Olympiad problems

1999 Korea Junior Math Olympiad, 2

Three integers are given. $A$ denotes the sum of the integers, $B$ denotes the sum of the square of the integers and $C$ denotes the sum of cubes of the integers(that is, if the three integers are $x, y, z$, then $A=x+y+z$, $B=x^2+y^2+z^2$, $C=x^3+y^3+z^3$). If $9A \geq B+60$ and $C \geq 360$, find $A, B, C$.

2022 Caucasus Mathematical Olympiad, 3

Tags: combinatorics , combinatorial geometry

Do there exist 100 points on the plane such that the pairwise distances between them are pairwise distinct consecutive integer numbers larger than 2022?

1992 Canada National Olympiad, 2

Tags: inequalities

For $ x,y,z \geq 0,$ establish the inequality \[ x(x\minus{}z)^2 \plus{} y(y\minus{}z)^2 \geq (x\minus{}z)(y\minus{}z)(x\plus{}y\minus{}z)\] and determine when equality holds.

2005 Bulgaria National Olympiad, 2

Tags: geometry , circumcircle , incenter , ratio , trigonometry , power of a point , radical axis

Consider two circles $k_{1},k_{2}$ touching externally at point $T$. a line touches $k_{2}$ at point $X$ and intersects $k_{1}$ at points $A$ and $B$. Let $S$ be the second intersection point of $k_{1}$ with the line $XT$ . On the arc $\widehat{TS}$ not containing $A$ and $B$ is chosen a point $C$ . Let $\ CY$ be the tangent line to $k_{2}$ with $Y\in k_{2}$ , such that the segment $CY$ does not intersect the segment $ST$ . If $I=XY\cap SC$ . Prove that : (a) the points $C,T,Y,I$ are concyclic. (b) $I$ is the excenter of triangle $ABC$ with respect to the side $BC$.

2012 China Second Round Olympiad, 4

Tags: ceiling function , floor function , inequalities

Let $S_n=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$, where $n$ is a positive integer. Prove that for any real numbers $a,b,0\le a\le b\le 1$, there exist infinite many $n\in\mathbb{N}$ such that \[a<S_n-[S_n]<b\] where $[x]$ represents the largest integer not exceeding $x$.

2024 Czech-Polish-Slovak Junior Match, 4

Tags: divisibility , floor function , number theory

How many positive integers $n<2024$ are divisible by $\lfloor \sqrt{n}\rfloor-1$?

2015 Online Math Open Problems, 17

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Let $A,B,M,C,D$ be distinct points on a line such that $AB=BM=MC=CD=6.$ Circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ and radius $4$ and $9$ are tangent to line $AD$ at $A$ and $D$ respectively such that $O_1,O_2$ lie on the same side of line $AD.$ Let $P$ be the point such that $PB\perp O_1M$ and $PC\perp O_2M.$ Determine the value of $PO_2^2-PO_1^2.$ [i]Proposed by Ray Li[/i]

1992 Vietnam Team Selection Test, 1

Tags: geometry

In the plane let a finite family of circles be given satisfying the condition: every two circles, either are outside each other, either touch each other from outside and each circle touch at most 6 other circles. Suppose that every circle which does not touch 6 other circles be assigned a real number. Show that there exist at most one assignment to each remaining circle a real number equal to arithmetic mean of 6 numbers assigned to 6 circles which touch it.

2009 Tournament Of Towns, 1

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Is it possible to cut a square into nine squares and colour one of them white, three of them grey and ve of them black, such that squares of the same colour have the same size and squares of different colours will have different sizes? [i](3 points)[/i]

2017 Saudi Arabia Pre-TST + Training Tests, 1

Tags: number theory , divide

Let $m, n, k$ and $l$ be positive integers with $n \ne 1$ such that $n^k + mn^l + 1$ divides $n^{k+l }- 1$. Prove that either $m = 1$ and $l = 2k$, or $l | k$ and $m =\frac{n^{k-l} - 1}{n^l - 1}$.

2014 Germany Team Selection Test, 1

Tags: algorithm , combinatorics , additive combinatorics

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

2023 China Northern MO, 2

Tags: algebra , inequality , inequalities

Let $ a,b,c \in (0,1) $ and $ab+bc+ca=4abc .$ Prove that $$\sqrt{a+b+c}\geq \sqrt{1-a}+\sqrt{1-b}+\sqrt{1-c}$$

2021 Peru Cono Sur TST., P4

Tags: combinatorics

Let $n\ge 5$ be an integer. Consider $2n-1$ subsets $A_1, A_2, A_3, \ldots , A_{2n-1}$ of the set $\{ 1, 2, 3,\ldots , n \}$, these subsets have the property that each of them has $2$ elements (that is that is, for $1 \le i \le 2n-1$ it is true that $A_i$ has $2$ elements). Show that it is always possible to select $n$ of these subsets in such a way that the union of these $n$ subsets has at most $\frac{2}{3}n + 1$ elements in total.

2010 China Team Selection Test, 2

Tags: number theory , prime number , factorial

Prove that there exists a sequence of unbounded positive integers $a_1\leq a_2\leq a_3\leq\cdots$, such that there exists a positive integer $M$ with the following property: for any integer $n\geq M$, if $n+1$ is not prime, then any prime divisor of $n!+1$ is greater than $n+a_n$.

2024 Harvard-MIT Mathematics Tournament, 3

Tags: combinatorics

Compute the number of ways there are to assemble $2$ red unit cubes and $25$ white unit cubes into a $3 \times 3 \times 3$ cube such that red is visible on exactly $4$ faces of the larger cube. (Rotations and reflections are considered distinct.)

1974 All Soviet Union Mathematical Olympiad, 192

Tags: geometry , trapezoid , minimum , tangent circle

Given two circles with the radiuses $R$ and $r$, touching each other from the outer side. Consider all the trapezoids, such that its lateral sides touch both circles, and its bases touch different circles. Find the shortest possible lateral side.

2024 Brazil Team Selection Test, 1

Tags: geometry

Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P$ and $Q$ lie between $B$ and $C$. Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y \neq X$. Prove that the points $A, X$, and $Y$ are collinear.

2017 CMIMC Computer Science, 4

Tags: computer science

How many complete directed graphs with vertex set $V=\{1,2,3,4,5,6\}$ contain no $3$-cycles? A graph is $\textit{directed}$ if all edges have a direction (e.g. from $u$ to $v$ rather than between $u$ and $v$), and $\textit{complete}$ if every pair of vertices has an edge between them. Further, a $\textit{3-cycle}$ in a directed graph is a triple $(u,v,w)$ of vertices such that there are edges from $u$ to $v$, $v$ to $w$, and $w$ to $u$.

2011 Puerto Rico Team Selection Test, 3

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The number $1234$ is written on the board. A play consists of subtracting a non-zero digit of that number from that number, and replacing the number by the result. The player who writes the number (not digit) zero wins. Determine if there is a winning strategy for one of the two players who play consecutively. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

DMM Individual Rounds, 2021 Tie

Tags: combinatorics

You are standing on one of the faces of a cube. Each turn, you randomly choose another face that shares an edge with the face you are standing on with equal probability, and move to that face. Let $F(n)$ the probability that you land on the starting face after $n$ turns. Supposed that $F(8) = \frac{43}{256}$ , and F(10) can be expressed as a simplified fraction $\frac{a}{b}$. Find $a+b$.

2014 South East Mathematical Olympiad, 6

Tags: exterior angle , geometry

Let $\omega_{1}$ be a circle with centre $O$. $P$ is a point on $\omega_{1}$. $\omega_{2}$ is a circle with centre $P$, with radius smaller than $\omega_{1}$. $\omega_{1}$ meets $\omega_{2}$ at points $T$ and $Q$. Let $TR$ be a diameter of $\omega_{2}$. Draw another two circles with $RQ$ as the radius, $R$ and $P$ as the centres. These two circles meet at point $M$, with $M$ and $Q$ lie on the same side of $PR$. A circle with centre $M$ and radius $MR$ intersects $\omega_{2}$ at $R$ and $N$. Prove that a circle with centre $T$ and radius $TN$ passes through $O$.

2023 May Olympiad, 1

Tags: number theory , algebra

Juanita wrote the numbers from $1$ to $13$ , calculated the sum of all the digits he had written and obtained $$1+2+3+4+5+6+7+8+9+(1+0)+(1+1)+(1+2)+(1+3)=55.$$ His brother Ariel wrote the numbers from $1$ to $100$ and calculated the sum of all the digits written. Find the value of Ariel's sum.

2025 Romanian Master of Mathematics, 5

Tags: geometry

Let triangle $ABC$ be an acute triangle with $AB<AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. Let $\Gamma$ be the circle $BOC$. The line $AO$ and the circle of radius $AO$ centered at $A$ cross $\Gamma$ at $A’$ and $F$, respectively. Prove that $\Gamma$ , the circle on diameter $AA’$ and circle $AFH$ are concurrent. [i]Proposed by Romania, Radu-Andrew Lecoiu[/i]

1952 Putnam, B1

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A mathematical moron is given two sides and the included angle of a triangle and attempts to use the Law of Cosines: $a^2 = b^2 + c^2 - 2bc \cos A,$ to find the third side $a.$ He uses logarithms as follows. He finds $\log b$ and doubles it; adds to that the double of $\log c;$ subtracts the sum of the logarithms of $2, b, c,$ and $\cos A;$ divides the result by $2;$ and takes the anti-logarithm. Although his method may be open to suspicion his computation is accurate. What are the necessary and sufficient conditions on the triangle that this method should yield the correct result?

1989 Spain Mathematical Olympiad, 1

Tags: probability , combinatorics

An exam at a university consists of one question randomly selected from the$ n$ possible questions. A student knows only one question, but he can take the exam $n$ times. Express as a function of $n$ the probability $p_n$ that the student will pass the exam. Does $p_n$ increase or decrease as $n$ increases? Compute $lim_{n\to \infty}p_n$. What is the largest lower bound of the probabilities $p_n$?