometry Olympiad, 5
Let $ABC$ be a triangle, $I$ and $O$ be its incenter and circumcenter respectively. $A'$ is symmetric to $O$ with respect to line $AI$. Points $B'$ and $C'$ are defined similarly. Prove that the nine-point centers of triangles $ABC$ and $A'B'C'$ coincide.
2009 Math Prize For Girls Problems, 12
Jenny places 100 pennies on a table, 30 showing heads and 70 showing tails. She chooses 40 of the pennies at random (all different) and turns them over. That is, if a chosen penny was showing heads, she turns it to show tails; if a chosen penny was showing tails, she turns it to show heads. At the end, what is the expected number of pennies showing heads?
2016 Israel National Olympiad, 2
We are given a cone with height 6, whose base is a circle with radius $\sqrt{2}$. Inside the cone, there is an inscribed cube: Its bottom face on the base of the cone, and all of its top vertices lie on the cone. What is the length of the cube's edge?
[img]https://i.imgur.com/AHqHHP6.png[/img]