Found problems: 85335
2022 Mexican Girls' Contest, 1
Let $ABCD$ be a quadrilateral, $E$ the midpoint of side $BC$, and $F$ the midpoint of side $AD$. Segment $AC$ intersects segment $BF$ at $M$ and segment $DE$ at $N$. If quadrilateral $MENF$ is also known to be a parallelogram, prove that $ABCD$ is also a parallelogram.
1995 All-Russian Olympiad, 6
In an acute-angled triangle ABC, points $A_2$, $B_2$, $C_2$ are the midpoints of the altitudes $AA_1$, $BB_1$, $CC_1$, respectively. Compute the sum of angles $B_2A_1C_2$, $C_2B_1A_2$ and $A_2C_1B_2$.
[i]D. Tereshin[/i]
2018 BMT Spring, 1
Bob has $3$ different fountain pens and $11$ different ink colors. How many ways can he fill his fountain pens with ink if he can only put one ink in each pen?
2019 HMNT, 8
Compute the number of ordered pairs of integers $(x,y)$ such that $x^2 + y^2 < 2019$ and
$$x^2 + min(x,y) = y^2 + max(x, y) .$$
2002 Hungary-Israel Binational, 1
Find the greatest exponent $k$ for which $2001^{k}$ divides $2000^{2001^{2002}}+2002^{2001^{2000}}$.
2012 Argentina National Olympiad, 1
Determine if there are triplets ($x,y,z)$ of real numbers such that
$$\begin{cases} x+y+z=7 \\ xy+yz+zx=11\end{cases}$$
If the answer is affirmative, find the minimum and maximum values of $z$ in such a triplet.
2008 Czech and Slovak Olympiad III A, 1
Find all pairs of real numbers $(x,y)$ satisfying:
\[x+y^2=y^3,\]\[y+x^2=x^3.\]
2022 Yasinsky Geometry Olympiad, 3
In an isosceles right triangle $ABC$ with a right angle $C$, point $M$ is the midpoint of leg $AC$. At the perpendicular bisector of $AC$, point $D$ was chosen such that $\angle CDM = 30^o$, and $D$ and $B$ lie on different sides of $AC$. Find the angle $\angle ABD$.
(Volodymyr Petruk)
1998 Vietnam Team Selection Test, 1
Let $f(x)$ be a real function such that for each positive real $c$ there exist a polynomial $P(x)$ (maybe dependent on $c$) such that $| f(x) - P(x)| \leq c \cdot x^{1998}$ for all real $x$. Prove that $f$ is a real polynomial.
2023 IFYM, Sozopol, 7
The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at points $A_1$, $B_1$, and $C_1$. The line through the midpoints of segments $AB_1$ and $AC_1$ intersects the tangent at $A$ to the circumcircle of triangle $ABC$ at point $A_2$. Points $B_2$ and $C_2$ are defined similarly. Prove that points $A_2$, $B_2$, and $C_2$ lie on a line.
2003 All-Russian Olympiad, 4
Find the greatest natural number $N$ such that, for any arrangement of the numbers $1, 2, \ldots, 400$ in a chessboard $20 \times 20$, there exist two numbers in the same row or column, which differ by at least $N.$
2019 CCA Math Bonanza, L1.3
Points $P$ and $Q$ are chosen on diagonal $AC$ of square $ABCD$ such that $AB=AP=CQ=1$. What is the measure of $\angle{PBQ}$ in degrees?
[i]2019 CCA Math Bonanza Lightning Round #1.3[/i]
2021 AMC 10 Fall, 8
The largest prime factor of $16384$ is $2$, because $16384 = 2^{14}$. What is the sum of the digits of the largest prime factor of $16383$?
$\textbf{(A) }3\qquad\textbf{(B) }7\qquad\textbf{(C) }10\qquad\textbf{(D) }16\qquad\textbf{(E) }22$
2024 Moldova EGMO TST, 7
$ \frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+...+\sqrt{10+\sqrt{99}}}{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+...+\sqrt{10-\sqrt{99}}}=? $
1935 Moscow Mathematical Olympiad, 007
Find four consecutive terms $a, b, c, d$ of an arithmetic progression and four consecutive terms $a_1, b_1, c_1, d_1$ of a geometric progression such that $$\begin{cases}a + a_1 = 27 \\\ b + b_1 = 27 \\ c + c_1 = 39 \\ d + d_1 = 87\end{cases}$$.
2020 Purple Comet Problems, 16
Find the maximum possible value of $$\left( \frac{a^3}{b^2c}+\frac{b^3}{c^2a}+\frac{c^3}{a^2b} \right)^2$$ where $a, b$, and $c$ are nonzero real numbers satisfying $$a \sqrt[3]{\frac{a}{b}}+b\sqrt[3]{\frac{b}{c}}+c\sqrt[3]{\frac{c}{a}}=0$$
1994 All-Russian Olympiad Regional Round, 11.3
A circle with center $O$ is tangent to the sides $AB$, $BC$, $AC$ of a triangle $ABC$ at points $E,F,D$ respectively. The lines $AO$ and $CO$ meet $EF$ at points $N$ and $M$. Prove that the circumcircle of triangle $OMN$ and points $O$ and $D$ lie on a line.
2010 District Olympiad, 4
Let $ f: [0,1]\rightarrow \mathbb{R}$ a derivable function such that $ f(0)\equal{}f(1)$, $ \int_0^1f(x)dx\equal{}0$ and $ f^{\prime}(x) \neq 1\ ,\ (\forall)x\in [0,1]$.
i)Prove that the function $ g: [0,1]\rightarrow \mathbb{R}\ ,\ g(x)\equal{}f(x)\minus{}x$ is strictly decreasing.
ii)Prove that for each integer number $ n\ge 1$, we have:
$ \left|\sum_{k\equal{}0}^{n\minus{}1}f\left(\frac{k}{n}\right)\right|<\frac{1}{2}$
2004 National Olympiad First Round, 3
At most how many elements does a set have such that all elements are less than $102$ and it doesn't contain the sum of any two elements?
$
\textbf{(A)}\ 49
\qquad\textbf{(B)}\ 50
\qquad\textbf{(C)}\ 51
\qquad\textbf{(D)}\ 54
\qquad\textbf{(E)}\ 62
$
1969 IMO Shortlist, 53
$(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$
2018 Math Prize for Girls Problems, 8
A mustache is created by taking the set of points $(x, y)$ in the $xy$-coordinate plane that satisfy $4 + 4 \cos(\pi x/24) \le y \le 6 + 6\cos(\pi x/24)$ and $-24 \le x \le 24$. What is the area of the mustache?
2011 LMT, 1
A positive integer is randomly selected from among the first $2011$ primes. What is the probability that it is even?
2010 USA Team Selection Test, 6
Let $T$ be a finite set of positive integers greater than 1. A subset $S$ of $T$ is called [i]good[/i] if for every $t \in T$ there exists some $s \in S$ with $\gcd(s, t) > 1$. Prove that the number of good subsets of $T$ is odd.
2025 Kyiv City MO Round 2, Problem 1
Mykhailo drew a triangular grid with side \( n \) for \( n \geq 2 \). It is formed from an equilateral triangle \( T \) with side length \( n \), by dividing each side into \( n \) equal parts. Then lines are drawn parallel to the sides of triangle \( T \), dividing it into \( n^2 \) equilateral triangles with side length \( 1 \), which we will call \textbf{cells}.
Next, Oleksii writes some positive integer into each cell. Mykhailo receives 1 candy for each cell, where the number written is equal to the sum of all the numbers in the adjacent cells. Oleksii wants to arrange the numbers in such a way that Mykhailo receives the maximum number of candies. How many candies can Mykhailo receive under such conditions?
In the figure below, an example is shown for \( n = 4 \) with 16 cells and numbers written inside them. For the numbers arranged as in the figure, Mykhailo receives 5 candies for the numbers \( 2 \) (the topmost cell), \( 8 \), \( 13 \), \( 12 \), and \( 11 \).
[img]https://i.ibb.co/LrLks9q/Kyiv-MO-2025-R2-7-1.png[/img]
[i]Proposed by Mykhailo Shtandenko[/i]
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.