Found problems: 85335
Ukrainian From Tasks to Tasks - geometry, 2014.15
Construct a right triangle given the hypotenuse and the median drawn to the leg.
2019 Belarus Team Selection Test, 5.1
A function $f:\mathbb N\to\mathbb N$, where $\mathbb N$ is the set of positive integers, satisfies the following condition: for any positive integers $m$ and $n$ ($m>n$) the number $f(m)-f(n)$ is divisible by $m-n$.
Is the function $f$ necessarily a polynomial? (In other words, is it true that for any such function there exists a polynomial $p(x)$ with real coefficients such that $f(n)=p(n)$ for all positive integers $n$?)
[i](Folklore)[/i]
2003 South africa National Olympiad, 3
The first four digits of a certain positive integer $n$ are $1137$. Prove that the digits of $n$ can be shuffled in such a way that the new number is divisible by 7.
2015 BMT Spring, 10
Evaluate
$$\int^{\pi/2}_0\ln(4\sin x)dx.$$
1991 IMTS, 4
Show that an arbitary triangle can be dissected by straight line segments into three parts in three different ways so that each part has a line of symmetry.
2022 Indonesia TST, N
Prove that there exists a set $X \subseteq \mathbb{N}$ which contains exactly 2022 elements such that for every distinct $a, b, c \in X$ the following equality:
\[ \gcd(a^n+b^n, c) = 1 \] is satisfied for every positive integer $n$.
2019 IMC, 10
$2019$ points are chosen at random, independently, and distributed uniformly in the unit disc $\{(x,y)\in\mathbb R^2: x^2+y^2\le 1\}$. Let $C$ be the convex hull of the chosen points. Which probability is larger: that $C$ is a polygon with three vertices, or a polygon with four vertices?
[i]Proposed by Fedor Petrov, St. Petersburg State University[/i]
2000 Switzerland Team Selection Test, 2
Real numbers $a_1,a_2,...,a_{16}$ satisfy the conditions $\sum_{i=1}^{16}a_i = 100$ and $\sum_{i=1}^{16}a_i^2 = 1000$ .
What is the greatest possible value of $a_16$?
Kvant 2022, M2690
Vasya has $n{}$ candies of several types, where $n>145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n{}$.
[i]Proposed by A. Antropov[/i]
2020 New Zealand MO, 6
Let $\vartriangle ABC$ be an acute triangle with $AB > AC$. Let $P$ be the foot of the altitude from $C$ to $AB$ and let $Q$ be the foot of the altitude from $B$ to $AC$. Let $X$ be the intersection of $PQ$ and $BC$. Let the intersection of the circumcircles of triangle $\vartriangle AXC$ and triangle $\vartriangle PQC$ be distinct points: $C$ and $Y$ . Prove that $PY$ bisects $AX$.
2024 LMT Fall, 8
The LHS Math Team is doing Karaoke. William sings every song, David sings every other song, Peter sings every third song, and Muztaba sings every fourth song. If they sing $600$ songs, find the average number of people singing each song.
2012 Balkan MO, 3
Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$
Prove that there is a subset $Y$ of $P_n$ such that $0 \leq y-S_Y < 2^n$
2020 Greece Team Selection Test, 3
The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).
1935 Moscow Mathematical Olympiad, 021
Denote by $M(a, b, c, . . . , k)$ the least common multiple and by $D(a, b, c, . . . , k)$ the greatest common divisor of $a, b, c, . . . , k$. Prove that:
a) $M(a, b)D(a, b) = ab$,
b) $\frac{M(a, b, c)D(a, b)D(b, c)D(a, c)}{D(a, b, c)}= abc$.
2016 PUMaC Algebra Individual A, A3
For positive real numbers $x$ and $y$, let $f(x, y) = x^{\log_2y}$. The sum of the solutions to the equation
\[4096f(f(x, x), x) = x^{13}\]
can be written in simplest form as $\tfrac{m}{n}$. Compute $m + n$.
2024 Turkey Olympic Revenge, 5
Let $a$ be a positive real number. Prove that
a) There exists $n\in \mathbb{N}$ with $\frac{\sigma(\varphi(n))}{\varphi(\sigma(n))} > a$.
b) There exists $n\in \mathbb{N}$ with $\frac{\sigma(\varphi(n))}{\varphi(\sigma(n))} < a$.
(As usual, $\sigma(n) = \sum_{d\mid n} d$ and $\varphi(n)$ is the number of integers $1\le m\le n$ which are coprime with $n$.)
Proposed by [i]Deniz Can Karaçelebi[/i]
1970 Putnam, B6
Show that if a circumscribable quadrilateral of sides $a,b,c,d$ has area $A= \sqrt{abcd},$ then it is also inscribable.
2006 Petru Moroșan-Trident, 3
In an acute-angled triangle $ ABC $ consider $ A_1,B_1,C_1 $ to be the symmetric points of the orthocenter of $ ABC $ to the sides $ BC,AC,AB, $ respectively. Show that if the centroids of the triangles $ ABC,A_1B_1C_1 $ are the same, then $ ABC $ is equilateral.
[i]Carmen Botea[/i]
1984 IMO Longlists, 25
Prove that the product of five consecutive positive integers cannot be the square of an integer.
2024 Saint Petersburg Mathematical Olympiad, 6
Polynomial $P(x)$ with integer coefficients is given. For some positive integer $n$ numbers $P(0),P(1),\dots,P(2^n+1)$ are all divisible by $2^{2^n}$. Prove that values of $P(x)$ in all integer points are divisible by $2^{2^n}$.
2012 Dutch IMO TST, 5
Find all functions $f : R \to R$ satisfying $f(x + xy + f(y))=(f(x) + \frac12)(f(y) + \frac12 )$ for all $x, y \in R$.
2023 Switzerland - Final Round, 8
Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.
2000 Romania National Olympiad, 4
Let $ I $ be the center of the incircle of a triangle $ ABC. $ Shw that, if for any point $ M $ on the segment $ AB $ (extremities excluded) there exist two points $ N,P $ on $ BC, $ respectively, $ AC $ (both excluding the extremities) such that the center of mass of $ MNP $ coincides with $ I, $ then $ ABC $ is equilateral.
2024 Singapore MO Open, Q4
Alice and Bob play a game. Bob starts by picking a set $S$ consisting of $M$ vectors of length $n$ with entries either $0$ or $1$. Alice picks a sequence of numbers $y_1\le y_2\le\dots\le y_n$ from the interval $[0,1]$, and a choice of real numbers $x_1,x_2\dots,x_n\in \mathbb{R}$. Bob wins if he can pick a vector $(z_1,z_2,\dots,z_n)\in S$ such that $$\sum_{i=1}^n x_iy_i\le \sum_{i=1}^n x_iz_i,$$otherwise Alice wins. Determine the minimum value of $M$ so that Bob can guarantee a win.
[i]Proposed by DVDthe1st[/i]
1961 AMC 12/AHSME, 8
Let the two base angles of a triangle be $A$ and $B$, with $B$ larger than $A$. The altitude to the base divides the vertex angle $C$ into two parts, $C_1$ and $C_2$, with $C_2$ adjacent to side $a$. Then:
${{{ \textbf{(A)}\ C_1+C_2=A+B \qquad\textbf{(B)}\ C_1-C_2=B-A \qquad\textbf{(C)}\ C_1-C_2=A-B} \qquad\textbf{(D)}\ C_1+C_2=B-A}\qquad\textbf{(E)}\ C_1-C_2=A+B} $