Found problems: 85335
2016 IMO Shortlist, N7
Let $P=A_1A_2\cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by $n$. Prove that $2S$ is an integer divisible by $n$.
2013 Princeton University Math Competition, 3
How many tuples of integers $(a_0,a_1,a_2,a_3,a_4)$ are there, with $1\leq a_i\leq 5$ for each $i$, so that $a_0<a_1>a_2<a_3>a_4$?
May Olympiad L2 - geometry, 2018.5
Each point on a circle is colored with one of $10$ colors. Is it true that for any coloring there are $4$ points of the same color that are vertices of a quadrilateral with two parallel sides (an isosceles trapezoid or a rectangle)?
2013 Purple Comet Problems, 7
Find the least six-digit palindrome that is a multiple of $45$. Note that a palindrome is a number that reads the same forward and backwards such as $1441$ or $35253$.
1940 Putnam, A7
If $\sum_{i=1}^{\infty} u_{i}^{2}$ and $\sum_{i=1}^{\infty} v_{i}^{2}$ are convergent series of real numbers, prove that
$$\sum_{i=1}^{\infty}(u_{i}-v_{i})^{p}$$
is convergent, where $p\geq 2$ is an integer.
2000 Croatia National Olympiad, Problem 4
If $n\ge2$ is an integer, prove the equality
$$\lfloor\log_2n\rfloor+\lfloor\log_3n\rfloor+\ldots+\lfloor\log_nn\rfloor=\left\lfloor\sqrt n\right\rfloor+\left\lfloor\sqrt[3]n\right\rfloor+\ldots+\left\lfloor\sqrt[n]n\right\rfloor.$$
2002 JBMO ShortLists, 5
Let $ a,b,c$ be positive real numbers. Prove the inequality:
$ \frac {a^3}{b^2} \plus{} \frac {b^3}{c^2} \plus{} \frac {c^3}{a^2}\ge \frac {a^2}{b} \plus{} \frac {b^2}{c} \plus{} \frac {c^2}{a}$
2014 Saudi Arabia IMO TST, 1
Let $a_1,\dots,a_n$ be a non increasing sequence of positive real numbers. Prove that \[\sqrt{a_1^2+a_2^2+\cdots+a_n^2}\le a_1+\frac{a_2}{\sqrt{2}+1}+\cdots+\frac{a_n}{\sqrt{n}+\sqrt{n-1}}.\] When does equality hold?
2011 239 Open Mathematical Olympiad, 5
Prove that there exist 1000 consecutive numbers such that none of them is divisible by its sum of the digits
2008 Indonesia TST, 2
Let $S = \{1, 2, 3, ..., 100\}$ and $P$ is the collection of all subset $T$ of $S$ that have $49$ elements, or in other words: $$P = \{T \subset S : |T| = 49\}.$$ Every element of $P$ is labelled by the element of $S$ randomly (the labels may be the same). Show that there exist subset $M$ of $S$ that has $50$ members such that for every $x \in M$, the label of $M -\{x\}$ is not equal to $x$
1999 May Olympiad, 1
A three-digit natural number is called [i]tricubic [/i] if it is equal to the sum of the cubes of its digits.
Find all pairs of consecutive numbers such that both are tricubic.
2016 Latvia Baltic Way TST, 9
The numbers from$ 1$ to $2016$ are divided into three (disjoint) subsets $A, B$ and $C$, each one contains exactly $672$ numbers. Prove that you can find three numbers, each from a different subset, such that the sum of two of them is equal to the third.
[hide=original wording]Skaitļi no 1 līdz 2016 ir sadalīti trīs (nešķeļošās) apakškopās A, B un C, katranotām satur tieši 672 skaitļus. Pierādīt, ka var atrast trīs tādus skaitļus, katru no citas apakškopas, ka divu no tiem summa ir vienāda ar trešo.
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2008 AMC 10, 9
A quadratic equation $ ax^2\minus{}2ax\plus{}b\equal{}0$ has two real solutions. What is the average of the solutions?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ \frac{b}{a} \qquad
\textbf{(D)}\ \frac{2b}{a} \qquad
\textbf{(E)}\ \sqrt{2b\minus{}a}$
MIPT student olimpiad autumn 2024, 4
The ellipsoid $E$ is contained in the simplex $S$, which is located in the unit ball
B space $R^n$. Prove that the sum of the principal semi-axes of the ellipsoid $E$ is no more than
units.
2002 Tuymaada Olympiad, 3
Is there a quadratic trinomial with integer coefficients, such that all values which are natural to be natural powers of two?
1964 Miklós Schweitzer, 9
Let $ E$ be the set of all real functions on $ I\equal{}[0,1]$. Prove that one cannot define a topology on $ E$ in which $ f_n\rightarrow f$ holds if and only if $ f_n$ converges to $ f$ almost everywhere.
2005 Kurschak Competition, 2
A and B play tennis. The player to first win at least four points and at least two more than the other player wins. We know that A gets a point each time with probability $p\le \frac12$, independent of the game so far. Prove that the probability that A wins is at most $2p^2$.
2009 National Olympiad First Round, 18
$ 1 \le n \le 455$ and $ n^3 \equiv 1 \pmod {455}$. The number of solutions is ?
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None}$
2021 AMC 10 Fall, 13
A square with side length $3$ is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length $2$ has two vertices on the other square and the other two on sides of the triangle, as shown. What is the area of the triangle?
[asy]
//diagram by kante314
draw((0,0)--(8,0)--(4,8)--cycle, linewidth(1.5));
draw((2,0)--(2,4)--(6,4)--(6,0)--cycle, linewidth(1.5));
draw((3,4)--(3,6)--(5,6)--(5,4)--cycle, linewidth(1.5));
[/asy]
$(\textbf{A})\: 19\frac14\qquad(\textbf{B}) \: 20\frac14\qquad(\textbf{C}) \: 21 \frac34\qquad(\textbf{D}) \: 22\frac12\qquad(\textbf{E}) \: 23\frac34$
2017 Harvard-MIT Mathematics Tournament, 9
The Fibonacci sequence is defined as follows: $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ for all integers $n\ge 2$. Find the smallest positive integer $m$ such that $F_m\equiv 0 \pmod {127}$ and $F_{m+1}\equiv 1\pmod {127}$.
2010 Switzerland - Final Round, 2
Let $ \triangle{ABC}$ be a triangle with $ AB\not\equal{}AC$. The incircle with centre $ I$ touches $ BC$, $ CA$, $ AB$ at $ D$, $ E$, $ F$, respectively. Furthermore let $ M$ the midpoint of $ EF$ and $ AD$ intersect the incircle at $ P\not\equal{}D$.
Show that $ PMID$ ist cyclic.
2023 AIME, 5
Let $P$ be a point on the circumcircle of square $ABCD$ such that $PA \cdot PC = 56$ and $PB \cdot PD = 90.$ What is the area of square $ABCD?$
2016 CCA Math Bonanza, T7
A [i]cuboctahedron[/i], shown below, is a polyhedron with 8 equilateral triangle faces and 6 square faces. Each edge has the same length and each of the 24 vertices borders 2 squares and 2 triangles. An \textit{octahedron} is a regular polyhedron with 6 vertices and 8 equilateral triangle faces. Compute the sum of the volumes of an octahedron with side length 5 and a cuboctahedron with side length 5.
[img]http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMi82LzBmNjM1OTM2M2ExYTQzOTFhODEwODkwM2FiYmM1MTljOGQzNmJhLmpwZw==&rn=Q3Vib2N0YWhlZHJvbi5qcGc=[/img]
[i]2016 CCA Math Bonanza Team #7[/i]
2009 HMNT, 1-3
[u]Down the In finite Corridor[/u]
Consider an isosceles triangle $T$ with base $10$ and height $12$. Defi ne a sequence $\omega_1$, $\omega_2$,$...$of circles such that $\omega_1$ is the incircle of $T$ and $\omega_{i+1}$ is tangent to $\omega_i$ and both legs of the isosceles triangle for $i > 1$.
[b]p1.[/b] Find the radius of $\omega_1$.
[b]p2.[/b] Find the ratio of the radius of $\omega_{i+1}$ to the radius of $\omega_i$.
[b]p3.[/b] Find the total area contained in all the circles.
1970 IMO Longlists, 5
Prove that $\sqrt[n]{\sum_{i=1}^{n}{\frac{i}{n+1}}}\ge 1$ for $2 \le n \in \mathbb{N}$.