Found problems: 85335
1996 French Mathematical Olympiad, Problem 4
(a) A function $f$ is defined by $f(x)=x^x$ for all $x>0$. Find the minimum value of $f$.
(b) If $x$ and $y$ are two positive real numbers, show that $x^y+y^x>1$.
2009 Tournament Of Towns, 2
Several points on the plane are given, no three of them lie on the same line. Some of these points are connected by line segments. Assume that any line that does not pass through any of these points intersects an even number of these segments. Prove that from each point exits an even number of the segments.
2021 Bulgaria National Olympiad, 3
Find all $f:R^+ \rightarrow R^+$ such that
$f(f(x) + y)f(x) = f(xy + 1)\ \ \forall x, y \in R^+$
@below: [url]https://artofproblemsolving.com/community/c6h2254883_2020_imoc_problems[/url]
[quote]Feel free to start individual threads for the problems as usual[/quote]
1977 AMC 12/AHSME, 9
[asy]
size(120);
path c = Circle((0, 0), 1);
pair A = dir(20), B = dir(130), C = dir(240), D = dir(330);
draw(c);
pair F = 3(A-B) + B;
pair G = 3(D-C) + C;
pair E = intersectionpoints(B--F, C--G)[0];
draw(B--E--C--A);
label("$A$", A, NE);
label("$B$", B, NW);
label("$C$", C, SW);
label("$D$", D, SE);
label("$E$", E, E);
//Credit to MSTang for the diagram[/asy]
In the adjoining figure $\measuredangle E=40^\circ$ and arc $AB$, arc $BC$, and arc $CD$ all have equal length. Find the measure of $\measuredangle ACD$.
$\textbf{(A) }10^\circ\qquad\textbf{(B) }15^\circ\qquad\textbf{(C) }20^\circ\qquad\textbf{(D) }\left(\frac{45}{2}\right)^\circ\qquad \textbf{(E) }30^\circ$
2003 All-Russian Olympiad Regional Round, 8.4
Prove that an arbitrary triangle can be cut into three polygons, one of which must be an obtuse triangle, so that they can then be folded into a rectangle. (Turning over parts is possible).
2009 Indonesia TST, 2
Find the formula to express the number of $ n\minus{}$series of letters which contain an even number of vocals (A,I,U,E,O).
2005 All-Russian Olympiad, 4
Integers $x>2,\,y>1,\,z>0$ satisfy an equation $x^y+1=z^2$. Let $p$ be a number of different prime divisors of $x$, $q$ be a number of different prime divisors of $y$. Prove that $p\geq q+2$.
2024 Korea Junior Math Olympiad (First Round), 11.
There is a square $ ABCD. $
$ P $ is on $\bar{AB}$ , and $Q$ is on $ \bar{AD} $
They follow $ \bar{AP}=\bar{AQ}=\frac{\bar{AB}}{5} $
Let $ H $ be the foot of the perpendicular point from $ A $ to $ \bar{PD} $
If $ |\triangle APH|=20 $, Find the area of $ \triangle HCQ $.
2018 Estonia Team Selection Test, 12
We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$.
Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.
2018 CMIMC Individual Finals, 1
For all real numbers $r$, denote by $\{r\}$ the fractional part of $r$, i.e. the unique real number $s\in[0,1)$ such that $r-s$ is an integer. How many real numbers $x\in[1,2)$ satisfy the equation $\left\{x^{2018}\right\} = \left\{x^{2017}\right\}?$
2024 AMC 12/AHSME, 12
The first three terms of a geometric sequence are the integers $a,\,720,$ and $b,$ where $a<720<b.$ What is the sum of the digits of the least possible value of $b?$
$\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 21$
2016 Croatia Team Selection Test, Problem 4
Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime.
Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.
1995 Iran MO (2nd round), 2
Let $ABC$ be an acute triangle and let $\ell$ be a line in the plane of triangle $ABC.$ We've drawn the reflection of the line $\ell$ over the sides $AB, BC$ and $AC$ and they intersect in the points $A', B'$ and $C'.$ Prove that the incenter of the triangle $A'B'C'$ lies on the circumcircle of the triangle $ABC.$
2019 MOAA, 2
The lengths of the two legs of a right triangle are the two distinct roots of the quadratic $x^2 - 36x + 70$. What is the length of the triangle’s hypotenuse?
2016-2017 SDML (Middle School), 7
Point $P$ is selected at random from the interior of the pentagon with vertices $A = (0, 2), B = (4, 0), C = (2\pi + 1, 0), D = (2\pi + 1, 4),$ and $E = (0, 4)$. What is the probability that $\angle ABP$ is obtuse? Express your answer as a common fraction.
India EGMO 2025 TST, 3
Let $\Delta ABC$ be an acute angled scalene triangle with circumcircle $\omega$. Let $O$ and $H$ be the circumcenter and orthocenter of $\Delta ABC,$ respectively. Let $E,F$ and $Q$ be points on segments $AB,AC$ and $\omega$, respectively, such that
$$\angle BHE=\angle CHF=\angle AQH=90^\circ.$$
Prove that $OQ$ and $AH$ intersect on the circumcircle of $\Delta AEF$.
[i] Proposed by Antareep Nath [/i]
2017 JBMO Shortlist, A3
let $a\le b\le c \le d$ show that:
$$ab^3+bc^3+cd^3+da^3\ge a^2b^2+b^2c^2+c^2d^2+d^2a^2$$
2015 Argentina National Olympiad, 4
An segment $S$ of length $50$ is covered by several segments of length $1$ , all of them contained in $S$. If any of these unit segments were removed, $S$ would no longer be completely covered. Find the maximum number of unit segments with this property.
Clarification: Assume that the segments include their endpoints.
2023 Quang Nam Province Math Contest (Grade 11), Problem 3
Given a polynomial $P(x)$ with real coefficents satisfying:$$P(x).P(x+1)=P(x^2+x+1),\forall x\in \mathbb{R}.$$
Prove that: $\deg(P)$ is an even number and find $P(x).$
1988 All Soviet Union Mathematical Olympiad, 467
The quadrilateral $ABCD$ is inscribed in a fixed circle. It has $AB$ parallel to $CD$ and the length $AC$ is fixed, but it is otherwise allowed to vary. If $h$ is the distance between the midpoints of $AC$ and $BD$ and $k$ is the distance between the midpoints of $AB$ and $CD$, show that the ratio $h/k$ remains constant.
2023 Vietnam National Olympiad, 2
Given are the integers $a , b , c, \alpha, \beta$ and the sequence $(u_n)$ is defined by $u_1=\alpha, u_2=\beta, u_{n+2}=au_{n+1}+bu_n+c$ for all $n \geq 1$.
a) Prove that if $a = 3 , b= -2 , c = -1$ then there are infinitely many pairs of integers $(\alpha ; \beta)$ so that $u_{2023}=2^{2022}$.
b) Prove that there exists a positive integer $n_0$ such that only one of the following two statements is true:
i) There are infinitely many positive integers $m$, such that $u_{n_0}u_{n_0+1}\ldots u_{n_0+m}$ is divisible by $7^{2023}$ or $17^{2023}$
ii) There are infinitely many positive integers $k$ so that $u_{n_0}u_{n_0+1}\ldots u_{n_0+k}-1$ is divisible by $2023$
VI Soros Olympiad 1999 - 2000 (Russia), 10.1
For real numbers $x,y, \in [1,2]$, prove the inequality $3(x + y)\ge 2xy + 4$
2004 VTRMC, Problem 5
Let $f(x)=\int^x_0\sin(t^2-t+x)dt$. Compute $f''(x)+f(x)$ and deduce that $f^{(12)}(0)+f^{(10)}(0)=0$.
2024 AIME, 12
Define $f(x)=|| x|-\tfrac{1}{2}|$ and $g(x)=|| x|-\tfrac{1}{4}|$. Find the number of intersections of the graphs of $$y=4 g(f(\sin (2 \pi x))) \quad\text{ and }\quad x=4 g(f(\cos (3 \pi y))).$$
2003 AIME Problems, 9
An integer between 1000 and 9999, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?