Olympiad problems

1955 Poland - Second Round, 2

Find the natural number $ n $ knowing that the sum $$ 1 + 2 + 3 + \ldots + n$$ is a three-digit number with identical digits.

2011 Mediterranean Mathematics Olympiad, 4

Tags: incenter , trigonometry , angle bisector , geometry

Let $D$ be the foot of the internal bisector of the angle $\angle A$ of the triangle $ABC$. The straight line which joins the incenters of the triangles $ABD$ and $ACD$ cut $AB$ and $AC$ at $M$ and $N$, respectively. Show that $BN$ and $CM$ meet on the bisector $AD$.

1986 IMO Shortlist, 19

Tags: geometry , 3d geometry , tetrahedron , circumcircle , minimization

A tetrahedron $ABCD$ is given such that $AD = BC = a; AC = BD = b; AB\cdot CD = c^2$. Let $f(P) = AP + BP + CP + DP$, where $P$ is an arbitrary point in space. Compute the least value of $f(P).$

2007 Iran MO (2nd Round), 2

Tags: geometry , 3d geometry , combinatorics

Two vertices of a cube are $A,O$ such that $AO$ is the diagonal of one its faces. A $n-$run is a sequence of $n+1$ vertices of the cube such that each $2$ consecutive vertices in the sequence are $2$ ends of one side of the cube. Is the $1386-$runs from $O$ to itself less than $1386-$runs from $O$ to $A$ or more than it?

2006 France Team Selection Test, 1

Tags: circumcircle , projective geometry , cyclic quadrilateral , angle bisector , geometry

Let $ABCD$ be a square and let $\Gamma$ be the circumcircle of $ABCD$. $M$ is a point of $\Gamma$ belonging to the arc $CD$ which doesn't contain $A$. $P$ and $R$ are respectively the intersection points of $(AM)$ with $[BD]$ and $[CD]$, $Q$ and $S$ are respectively the intersection points of $(BM)$ with $[AC]$ and $[DC]$. Prove that $(PS)$ and $(QR)$ are perpendicular.

2019 Jozsef Wildt International Math Competition, W. 22

Tags: gamma function , number theory

Let $A$ and $B$ the series: $$A=\sum \limits_{n=1}^{\infty}\frac{C_{2n}^1}{C_{2n}^0+C_{2n}^1+\cdots +C_{2n}^{2n}},\ B=\sum \limits_{n=1}^{\infty}\frac{\Gamma \left(n+\frac{1}{2}\right) }{\Gamma \left(n+\frac{5}{2}\right)}$$Study if $\frac{A}{B}$ is irrational number.

2006 Harvard-MIT Mathematics Tournament, 9

Tags:

Eight celebrities meet at a party. It so happens that each celebrity shakes hands with exactly two others. A fan makes a list of all unordered pairs of celebrities who shook hands with each other. If order does not matter, how many different lists are possible?

2022 Thailand TST, 1

Tags: number theory

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

2023 Belarus Team Selection Test, 1.3

Tags: number theory , prime factorization , function , prime number , prime

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

2000 May Olympiad, 4

Tags: combinatorics , combinatorial geometry

There is a cube of $3 \times 3 \times 3$ formed by the union of $27$ cubes of $1 \times 1 \times 1$. Some cubes are removed in such a way that those that remain continue to form a solid made up of cubes that are united by at least one facing the rest of the solid. When a cube is removed, those that remain do so in the same place they were. What is the maximum number of cubes that can be removed so that the area of the resulting solid is equal to the area of the original cube?

2001 Tournament Of Towns, 2

Tags: perimeter , geometry

Let $n\ge3$ be an integer. A circle is divided into $2n$ arcs by $2n$ points. Each arc has one of three possible lengths, and no two adjacent arcs have the same lengths. The $2n$ points are colored alternately red and blue. Prove that the $n$-gon with red vertices and the $n$-gon with blue vertices have the same perimeter and the same area.

2024 India Iran Friendly Math Competition, 5

Tags: algorithm , combinatorics

Let $n \geq k$ be positive integers and let $a_1, \dots, a_n$ be a non-increasing list of positive real numbers. Prove that there exists $k$ sets $B_1, \dots, B_k$ which partition the set $\{1, 2, \dots, n\}$ such that $$\min_{1 \le j \le k} \left(\sum_{i \in B_j} a_i \right) \geq \min_{1 \le j \le k} \left(\frac{1}{2k+1-2j} \cdot \sum^n_{i=j} a_i\right).$$ [i]Proposed by Navid Safaei[/i]

2024 Austrian MO National Competition, 2

Tags: semicircle , touching circles , cyclic quadrilateral , geometry

Let $h$ be a semicircle with diameter $AB$. The two circles $k_1$ and $k_2$, $k_1 \ne k_2$, touch the segment $AB$ at the points $C$ and $D$, respectively, and the semicircle $h$ fom the inside at the points $E$ and $F$, respectively. Prove that the four points $C$, $D$, $E$ and $F$ lie on a circle. [i](Walther Janous)[/i]

2022 Brazil Undergrad MO, 6

Tags: number theory , prime number

Let $p \equiv 3 \,(\textrm{mod}\, 4)$ be a prime and $\theta$ some angle such that $\tan(\theta)$ is rational. Prove that $\tan((p+1)\theta)$ is a rational number with numerator divisible by $p$, that is, $\tan((p+1)\theta) = \frac{u}{v}$ with $u, v \in \mathbb{Z}, v >0, \textrm{mdc}(u, v) = 1$ and $u \equiv 0 \,(\textrm{mod}\,p) $.

2025 Harvard-MIT Mathematics Tournament, 14

Tags: guts

A parallelogram $P$ can be folded over a straight line so that the resulting shape is a regular pentagon with side length $1.$ Compute the perimeter of $P.$

2017 AMC 12/AHSME, 10

Tags: probability

Chloé chooses a real number uniformly at random from the interval $[0, 2017]$. Independently, Laurent chooses a real number uniformly at random from the interval $[0,4034]$. What is the probability that Laurent's number is greater than Chloé's number? $\textbf{(A)}~\frac12 \qquad \textbf{(B)}~\frac23 \qquad \textbf{(C)}~\frac34 \qquad \textbf{(D)}~\frac56\qquad \textbf{(E)}~\frac78$

2020 Taiwan APMO Preliminary, P7

Tags: geometry

[$XYZ$] denotes the area of $\triangle XYZ$ We have a $\triangle ABC$,$BC=6,CA=7,AB=8$ (1)If $O$ is the circumcenter of $\triangle ABC$, find [$OBC$]:[$OCA$]:[$OAB$] (2)If $H$ is the orthocenter of $\triangle ABC$, find [$HBC$]:[$HCA$]:[$HAB$]

2013 CHMMC (Fall), 1

Tags: geometry

In the diagram below, point $A$ lies on the circle centered at $O$. $AB$ is tangent to circle $O$ with $\overline{AB} = 6$. Point $C$ is $\frac{2\pi}{3}$ radians away from point $A$ on the circle, with $BC$ intersecting circle $O$ at point $D$. The length of $BD$ is $3$. Compute the radius of the circle. [img]https://cdn.artofproblemsolving.com/attachments/7/8/baa528c776eb50455f31ae50a3ec28efc291e8.png[/img]

1998 Nordic, 2

Tags: geometry , rectangle

Let $C_1$ and $C_2$ be two circles intersecting at $A $ and $B$. Let $S$ and $T $ be the centres of $C_1 $ and $C_2$, respectively. Let $P$ be a point on the segment $AB$ such that $ |AP|\ne |BP|$ and $P\ne A, P \ne B$. We draw a line perpendicular to $SP$ through $P$ and denote by $C$ and $D$ the points at which this line intersects $C_1$. We likewise draw a line perpendicular to $TP$ through $P$ and denote by $E$ and F the points at which this line intersects $C_2$. Show that $C, D, E,$ and $F$ are the vertices of a rectangle.

2024 Nordic, 4

Tags: combinatorics , set

Alice and Bob are playing a game. First, Alice chooses a partition $\mathcal{C}$ of the positive integers into a (not necessarily finite) set of sets, such that each positive integer is in exactly one of the sets in $\mathcal{C}$. Then Bob does the following operation a finite number of times. Choose a set $S \in \mathcal{C}$ not previously chosen, and let $D$ be the set of all positive integers dividing at least one element in $S$. Then add the set $D \setminus S$ (possibly the empty set) to $\mathcal{C}$. Bob wins if there are two equal sets in $\mathcal{C}$ after he has done all his moves, otherwise, Alice wins. Determine which player has a winning strategy.

2022 Olimphíada, 2

Tags: algebra , sequence

We say that a real $a\geq-1$ is philosophical if there exists a sequence $\epsilon_1,\epsilon_2,\dots$, with $\epsilon_i \in\{-1,1\}$ for all $i\geq1$, such that the sequence $a_1,a_2,a_3,\dots$, with $a_1=a$, satisfies $$a_{n+1}=\epsilon_{n}\sqrt{a_{n}+1},\forall n\geq1$$ and is periodic. Find all philosophical numbers.

V Soros Olympiad 1998 - 99 (Russia), 10.1

Tags: number theory , perfect power

Find some natural number $a$ such that $2a$ is a perfect square, $3a$ is a perfect cube, $5a$ is the fifth power of some natural number.

2019 Tournament Of Towns, 2

Tags: line , game , game strategy , combinatorics

$2019$ point grasshoppers sit on a line. At each move one of the grasshoppers jumps over another one and lands at the point the same distance away from it. Jumping only to the right, the grasshoppers are able to position themselves so that some two of them are exactly $1$ mm apart. Prove that the grasshoppers can achieve the same, jumping only to the left and starting from the initial position. (Sergey Dorichenko)

2004 All-Russian Olympiad Regional Round, 11.2

Tags: geometry , equal circles , tangent circle

Three circles $\omega_1$, $\omega_2$, $\omega_3$ of radius $r$ pass through the point$ S$ and internally touch the circle $\omega$ of radius $R$ ($R > r$) at points $T_1$, $T_2$, $T_3$ respectively. Prove that the line $T_1T_2$ passes through the second (different from $S$) intersection point of the circles $\omega_1$ and $\omega_2$.

2013 ISI Entrance Examination, 8

Tags: parabola , analytic geometry , conic

Let $ABCD$ be a square such that $AB$ lies along the line $y=x+8,$ and $C$ and $D$ lie on the parabola $y=x^2.$ Find all possible values of sidelength of the square.