Found problems: 85335
2011 Math Prize for Girls Olympiad, 4
Let $M$ be a matrix with $r$ rows and $c$ columns. Each entry of $M$ is a nonnegative integer. Let $a$ be the average of all $rc$ entries of $M$. If $r > {(10 a + 10)}^c$, prove that $M$ has two identical rows.
2004 China Team Selection Test, 3
Let $a, b, c$ be sides of a triangle whose perimeter does not exceed $2 \cdot \pi.$, Prove that $\sin a, \sin b, \sin c$ are sides of a triangle.
1999 Brazil Team Selection Test, Problem 4
Assume that it is possible to color more than half of the surfaces of a given polyhedron so that no two colored surfaces have a common edge.
(a) Describe one polyhedron with the above property.
(b) Prove that one cannot inscribe a sphere touching all the surfaces of a polyhedron with the above property.
1995 Belarus Team Selection Test, 3
If $0<a,b<1$ and $p,q\geq 0 ,\ p+q=1$ are real numbers , then prove that: \[a^pb^q+(1-a)^p(1-b)^q\le 1\]
1995 Italy TST, 2
Twenty-one rectangles of size $3\times 1$ are placed on an $8\times 8$ chessboard, leaving only one free unit square. What position can the free square lie at?
2021 The Chinese Mathematics Competition, Problem 1
Evaluate $\lim_{x \to +\infty}\sqrt{x^2+x+1}\frac{x-ln(e^x+x)}{x}$.
2014 JBMO Shortlist, 2
In a country with $n$ towns, all the direct flights are of double destinations (back and forth). There are $r>2014$ rootes between different pairs of towns, that include no more than one intermediate stop (direction of each root matters). Find the minimum possible value of $n$ and the minimum possible $r$ for that value of $n$.
2016 China Team Selection Test, 2
In the coordinate plane the points with both coordinates being rational numbers are called rational points. For any positive integer $n$, is there a way to use $n$ colours to colour all rational points, every point is coloured one colour, such that any line segment with both endpoints being rational points contains the rational points of every colour?
2009 Estonia Team Selection Test, 2
Call a finite set of positive integers [i]independent [/i] if its elements are pairwise coprime, and [i]nice [/i] if the arithmetic mean of the elements of every non-empty subset of it is an integer.
a) Prove that for any positive integer $n$ there is an $n$-element set of positive integers which is both independent and nice.
b) Is there an infinite set of positive integers whose every independent subset is nice and which has an $n$-element independent subset for every positive integer $n$?
1999 Romania National Olympiad, 3
Let $f:\mathbb{R} \to \mathbb{R}$ be a monotonic function and $a,b,c,d$ be real numbers with $a$ and $c$ nonzero. Prove that if the equalities [center]$\int\limits_x^{x+\sqrt{3}} f(t) \mathrm{d}t=ax+b$ and $\int\limits_x^{x+\sqrt{2}} f(t) \mathrm{d}t=cx+d$[/center] hold for every real number $x,$ then $f$ is a polynomial function of degree one.
2016 AMC 8, 13
Two different numbers are randomly selected from the set ${ - 2, -1, 0, 3, 4, 5}$ and multiplied together. What is the probability that the product is $0$?
$\textbf{(A) }\dfrac{1}{6}\qquad\textbf{(B) }\dfrac{1}{5}\qquad\textbf{(C) }\dfrac{1}{4}\qquad\textbf{(D) }\dfrac{1}{3}\qquad \textbf{(E) }\dfrac{1}{2}$
2006 India IMO Training Camp, 2
the positive divisors $d_1,d_2,\cdots,d_k$ of a positive integer $n$ are ordered
\[1=d_1<d_2<\cdots<d_k=n\]
Suppose $d_7^2+d_{15}^2=d_{16}^2$. Find all possible values of $d_{17}$.
1968 Yugoslav Team Selection Test, Problem 3
Each side of a triangle $ABC$ is divided into three equal parts, and the middle segment in each of the sides is painted green. In the exterior of $\triangle ABC$ three equilateral triangles are constructed, in such a way that the three green segments are sides of these triangles. Denote by $A',B',C'$ the vertices of these new equilateral triangles that don’t belong to the edges of $\triangle ABC$, respectively. Let $A'',B'',C''$ be the points symmetric to $A',B',C'$ with respect to $BC,CA,AB$.
(a) Prove that $\triangle A'B'C'$ and $\triangle A''B''C''$ are equilateral.
(b) Prove that $ABC,A'B'C'$, and $A''B''C''$ have a common centroid.
2002 Baltic Way, 16
Find all nonnegative integers $m$ such that
\[a_m=(2^{2m+1})^2+1 \]
is divisible by at most two different primes.
Swiss NMO - geometry, 2018.4
Let $D$ be a point inside an acute triangle $ABC$, such that $\angle BAD = \angle DBC$ and $\angle DAC = \angle BCD$. Let $P$ be a point on the circumcircle of the triangle $ADB$. Suppose $P$ are itself outside the triangle $ABC$. A line through $P$ intersects the ray $BA$ in $X$ and ray $CA$ in $Y$, so that $\angle XPB = \angle PDB$. Show that $BY$ and $CX$ intersect on $AD$.
2014 CIIM, Problem 2
Let $n$ be an integer and $p$ a prime greater than 2. Show that: $$(p-1)^nn!|(p^n-1)(p^n-p)(p^n-p^2)\cdots(p^n-p^{n-1}).$$
2016 Ecuador NMO (OMEC), 3
Let $A, B, C, D$ be four different points on a line $\ell$, such that $AB = BC = CD$. In one of the semiplanes determined by the line $\ell$, the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points on the plane such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the measure of the angle $\angle MBN$.
ICMC 6, 5
A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of $T$?
[i]Proposed by Dylan Toh[/i]
2018 Regional Olympiad of Mexico West, 2
Let $a,b,c,d, e$ be real numbers such that they simultaneously satisfy the following equations
$$a+b+c+d+e=8$$
$$a^2+b^2+c^2+d^2+e^2=16$$
Determine the smallest and largest value that $a$ can take.
2004 AIME Problems, 11
A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F$, in such a way that the ratio between the areas of the painted surfaces of $C$ and $F$ and the ratio between the volumes of $C$ and $F$ are both equal to $k$. Given that $k=m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
2022 Saint Petersburg Mathematical Olympiad, 4
We will say that a set of real numbers $A = (a_1,... , a_{17})$ is stronger than the set of real numbers $B = (b_1, . . . , b_{17})$, and write $A >B$ if among all inequalities $a_i > b_j$ the number of true inequalities is at least $3$ times greater than the number of false. Prove that there is no chain of sets $A_1, A_2, . . . , A_N$ such that $A_1>A_2> \cdots A_N>A_1$.
Remark: For 11.4, the constant $3$ is changed to $2$ and $N=3$ and $17$ is changed to $m$ and $n$ in the definition (the number of elements don't have to be equal).
2014 Saudi Arabia IMO TST, 2
Determine all functions $f:[0,\infty)\rightarrow\mathbb{R}$ such that $f(0)=0$ and \[f(x)=1+5f\left(\left\lfloor{\frac{x}{2}\right\rfloor}\right)-6f\left(\left\lfloor{\frac{x}{4}\right\rfloor}\right)\] for all $x>0$.
2016 Greece Team Selection Test, 4
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.
2024 ELMO Shortlist, C4
Let $n \geq 2$ be a positive integer. Let $\mathcal{R}$ be a connected set of unit squares on a grid. A [i]bar[/i] is a rectangle of length or width $1$ which is fully contained in $\mathcal{R}$. A bar is [i]special[/i] if it is not fully contained within any larger bar. Given that $\mathcal{R}$ contains special bars of sizes $1 \times 2,1 \times 3,\ldots,1 \times n$, find the smallest possible number of unit squares in $\mathcal{R}$.
[i]Srinivas Arun[/i]
2019 Taiwan APMO Preliminary Test, P2
Put $1,2,....,2018$ (2018 numbers) in a row randomly and call this number $A$. Find the remainder of $A$ divided by $3$.