This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

2005 Junior Balkan Team Selection Tests - Romania, 11
Tags: circumcircle , geometric transformation , reflection , homothety , ratio , geometry
Three circles $\mathcal C_1(O_1)$, $\mathcal C_2(O_2)$ and $\mathcal C_3(O_3)$ share a common point and meet again pairwise at the points $A$, $B$ and $C$. Show that if the points $A$, $B$, $C$ are collinear then the points $Q$, $O_1$, $O_2$ and $O_3$ lie on the same circle.

1985 IMO Longlists, 72
Tags: circumcircle , geometric transformation , reflection , geometry
Construct a triangle $ABC$ given the side $AB$ and the distance $OH$ from the circumcenter $O$ to the orthocenter $H$, assuming that $OH$ and $AB$ are parallel.

1969 AMC 12/AHSME, 2
Tags: ratio
If an item is sold for $x$ dollars, there is a loss of $15\%$ based on the cost. If, however, the same item is sold for $y$ dollars, there is a profit of $15\%$ based on the cost. The ratio $y:x$ is: $\textbf{(A) }23:17\qquad \textbf{(B) }17y:23\qquad \textbf{(C) }23x:17\qquad$ $\textbf{(D) }\text{dependent upon the cost}\qquad \textbf{(E) }\text{none of these.}$

2017 Purple Comet Problems, 16
Tags: algebra , sequence
Let $a_1 = 1 +\sqrt2$ and for each $n \ge 1$ de ne $a_{n+1} = 2 -\frac{1}{a_n}$. Find the greatest integer less than or equal to the product $a_1a_2a_3 ... a_{200}$.

1986 IMO Shortlist, 6
Tags: number theory , prime factorization , divisor
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 $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.