Found problems: 85335
2014 National Olympiad First Round, 24
If the integers $1,2,\dots,n$ can be divided into two sets such that each of the two sets does not contain the arithmetic mean of its any two elements, what is the largest possible value of $n$?
$
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 8
\qquad\textbf{(C)}\ 9
\qquad\textbf{(D)}\ 10
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2022 Mexican Girls' Contest, 5
A biologist found a pond with frogs. When classifying them by their mass, he noticed the following:
[i]The $50$ lightest frogs represented $30\%$ of the total mass of all the frogs in the pond, while the $44$ heaviest frogs represented $27\%$ of the total mass.
[/i]As fate would have it, the frogs escaped and the biologist only has the above information. How many frogs were in the pond?
2018 Hanoi Open Mathematics Competitions, 9
Let $ABC$ be acute, non-isosceles triangle, inscribed in the circle $(O)$. Let $D$ be perpendicular projection of $A$ onto $BC$, and $E, F$ be perpendicular projections of $D$ onto $CA,AB$ respectively.
(a) Prove that $AO \perp EF$.
(b) The line $AO$ intersects $DE,DF$ at $I,J$ respectively. Prove that $\vartriangle DIJ$ and $\vartriangle ABC$ are similar.
(c) Prove that circumcenter of $\vartriangle DIJ$ is equidistant from $B$ and $C$
2014 NIMO Problems, 2
Let $0^{\circ}\leq\alpha,\beta,\gamma\leq90^{\circ}$ be angles such that \[\sin\alpha-\cos\beta=\tan\gamma\] \[\sin\beta-\cos\alpha=\cot\gamma\]
Compute the sum of all possible values of $\gamma$ in degrees.
[i]Proposed by Michael Ren[/i]
1986 China Team Selection Test, 2
Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent:
[b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$
[b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.
2023 Israel TST, P3
In triangle $ABC$ the orthocenter is $H$ and the foot of the altitude from $A$ is $D$. Point $P$ satisfies $AP=HP$, and the line $PA$ is tangent to $(ABC)$. Line $PD$ intersects lines $AB, AC$ at points $X,Y$ respectively.
Prove that $\angle YHX = \angle BAC$ or $\angle YHX+\angle BAC= 180^\circ$.
1984 IMO Longlists, 40
Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.
2024 Romania EGMO TST, P4
Find all composite positive integers $a{}$ for which there exists a positive integer $b\geqslant a$ with the same number of divisors as $a{}$ with the following property: if $a_1<\cdots<a_n$ and $b_1<\cdots<b_n$ are the proper divisors of $a{}$ and $b{}$ respectively, then $a_i+b_i, 1\leqslant i\leqslant n$ are the proper divisors of some positive integer $c.{}$
2021 Durer Math Competition Finals, 1
Given a right angled triangle $ABC$ in which $\angle ACB = 90^o$. Let $D$ be an inner point of $AB$, and let $E$ be an inner point of $AC$. It is known that $\angle ADE = 90^o$, and that the length of the segment $AD$ is $8$, the length of the segment $DE$ is $15$, and the length of segment $CE$ is $3$. What is the area of triangle $ABC$?
1999 Mongolian Mathematical Olympiad, Problem 1
Suppose that a function $f:\mathbb R\to\mathbb R$ is such that for any real $h$ there exist at most $19990509$ different values of $x$ for which $f(x)\ne f(x+h)$. Prove that there is a set of at most $9995256$ real numbers such that $f$ is constant outside of this set.
2009 Korea Junior Math Olympiad, 2
In an acute triangle $\triangle ABC$, let $A',B',C'$ be the reflection of $A,B,C$ with respect to $BC,CA,AB$. Let $D = B'C \cap BC'$, $E = CA' \cap C'A$, $F = A'B \cap AB'$. Prove that $AD,BE,CF$ are concurrent
2018 May Olympiad, 2
A thousand integer divisions are made: $2018$ is divided by each of the integers from $ 1$ to $1000$. Thus, a thousand integer quotients are obtained with their respective remainders. Which of these thousand remainders is the bigger?
2019 Turkey MO (2nd round), 1
$a, b, c$ are positive real numbers such that $$(\sqrt {ab}-1)(\sqrt {bc}-1)(\sqrt {ca}-1)=1$$
At most, how many of the numbers: $$a-\frac {b}{c}, a-\frac {c}{b}, b-\frac {a}{c}, b-\frac {c}{a}, c-\frac {a}{b}, c-\frac {b}{a}$$ can be bigger than $1$?
2001 Paraguay Mathematical Olympiad, 1
In a warehouse there are many empty cans of $4$ colors: red, green, Blue and yellow. Some boys play to build towers in which no two cans of the same color, with a can in each floor and at any height. How many different towers can be built?
2016 Grand Duchy of Lithuania, 1
Let $a, b$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove that
$$\frac{a}{a+b^2}+\frac{b}{b+c^2}+\frac{c}{c+a^2} \le \frac{1}{4} \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)$$
1992 Austrian-Polish Competition, 5
Given a circle $k$ with center $M$ and radius $r$, let $AB$ be a fixed diameter of $k$ and let $K$ be a fixed point on the segment $AM$. Denote by $t$ the tangent of $k$ at $A$. For any chord $CD$ through $K$ other than $AB$, denote by $P$ and Q the intersection points of $BC$ and $BD$ with $t$, respectively. Prove that $AP\cdot AQ$ does not depend on $CD$.
1977 Canada National Olympiad, 4
Let
\[p(x) = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_1 x + a_0\]
and
\[q(x) = b_m x^m + a_{m - 1} x^{m - 1} + \dots + b_1 x + b_0\]
be two polynomials with integer coefficients. Suppose that all the coefficients of the product $p(x) \cdot q(x)$ are even but not all of them are divisible by 4. Show that one of $p(x)$ and $q(x)$ has all even coefficients and the other has at least one odd coefficient.
2021 Princeton University Math Competition, 4
Abby and Ben have a little brother Carl who wants candy. Abby has $7$ different pieces of candy and Ben has $15$ different pieces of candy. Abby and Ben then decide to give Carl some candy. As Ben wants to be a better sibling than Abby, so he decides to give two more pieces of candy to Carl than Abby does. Let $N$ be the number of ways Abby and Ben can give Carl candy. Compute the number of positive divisors of $N$.
1991 IberoAmerican, 5
Let $P(x,\, y)=2x^{2}-6xy+5y^{2}$. Let us say an integer number $a$ is a value of $P$ if there exist integer numbers $b$, $c$ such that $P(b,\, c)=a$.
a) Find all values of $P$ lying between 1 and 100.
b) Show that if $r$ and $s$ are values of $P$, then so is $rs$.
1990 AMC 12/AHSME, 17
How many of the numbers, $100,101,\ldots,999$, have three different digits in increasing order or in decreasing order?
$\text{(A)} \ 120 \qquad \text{(B)} \ 168 \qquad \text{(C)} \ 204 \qquad \text{(D)} \ 216 \qquad \text{(E)} \ 240$
2007 Peru IMO TST, 3
Let $T$ a set with 2007 points on the plane, without any 3 collinear points.
Let $P$ any point which belongs to $T$.
Prove that the number of triangles that contains the point $P$ inside and
its vertices are from $T$, is even.
1988 IMO Longlists, 22
In a triangle $ ABC,$ choose any points $ K \in BC, L \in AC, M \in AB, N \in LM, R \in MK$ and $ F \in KL.$ If $ E_1, E_2, E_3, E_4, E_5, E_6$ and $ E$ denote the areas of the triangles $ AMR, CKR, BKF, ALF, BNM, CLN$ and $ ABC$ respectively, show that
\[ E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}.
\]
1986 Bundeswettbewerb Mathematik, 1
There are $n$ points on a circle ($n > 1$). Denote them with $P_1,P_2, P_3, ..., P_n$ such that the polyline $P_1P_2P_3... P_n$ does not intersect itself. In how many ways is this possible?
Kyiv City MO 1984-93 - geometry, 1991.9.3
The point $M$ is the midpoint of the median $BD$ of the triangle $ABC$, the area of which is $S$. The line $AM$ intersects the side $BC$ at the point $K$. Determine the area of the triangle $BKM$.
2019 Belarusian National Olympiad, 10.8
Call a polygon on a Cartesian plane to be[i]integer[/i] if all its vertices are integer. A convex integer $14$-gon is cut into integer parallelograms with areas not greater than $C$.
Find the minimal possible $C$.
[i](A. Yuran)[/i]