Olympiad problems

1986 IMO Shortlist, 6

Find four positive integers each not exceeding $70000$ and each having more than $100$ divisors.

2007 Vietnam National Olympiad, 2

Tags: number theory

Let $x,y$ be integer number with $x,y\neq-1$ so that $\frac{x^{4}-1}{y+1}+\frac{y^{4}-1}{x+1}\in\mathbb{Z}$. Prove that $x^{4}y^{44}-1$ is divisble by $x+1$

2018 Bulgaria JBMO TST, Source

Tags: inequalities

For real numbers $a$ and $b$, define $$f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}.$$ Find the smallest possible value of the expression $$f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b).$$

1992 AMC 12/AHSME, 19

Tags: geometry , 3d geometry , tetrahedron , ratio

For each vertex of a solid cube, consider the tetrahedron determined by the vertex and the midpoints of the three edges that meet at that vertex. The portion of the cube that remains when these eight tetrahedra are cut away is called a [i]cuboctahedron[/i]. The ratio of the volume of the cuboctahedron to the volume of the original cube is closest to which of these? $ \textbf{(A)}\ 75\%\qquad\textbf{(B)}\ 78\%\qquad\textbf{(C)}\ 81\%\qquad\textbf{(D)}\ 84\%\qquad\textbf{(E)}\ 87\% $

2023 Harvard-MIT Mathematics Tournament, 9

Tags: number theory

For any positive integers $a$ and $b$ with $b > 1$, let $s_b(a)$ be the sum of the digits of $a$ when it is written in base $b$. Suppose $n$ is a positive integer such that $$\sum^{\lfloor \log_{23} n\rfloor}_{i=1} s_{20} \left( \left\lfloor \frac{n}{23^i} \right\rfloor \right)= 103 \,\,\, \text{and} \,\,\, \sum^{\lfloor \log_{20} n\rfloor}_{i=1} s_{23} \left( \left\lfloor \frac{n}{20^i} \right\rfloor \right)= 115$$ Compute $s_{20}(n) - s_{23}(n)$.

2021 Novosibirsk Oral Olympiad in Geometry, 3

Tags: geometry , equal segments

Prove that in a triangle one of the sides is twice as large as the other if and only if a median and an angle bisector of this triangle are perpendicular

2024 HMNT, 7

Tags: guts

Let $\mathcal{P}$ be a regular $10$-gon in the coordinate plane. Mark computes the number of distinct $x$-coordinates that vertices of $\mathcal{P}$ take. Across all possible placements of $\mathcal{P}$ in the plane, compute the sum of all possible answers Mark could get.

2022 Girls in Math at Yale, 10

Tags: college

How many ways are there to choose distinct positive integers $a, b, c, d$ dividing $15^6$ such that none of $a, b, c,$ or $d$ divide each other? (Order does not matter.) [i]Proposed by Miles Yamner and Andrew Wu[/i] (Note: wording changed from original to clarify)

2004 IMO Shortlist, 3

Tags: geometry , circumcircle , homothety , triangle

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

1981 National High School Mathematics League, 5

Tags: geometry , 3d geometry

Given a cube $ABCD-A'B'C'D'$, in the $12$ lines:$AB',BA',CD',DC',AD',DA',BC',CB',AC,BD,A'C',B'D'$, how many sets of lines are skew lines? $\text{(A)}30\qquad\text{(B)}60\qquad\text{(C)}24\qquad\text{(D)}48$

2002 Junior Balkan Team Selection Tests - Romania, 4

Tags: inequalities

0<a,b,c<1 ==> \sqrt (abc) + \sqrt (1-a)(1-b)(1-c) <1

2017 Brazil Team Selection Test, 1

Tags: combinatorics

We call a $5$-tuple of integers [i]arrangeable[/i] if its elements can be labeled $a, b, c, d, e$ in some order so that $a-b+c-d+e=29$. Determine all $2017$-tuples of integers $n_1, n_2, . . . , n_{2017}$ such that if we place them in a circle in clockwise order, then any $5$-tuple of numbers in consecutive positions on the circle is arrangeable. [i]Warut Suksompong, Thailand[/i]

2016 Bundeswettbewerb Mathematik, 1

Tags: consecutive , sum , summation , number theory

There are $\tfrac{n(n+1)}{2}$ distinct sums of two distinct numbers, if there are $n$ numbers. For which $n \ (n \geq 3)$ do there exist $n$ distinct integers, such that those sums are $\tfrac{n(n-1)}{2}$ consecutive numbers?

2023 Chile National Olympiad, 5

Tags: number theory

What is the smallest positive integer that is divisible by $225$ and that has ony the numbers one and zero as digits?

2019 New Zealand MO, 4

Tags: number theory , perfect square

Show that for all positive integers $k$, there exists a positive integer n such that $n2^k -7$ is a perfect square.

1973 Bulgaria National Olympiad, Problem 2

Tags: arithmetic sequence , algebra , sequence , geometric sequence

Let the numbers $a_1,a_2,a_3,a_4$ form an arithmetic progression with difference $d\ne0$. Prove that there are no exists geometric progressions $b_1,b_2,b_3,b_4$ and $c_1,c_2,c_3,c_4$ such that: $$a_1=b_1+c_1,a_2=b_2+c_2,a_3=b_3+c_3,a_4=b_4+c_4.$$

2014 Contests, 1

Tags: number theory

Find with proof all positive $3$ digit integers $\overline{abc}$ satisfying \[ b\cdot \overline{ac}=c \cdot \overline{ab} +10 \]

1987 All Soviet Union Mathematical Olympiad, 448

Tags: geometry , combinatorics , combinatorial geometry , odd , convex , broken line

Given two closed broken lines in the plane with odd numbers of edges. All the lines, containing those edges are different, and not a triple of them intersects in one point. Prove that it is possible to chose one edge from each line such, that the chosen edges will be the opposite sides of a convex quadrangle.

1974 Chisinau City MO, 74

Tags: algebra , cubic , parameter

Solve the equation: $x^3-2ax^2+(a^2-2\sqrt2 a -6)x + 2\sqrt2 a^2+ 8a + 4\sqrt2 =0$

2024 Harvard-MIT Mathematics Tournament, 15

Tags: guts

Let $a \star b=ab-2.$ Comute the remainder when $(((579\star569)\star559)\star\cdots\star19)\star9$ is divided by $100.$

2004 France Team Selection Test, 2

Tags: inequalities , geometry , incenter , trigonometry , inradius , perimeter , cyclic quadrilateral

Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

2022 Bulgarian Spring Math Competition, Problem 12.2

Tags: ratio , pyramid , geometry , 3d geometry

Let $ABCDV$ be a regular quadrangular pyramid with $V$ as the apex. The plane $\lambda$ intersects the $VA$, $VB$, $VC$ and $VD$ at $M$, $N$, $P$, $Q$ respectively. Find $VQ : QD$, if $VM : MA = 2 : 1$, $VN : NB = 1 : 1$ and $VP : PC = 1 : 2$.

2014 Contests, 3

Tags: geometry

In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$. Prove that $|AR|\cdot |BQ|=|P I|^2$

2014 Contests, 3

Tags: combinatorial geometry , combinatorics

We have $4n + 5$ points on the plane, no three of them are collinear. The points are colored with two colors. Prove that from the points we can form $n$ empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors, such that all points occurring as vertices of the $n$ triangles have the same color.

1957 AMC 12/AHSME, 6

Tags: geometry , 3d geometry , prism

An open box is constructed by starting with a rectangular sheet of metal $ 10$ in. by $ 14$ in. and cutting a square of side $ x$ inches from each corner. The resulting projections are folded up and the seams welded. The volume of the resulting box is: $ \textbf{(A)}\ 140x \minus{} 48x^2 \plus{} 4x^3 \qquad \textbf{(B)}\ 140x \plus{} 48x^2 \plus{} 4x^3\qquad \\\textbf{(C)}\ 140x \plus{} 24x^2 \plus{} x^3\qquad \textbf{(D)}\ 140x \minus{} 24x^2 \plus{} x^3\qquad \textbf{(E)}\ \text{none of these}$