Found problems: 85335
2013 IMO, 5
Let $\mathbb Q_{>0}$ be the set of all positive rational numbers. Let $f:\mathbb Q_{>0}\to\mathbb R$ be a function satisfying the following three conditions:
(i) for all $x,y\in\mathbb Q_{>0}$, we have $f(x)f(y)\geq f(xy)$;
(ii) for all $x,y\in\mathbb Q_{>0}$, we have $f(x+y)\geq f(x)+f(y)$;
(iii) there exists a rational number $a>1$ such that $f(a)=a$.
Prove that $f(x)=x$ for all $x\in\mathbb Q_{>0}$.
[i]Proposed by Bulgaria[/i]
2025 Harvard-MIT Mathematics Tournament, 3
Given that $x, y,$ and $z$ are positive real numbers such that $$x^{\log_2(yz)}=2^8\cdot3^4, \quad y^{\log_2(zx)}=2^9\cdot3^6, \quad \text{and}\quad z^{\log_2(xy)}=2^5 \cdot 3^{10},$$ compute the smallest possible value of $xyz.$
2012 Harvard-MIT Mathematics Tournament, 10
Let $C$ denote the set of points $(x, y) \in R^2$ such that $x^2 + y^2 \le1$. A sequence $A_i = (x_i, y_i), |i \ge¸ 0$ of points in $R^2$ is ‘centric’ if it satisfies the following properties:
$\bullet$ $A_0 = (x_0, y_0) = (0, 0)$, $A_1 = (x_1, y_1) = (1, 0)$.
$\bullet$ For all $n\ge 0$, the circumcenter of triangle $A_nA_{n+1}A_{n+2}$ lies in $C$.
Let $K$ be the maximum value of $x^2_{2012} + y^2_{2012}$ over all centric sequences. Find all points $(x, y)$ such that $x^2 + y^2 = K$ and there exists a centric sequence such that $A_{2012} = (x, y)$.
2001 JBMO ShortLists, 9
Consider a convex quadrilateral $ABCD$ with $AB=CD$ and $\angle BAC=30^{\circ}$. If $\angle ADC=150^{\circ}$, prove that $\angle BCA= \angle ACD$.
2019 IMO Shortlist, N4
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.
2008 Moldova National Olympiad, 12.4
Define the sequence $ (a_p)_{p\ge0}$ as follows: $ a_p\equal{}\displaystyle\frac{\binom p0}{2\cdot 4}\minus{}\frac{\binom p1}{3\cdot5}\plus{}\frac{\binom p2}{4\cdot6}\minus{}\ldots\plus{}(\minus{}1)^p\cdot\frac{\binom pp}{(p\plus{}2)(p\plus{}4)}$.
Find $ \lim_{n\to\infty}(a_0\plus{}a_1\plus{}\ldots\plus{}a_n)$.
2020 Jozsef Wildt International Math Competition, W45
Let $a_1,a_2,a_3,a_4$ be strictly positive numbers. Then is the following inequality true:
$$4\left(a_1a_2^n+a_2a_3^n+a_3a_4^n+a_4a_1^n\right)^n\le\left(a_1^n+a_2^n+a_3^n+a_4^n\right)^{n+1}$$
for each $n\in\mathbb N$?
[i]Proposed by Mihály Bencze and Marius Drăgan[/i]
2002 China Team Selection Test, 2
Circles $ \omega_{1}$ and $ \omega_{2}$ intersect at points $ A$ and $ B.$ Points $ C$ and $ D$ are on circles $ \omega_{1}$ and $ \omega_{2},$ respectively,
such that lines $ AC$ and $ AD$ are tangent to circles $ \omega_{2}$ and $ \omega_{1},$ respectively.
Let $ I_{1}$ and $ I_{2}$ be the incenters of triangles $ ABC$ and $ ABD,$ respectively. Segments $ I_{1}I_{2}$ and $ AB$ intersect at $ E$.
Prove that: $ \frac {1}{AE} \equal{} \frac {1}{AC} \plus{} \frac {1}{AD}$
1999 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 2
There are 8 members in a a bridge committee (committee for making bridges). Of these 8 members, 3 are chosen to be in special "approval" committee with 1 of 3 members being the "boss." In how many ways can this happen?
2022 Singapore MO Open, Q3
Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$.
[i]Proposed by DVDthe1st[/i]
2011 Tournament of Towns, 3
Three pairwise intersecting rays are given. At some point in time not on every ray from its beginning a point begins to move with speed. It is known that these three points form a triangle at any time, and the center of the circumscribed circle of this the triangle also moves uniformly and on a straight line. Is it true, that all these triangles are similar to each other?
1998 AMC 12/AHSME, 21
In an $ h$-meter race, Sunny is exactly $ d$ meters ahead of Windy when Sunny finishes the race. The next time they race, Sunny sportingly starts $ d$ meters behind Windy, who is at the starting line. Both runners run at the same constant speed as they did in the first race. How many meters ahead is Sunny when Sunny finishes the second race?
$ \textbf{(A)}\ \frac {d}{h} \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ \frac {d^2}{h} \qquad \textbf{(D)}\ \frac {h^2}{d} \qquad \textbf{(E)}\ \frac {d^2}{h \minus{} d}$
2019 Dürer Math Competition (First Round), P2
a) 11 kayakers row on the Danube from Szentendre to Kopaszi-gát. They do not necessarily start at the same time, but we know that they all take the same route and that each kayaker rows with a constant speed. Whenever a kayaker passes another one, they do a high five. After they all arrive, everybody claims to have done precisely $10$ high fives in total. Show that it is possible for the kayakers to have rowed in such a way that this is true.
b) At a different occasion $13$ kayakers rowed in the same manner; now after arrival everybody claims to have done precisely$ 6$ high fives. Prove that at least one kayaker has miscounted.
2003 Iran MO (3rd Round), 25
Let $ A,B,C,Q$ be fixed points on plane. $ M,N,P$ are intersection points of $ AQ,BQ,CQ$ with $ BC,CA,AB$. $ D',E',F'$ are tangency points of incircle of $ ABC$ with $ BC,CA,AB$. Tangents drawn from $ M,N,P$ (not triangle sides) to incircle of $ ABC$ make triangle $ DEF$. Prove that $ DD',EE',FF'$ intersect at $ Q$.
Durer Math Competition CD 1st Round - geometry, 2023.C7
Let $ABCDE$ be a regular pentagon. We drew two circles around $A$ and $B$ with radius $AB$. Let $F$ mark the intersection of the two circles that is inside the pentagon. Let $G$ mark the intersection of lines $EF$ and $AD$. What is the degree measure of angle $AGE$?
1977 Swedish Mathematical Competition, 3
Show that the only integral solution to
\[\left\{ \begin{array}{l}
xy + yz + zx = 3n^2 - 1\\
x + y + z = 3n \\
\end{array} \right.
\]
with $x \geq y \geq z$ is $x=n+1$, $y=n$, $z=n-1$.
1966 Poland - Second Round, 6
Prove that the sum of the squares of the right-angled projections of the sides of a triangle onto the line $ p $ of the plane of this triangle does not depend on the position of the line $ p $ if and only if it the triangle is equilateral.
2017 BMT Spring, 13
Two points are located $10$ units apart, and a circle is drawn with radius $ r$ centered at one of the points. A tangent line to the circle is drawn from the other point. What value of $ r$ maximizes the area of the triangle formed by the two points and the point of tangency?
2015 South Africa National Olympiad, 2
Determine all pairs of real numbers $a$ and $x$ that satisfy the simultaneous equations $$5x^3 + ax^2 + 8 = 0$$ and $$5x^3 + 8x^2 + a = 0.$$
1964 Miklós Schweitzer, 8
Let $ F$ be a closed set in the $ n$-dimensional Euclidean space. Construct a function that is $ 0$ on $ F$, positive outside $ F$ , and whose partial derivatives all exist.
2019 Saudi Arabia JBMO TST, 3
Let $S$ be a set of real numbers such that:
i) $1$ is from $S$;
ii) for any $a, b$ from $S$ (not necessarily different), we have that $a-b$ is also from $S$;
iii) for any $a$ from $S$ ($a$ is different from $0$), we have that $1/a$ is from $S$.
Show that for every $a, b$ from $S$, we have that $ab$ is from $S$.
1982 Austrian-Polish Competition, 7
Find the triple of positive integers $(x,y,z)$ with $z$ least possible for which there are positive integers $a, b, c, d$ with the following properties:
(i) $x^y = a^b = c^d$ and $x > a > c$
(ii) $z = ab = cd$
(iii) $x + y = a + b$.
2008 F = Ma, 23
Consider two uniform spherical planets of equal density but unequal radius. Which of the following quantities is the same for both planets?
(a) The escape velocity from the planet’s surface.
(b) The acceleration due to gravity at the planet’s surface.
(c) The orbital period of a satellite in a circular orbit just above the planet’s surface.
(d) The orbital period of a satellite in a circular orbit at a given distance from the planet’s center.
(e) None of the above.
2013 JBMO Shortlist, 2
In a billiard with shape of a rectangle $ABCD$ with $AB=2013$ and $AD=1000$, a ball is launched along the line of the bisector of $\angle BAD$. Supposing that the ball is reflected on the sides with the same angle at the impact point as the angle shot , examine if it shall ever reach at vertex B.
1997 Croatia National Olympiad, Problem 4
On the sides of a triangle $ABC$ are constructed similar triangles $ABD,BCE,CAF$ with $k=AD/DB=BE/EC=CF/FA$ and $\alpha=\angle ADB=\angle BEC=\angle CFA$. Prove that the midpoints of the segments $AC,BC,CD$ and $EF$ form a parallelogram with an angle $\alpha$ and two sides whose ratio is $k$.