Found problems: 85335
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]
2024 New Zealand MO, 5
Determine the least real number $L$ such that $$\dfrac{1}{a}+\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}\leqslant L$$ for all quadruples $(a,b,c,d)$ of integers satisfying $1<a<b<c<d$.