Found problems: 85335
2022 CMIMC, 2.3 1.1
How many 4-digit numbers have exactly $9$ divisors from the set $\{1,2,3,4,5,6,7,8,9,10\}$?
[i]Proposed by Ethan Gu[/i]
2017 Putnam, A3
Let $a$ and $b$ be real numbers with $a<b,$ and let $f$ and $g$ be continuous functions from $[a,b]$ to $(0,\infty)$ such that $\int_a^b f(x)\,dx=\int_a^b g(x)\,dx$ but $f\ne g.$ For every positive integer $n,$ define
\[I_n=\int_a^b\frac{(f(x))^{n+1}}{(g(x))^n}\,dx.\]
Show that $I_1,I_2,I_3,\dots$ is an increasing sequence with $\displaystyle\lim_{n\to\infty}I_n=\infty.$
1991 USAMO, 4
Let $a = \frac{m^{m+1} + n^{n+1}}{m^m + n^n}$, where $m$ and $n$ are positive integers. Prove that $a^m + a^n \geq m^m + n^n$.
2013 Tournament of Towns, 1
There are six points on the plane such that one can split them into two triples each creating a triangle. Is it always possible to split these points into two triples creating two triangles with no common point (neither inside, nor on the boundary)?
2011 IMAR Test, 4
Given an integer number $n \ge 3$, show that the number of lists of jointly coprime positive integer numbers that sum to $n$ is divisible by $3$.
(For instance, if $n = 4$, there are six such lists: $(3, 1), (1, 3), (2, 1, 1), (1, 2, 1), (1, 1, 2)$ and $(1, 1, 1, 1)$.)
1998 IMC, 2
$S$ ist the set of all cubic polynomials $f$ with $|f(\pm 1)| \leq 1$ and $|f(\pm \frac{1}{2})| \leq 1$. Find $\sup_{f \in S} \max_{-1 \leq x \leq 1} |f''(x)|$ and all members of $f$ which give equality.
2014 IMO Shortlist, C1
Let $n$ points be given inside a rectangle $R$ such that no two of them lie on a line parallel to one of the sides of $R$. The rectangle $R$ is to be dissected into smaller rectangles with sides parallel to the sides of $R$ in such a way that none of these rectangles contains any of the given points in its interior. Prove that we have to dissect $R$ into at least $n + 1$ smaller rectangles.
[i]Proposed by Serbia[/i]
1999 Switzerland Team Selection Test, 9
Suppose that $P(x)$ is a polynomial with degree $10$ and integer coefficients.
Prove that, there is an infinite arithmetic progression (open to bothside) not contain value of $P(k)$ with $k\in\mathbb{Z}$
2014 Contests, 1
Let $k$ be a given circle and $A$ is a fixed point outside $k$. $BC$ is a diameter of $k$. Find the locus of the orthocentre of $\triangle ABC$ when $BC$ varies.
[i]Proposed by T. Vitanov, E. Kolev[/i]
2022 Kyiv City MO Round 1, Problem 3
Let $H$ and $O$ be the orthocenter and the circumcenter of the triangle $ABC$. Line $OH$ intersects the sides $AB, AC$ at points $X, Y$ correspondingly, so that $H$ belongs to the segment $OX$. It turned out that $XH = HO = OY$. Find $\angle BAC$.
[i](Proposed by Oleksii Masalitin)[/i]
2022/2023 Tournament of Towns, P7
Chameleons of five colors live on the island. When one chameleon bites another, the color of bitten chameleon changes to one of these five colors according to some rule, and the new color depends only on the color of the bitten and the color of the bitting. It is known that $2023$ red chameleons can agree on a sequence of bites between
themselves, after which they will all turn blue.
What is the smallest $k$ that can guarantee that $k$ red chameleons, biting only each other, can turn blue?
(For example, the rules might be: if a red chameleon bites a green one, the bitten one changes color to blue; if a green one bites a red one, the bitten one remains red, that is, "changes color to red"; if red bites red, the bitten one changes color to yellow, etc. The rules for changing colors may be different.)
2020 Online Math Open Problems, 4
An alien from the planet OMO Centauri writes the first ten prime numbers in arbitrary order as U, W, XW, ZZ, V, Y, ZV, ZW, ZY, and X. Each letter represents a nonzero digit. Each letter represents the same digit everywhere it appears, and different letters represent different digits. Also, the alien is using a base other than base ten. The alien writes another number as UZWX. Compute this number (expressed in base ten, with the usual, human digits).
[i]Proposed by Luke Robitaille & Eric Shen[/i]
2016 Dutch IMO TST, 1
Let $\triangle ABC$ be a acute triangle. Let $H$ the foot of the C-altitude in $AB$ such that $AH=3BH$, let $M$ and $N$ the midpoints of $AB$ and $AC$ and let $P$ be a point such that $NP=NC$ and $CP=CB$ and $B$, $P$ are located on different sides of the line $AC$. Prove that $\measuredangle APM=\measuredangle PBA$.
2023 Ukraine National Mathematical Olympiad, 11.5
Let's call a polynomial [i]mixed[/i] if it has both positive and negative coefficients ($0$ isn't considered positive or negative). Is the product of two mixed polynomials always mixed?
[i]Proposed by Vadym Koval[/i]
2006 Iran MO (3rd Round), 1
Let $A$ be a family of subsets of $\{1,2,\ldots,n\}$ such that no member of $A$ is contained in another. Sperner’s Theorem states that $|A|\leq{n\choose{\lfloor\frac{n}{2}\rfloor}}$. Find all the families for which the equality holds.
2024 Ukraine National Mathematical Olympiad, Problem 7
Find all polynomials $P(x)$ with integer coefficients, such that for any positive integer $n$ number $P(n)$ is a positive integer and a divisor of $n!$.
[i]Proposed by Mykyta Kharin[/i]
2014 ELMO Shortlist, 7
Let $ABC$ be a triangle inscribed in circle $\omega$ with center $O$, let $\omega_A$ be its $A$-mixtilinear incircle, $\omega_B$ be its $B$-mixtilinear incircle, $\omega_C$ be its $C$-mixtilinear incircle, and $X$ be the radical center of $\omega_A$, $\omega_B$, $\omega_C$. Let $A'$, $B'$, $C'$ be the points at which $\omega_A$, $\omega_B$, $\omega_C$ are tangent to $\omega$. Prove that $AA'$, $BB'$, $CC'$ and $OX$ are concurrent.
[i]Proposed by Robin Park[/i]
2021 Novosibirsk Oral Olympiad in Geometry, 4
It is known about two triangles that for each of them the sum of the lengths of any two of its sides is equal to the sum of the lengths of any two sides of the other triangle. Are triangles necessarily congruent?
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?
Kyiv City MO Seniors Round2 2010+ geometry, 2018.11.2
In the quadrilateral $ABCD $, $AB = BC $, the point $K $ is the midpoint of the side $CD $, the rays $BK $ and $AD $ intersect at the point $M $ , the circumscribed circle $ \Delta ABM $ intersects the line $AC $ for the second time at the point $P $. Prove that $\angle BKP = 90 {} ^ \circ $.
(Anton Trygub)
1995 National High School Mathematics League, 8
Consider the maximum value of circular cone inscribed to a sphere, the ratio of it to the volume of the sphere is________.
2021 Vietnam TST, 6
Let $n \geq 3$ be a positive integers and $p$ be a prime number such that $p > 6^{n-1} - 2^n + 1$. Let $S$ be the set of $n$ positive integers with different residues modulo $p$. Show that there exists a positive integer $c$ such that there are exactly two ordered triples $(x,y,z) \in S^3$ with distinct elements, such that $x-y+z-c$ is divisible by $p$.
2008 Paraguay Mathematical Olympiad, 5
Let $m,n,p$ be rational numbers such that $\sqrt{m} + \sqrt{n} + \sqrt{p}$ is a rational number. Prove that $\sqrt{m}, \sqrt{n}, \sqrt{p}$ are also rational numbers
2000 Estonia National Olympiad, 2
Let $PQRS$ be a cyclic quadrilateral with $\angle PSR = 90^o$, and let $H,K$ be the projections of $Q$ on the lines $PR$ and $PS$, respectively. Prove that the line $HK$ passes through the midpoint of the segment $SQ$.
2008 Hanoi Open Mathematics Competitions, 6
Let $P(x)$ be a polynomial such that $P(x^2 - 1) = x^4 - 3x^2 + 3$.
Find $P(x^2 + 1)$.