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

2019 Dutch IMO TST, 2
Tags: product , number theory
Let $n$ be a positive integer. Prove that $n^2 + n + 1$ cannot be written as the product of two positive integers of which the difference is smaller than $2\sqrt{n}$.

2017 Israel Oral Olympiad, 4
Tags: geometry , 3d geometry , octahedron , hypercube
What is the shortest possible side length of a four-dimensional hypercube that contains a regular octahedron with side 1?

LMT Team Rounds 2021+, 10
Tags: combinatorics
In a country with $5$ distinct cities, there may or may not be a road between each pair of cities. It’s possible to get from any city to any other city through a series of roads, but there is no set of three cities $\{A,B,C\}$ such that there are roads between $A$ and $B$, $B$ and $C$, and $C$ and $A$. How many road systems between the five cities are possible?

2018 ASDAN Math Tournament, 1
Tags: algebra test
Alice’s age in years is twice Eve’s age in years. In $10$ years, Eve will be as old as Alice is now. Compute Alice’s age in years now.

2010 All-Russian Olympiad Regional Round, 11.8
Tags: combinatorics
The numbers $1, 2,. . . , 10000, $ were placed in the cells of a $100 \times 100$ square, each once; in this case, numbers differing by $1$ are written in cells adjacent to each side. After that we calculated distances between the centers of every two cells whose numbers differ by exactly $5000$. Let $S$ be the minimum of these distances What is the largest value $S$ can take?

2024 IFYM, Sozopol, 3
Tags: geometry
Given a parallelogram \(ABCD\). Let \(\ell_1\) be the line through \(D\), parallel to \(AC\), and \(\ell_2\) the external bisector of \(\angle ACD\). The lines \(\ell_1\) and \(\ell_2\) intersect at \(E\). The lines \(\ell_1\) and \(AB\) intersect at \(F\), and the line \(\ell_2\) intersects the internal bisector of \(\angle BAC\) at \(X\). The line \(BX\) intersects the circumcircle of triangle \(EFX\) at a second point \(Y\). The internal bisector of \(\angle ACD\) intersects the circumcircle of triangle \(ACX\) at a second point \(Z\). Prove that the quadrilateral \(DXYZ\) is inscribed in a circle.

1993 AMC 8, 3
Tags:
Which of the following numbers has the largest prime factor? $\text{(A)}\ 39 \qquad \text{(B)}\ 51 \qquad \text{(C)}\ 77 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 121$

1994 Kurschak Competition, 3
Tags: floor function , combinatorics
Consider the sets $A_1,A_2,\dots,A_n$. Set $A_k$ is composed of $k$ disjoint intervals on the real axis ($k=1,2,\dots,n$). Prove that from the intervals contained by these sets, one can choose $\left\lfloor\frac{n+1}2\right\rfloor$ intervals such that they belong to pairwise different sets $A_k$, and no two of these intervals have a common point.

2023 AMC 12/AHSME, 1
Tags: word problem
Mrs. Jones is pouring orange juice for her 4 kids into 4 identical glasses. She fills the first 3 full, but only has enough orange juice to fill one third of the last glass. What fraction of a glass of orange juice does she need to pour from the 3 full glasses into the last glass so that all glasses have an equal amount of orange juice? $\textbf{(A) }\frac{1}{12}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{1}{6}\qquad\textbf{(D) }\frac{1}{8}\qquad\textbf{(E) }\frac{2}{9}$

2016 Regional Competition For Advanced Students, 1
Tags: number theory
Determine all positive integers $k$ and $n$ satisfying the equation $$k^2 - 2016 = 3^n$$ (Stephan Wagner)

2006 Tournament of Towns, 4
Tags: cyclic , angle bisector , geometry
Given triangle $ABC, BC$ is extended beyond $B$ to the point $D$ such that $BD = BA$. The bisectors of the exterior angles at vertices $B$ and $C$ intersect at the point $M$. Prove that quadrilateral $ADMC$ is cyclic. (4)

1949-56 Chisinau City MO, 18
Tags: algebra , trinomial
Prove that if the numbers $a, b, c$ are the lengths of the sides of some nondegenerate triangle, then the equation $$b^2x^2 + (b^2 + c^2 - a^2) x + c^2 = 0$$ has imaginary roots.

2019 Switzerland - Final Round, 7
Tags: geometry , equal angles , equal segments , angle
Let $ABC$ be a triangle with $\angle CAB = 2 \angle ABC$. Assume that a point $D$ is inside the triangle $ABC$ exists such that $AD = BD$ and $CD = AC$. Show that $\angle ACB = 3 \angle DCB$.

2004 Pre-Preparation Course Examination, 4
Tags: combinatorics
Let $ G$ be a simple graph. Suppose that size of largest independent set in $ G$ is $ \alpha$. Prove that: a) Vertices of $ G$ can be partitioned to at most $ \alpha$ paths. b) Suppose that a vertex and an edge are also cycles. Prove that vertices of $ G$ can be partitioned to at most $ \alpha$ cycles.

2009 Korea National Olympiad, 4
Tags: combinatorics
There are $n ( \ge 3) $ students in a class. Some students are friends each other, and friendship is always mutual. There are $ s ( \ge 1 ) $ couples of two students who are friends, and $ t ( \ge 1 ) $ triples of three students who are each friends. For two students $ x, y $ define $ d(x,y)$ be the number of students who are both friends with $ x $ and $ y $. Prove that there exist three students $ u, v, w $ who are each friends and satisfying \[ d(u,v) + d(v,w) + d(w,u) \ge \frac{9t}{s} . \]

2020 CCA Math Bonanza, I11
Tags:
Points $C$, $A$, $D$, $M$, $E$, $B$, $F$ lie on a line in that order such that $CA = AD = EB = BF = 1$ and $M$ is the midpoint of $DB$. Let $X$ be a point such that a quarter circle arc exists with center $D$ and endpoints $C$, $X$. Suppose that line $XM$ is tangent to the unit circle centered at $B$. Compute $AB$. [i]2020 CCA Math Bonanza Individual Round #11[/i]

2020 LMT Spring, 6
Tags:
Let $\triangle ABC$ be a triangle such that $AB=6, BC=8,$ and $AC=10$. Let $M$ be the midpoint of $BC$. Circle $\omega$ passes through $A$ and is tangent to $BC$ at $M$. Suppose $\omega$ intersects segments $AB$ and $AC$ again at points $X$ and $Y$, respectively. If the area of $AXY$ can be expressed as $\frac{p}{q}$ where $p, q$ are relatively prime integers, compute $p+q$.

2003 Bosnia and Herzegovina Junior BMO TST, 2
Tags: algebra
Solve in the set of rational numbers the equation $$2\sqrt{3(x + 1)^2} -3 \sqrt{2(y - 2)^2}= 4\sqrt2 + 5|\sqrt2 - \sqrt3|$$

2005 Germany Team Selection Test, 2
Tags: circumcircle , angle bisector , geometry
Let $ABC$ be a triangle satisfying $BC < CA$. Let $P$ be an arbitrary point on the side $AB$ (different from $A$ and $B$), and let the line $CP$ meet the circumcircle of triangle $ABC$ at a point $S$ (apart from the point $C$). Let the circumcircle of triangle $ASP$ meet the line $CA$ at a point $R$ (apart from $A$), and let the circumcircle of triangle $BPS$ meet the line $CB$ at a point $Q$ (apart from $B$). Prove that the excircle of triangle $APR$ at the side $AP$ is identical with the excircle of triangle $PQB$ at the side $PQ$ if and only if the point $S$ is the midpoint of the arc $AB$ on the circumcircle of triangle $ABC$.

2022 VN Math Olympiad For High School Students, Problem 6
Tags: geometry , perpendicular bisector
Let $ABC$ be a triangle with $\angle A,\angle B,\angle C <120^{\circ}$, $T$ is its [i]Fermat-Torricelli[/i] point. Let $G$ be the centroid of $\triangle ABC$. Prove that: the distances from $G$ to the perpendicular bisectors of $TA, TB, TC$ are the same.

2006 Estonia Math Open Junior Contests, 1
Tags: algebra , equation
The paper is written on consecutive integers $1$ through $n$. Then are deleted all numbers ending in $4$ and $9$ and the rest alternating between $-$ and $+$. Finally, an opening parenthesis is added after each character and at the end of the expression the corresponding number of parentheses: $1 - (2 + 3 - (5 + 6 - (7 + 8 - (10 +...))))$. Find all numbers $n$ such that the value of this expression is $13$.

2016 AMC 12/AHSME, 20
Tags:
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A?$ $\textbf{(A)}\ 385 \qquad \textbf{(B)}\ 665 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 1140 \qquad \textbf{(E)}\ 1330$

2009 Junior Balkan MO, 3
Tags: symmetry , inequalities
Let $ x$, $ y$, $ z$ be real numbers such that $ 0 < x,y,z < 1$ and $ xyz \equal{} (1 \minus{} x)(1 \minus{} y)(1 \minus{} z)$. Show that at least one of the numbers $ (1 \minus{} x)y,(1 \minus{} y)z,(1 \minus{} z)x$ is greater than or equal to $ \frac {1}{4}$

2018 Romania Team Selection Tests, 3
Tags: combinatorial geometry , combinatorics
Consider a 4-point configuration in the plane such that every 3 points can be covered by a strip of a unit width. Prove that: 1) the four points can be covered by a strip of length at most $\sqrt2$ and 2)if no strip of length less that $\sqrt2$ covers all the four points, then the points are vertices of a square of length $\sqrt2$

1998 AMC 8, 1
Tags:
For $x=7$, which of the following is the smallest? $ \text{(A)}\ \frac{6}{x}\qquad\text{(B)}\ \frac{6}{x+1}\qquad\text{(C)}\ \frac{6}{x-1}\qquad\text{(D)}\ \frac{x}{6}\qquad\text{(E)}\ \frac{x+1}{6} $