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

2016 LMT, 15
Tags:
For nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Let $g(n)=3^{f(n)}$. Compute \[\sum_{i=1}^{1000} g(i).\] [i]Proposed by Nathan Ramesh

2010 Contests, 4
Tags: inequalities
If $a,b,c\in (0,1)$ satisfy $a+b+c=2$ , prove that $\frac{abc}{(1-a)(1-b)(1-c)}\ge 8$

1956 Miklós Schweitzer, 1
Tags:
[b]1.[/b] Solve without use of determinants the following system of linear equations: $\sum_{j=0}{k} \binom{k+\alpha}{j} x_{k-j} =b_k$ ($k= 0,1, \dots , n$), where $\alpha$ is a fixed real number. [b](A. 7)[/b]

2017 Online Math Open Problems, 10
Tags:
Determine the value of $-1+2+3+4-5-6-7-8-9+...+10000$, where the signs change after each perfect square. [i]Proposed by Michael Ren

2005 Bundeswettbewerb Mathematik, 1
Tags: combinatorics , floor function
Two players $A$ and $B$ have one stone each on a $100 \times 100$ chessboard. They move their stones one after the other, and a move means moving one's stone to a neighbouring field (horizontally or vertically, not diagonally). At the beginning of the game, the stone of $A$ lies in the lower left corner, and the one of $B$ in the lower right corner. Player $A$ starts. Prove: Player $A$ is, independently from that what $B$ does, able to reach, after finitely many steps, the field $B$'s stone is lying on at that moment.

2010 VTRMC, Problem 5
Tags: geometry
Let $A,B$ be two circles in the plane with $B$ inside $A$. Assume that $A$ has radius $3$, $B$ has radius $1$, $P$ is a point on $A$, $Q$ is a point on $B$, and $A$ and $B$ touch so that $P$ and $Q$ are the same point. Suppose that $A$ is kept fixed and $B$ is rolled once round the inside of $A$ so that $Q$ traces out a curve starting and finishing at $P$. What is the area enclosed by this curve? [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOS84LzkwMDBjOTAwODk5M2QyM2IxMGUxZGE5OTI1NWU1ZDYwMDkyYTUwLnBuZw==&rn=VlRSTUMgMjAxMC5wbmc=[/img]

2022 CCA Math Bonanza, L2.1
Tags:
Given that a duck found that $5-2\sqrt{3}i$ is one of the roots of $-259 + 107x - 17x^2 + x^3$, what is the sum of the real parts of the other two roots? [i]2022 CCA Math Bonanza Lightning Round 2.1[/i]

2018 Stanford Mathematics Tournament, 8
Tags: geometry
Let $ABC$ be a right triangle with $\angle ACB = 90^o$, $BC = 16$, and $AC = 12$. Let the angle bisectors of $\angle BAC$ and $\angle ABC$ intersect $BC$ and $AC$ at $D$ and $E$ respectively. Let $AD$ and $BE$ intersect at $I$, and let the circle centered at $I$ passing through $C$ intersect $AB$ at $P$ and $Q$ such that $AQ < AP$. Compute the area of quadrilateral $DP QE$.

2014 ELMO Shortlist, 1
Tags: conic , asymptote , geometry , homothety , circumcircle , geometric transformation
Let $ABC$ be a triangle with symmedian point $K$. Select a point $A_1$ on line $BC$ such that the lines $AB$, $AC$, $A_1K$ and $BC$ are the sides of a cyclic quadrilateral. Define $B_1$ and $C_1$ similarly. Prove that $A_1$, $B_1$, and $C_1$ are collinear. [i]Proposed by Sammy Luo[/i]

2010 All-Russian Olympiad, 1
Tags: combinatorics
There are $24$ different pencils, $4$ different colors, and $6$ pencils of each color. They were given to $6$ children in such a way that each got $4$ pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors. P.S. for 10 grade gives same problem with $40$ pencils, $10$ of each color and $10$ children.

2002 IMC, 8
Tags: probabilistic method , probability
200 students participated in a math contest. They had 6 problems to solve. Each problem was correctly solved by at least 120 participants. Prove that there must be 2 participants such that every problem was solved by at least one of these two students.

2013 Online Math Open Problems, 33
Tags: combinatorial geometry , geometric transformation , reflection , geometry
Let $n$ be a positive integer. E. Chen and E. Chen play a game on the $n^2$ points of an $n \times n$ lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Each point can be marked only once. The game ends when no player can make a move, and the last player to make a move wins. Determine the number of values of $n$ between $1$ and $2013$ (inclusive) for which the first player can guarantee a win, regardless of the moves that the second player makes. [i]Ray Li[/i]

2017 ASDAN Math Tournament, 3
Tags:
For some integers $b$ and $c$, neither of the equations below have real solutions: \begin{align*} 2x^2+bx+c&=0\\ 2x^2+cx+b&=0. \end{align*} What is the largest possible value of $b+c$?

2017 SG Originals, N6
Tags: vieta jumping , number theory
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.

2008 ITest, 93
Tags:
For how many positive integers $n$, $1\leq n\leq 2008$, can the set \[\{1,2,3,\ldots,4n\}\] be divided into $n$ disjoint $4$-element subsets such that every one of the $n$ subsets contains the element which is the arithmetic mean of all the elements in that subset?

2019 Iran Team Selection Test, 2
Tags: combinatorial number theory , number theory , combinatorics
Hesam chose $10$ distinct positive integers and he gave all pairwise $\gcd$'s and pairwise ${\text lcm}$'s (a total of $90$ numbers) to Masoud. Can Masoud always find the first $10$ numbers, just by knowing these $90$ numbers? [i]Proposed by Morteza Saghafian [/i]

2019 BMT Spring, 5
Tags:
Let $ 2^{1110} \equiv n \bmod{1111} $ with $ 0 \leq n < 1111 $. Compute $ n $.

2024 HMNT, 4
Tags: guts
The number $17^6$ when written out in base $10$ contains $8$ distinct digits from $1,2,\ldots,9,$ with no repeated digits or zeroes. Compute the missing nonzero digit.

1967 German National Olympiad, 5
Tags: diophantine equation , diophantine , number theory
For each natural number $n$, determine the number $A(n)$ of all integer nonnegative solutions the equation $$5x + 2y + z = 10n.$$

Kvant 2022, M2724
Tags: sum of digits , number theory
In an infinite arithmetic progression of positive integers there are two integers with the same sum of digits. Will there necessarily be one more integer in the progression with the same sum of digits? [i]Proposed by A. Shapovalov[/i]

1996 Estonia National Olympiad, 1
Tags: sum , product , diophantine equation , diophantine , number theory
Find all pairs of integers $(x, y)$ such that ths sum of the fractions $\frac{19}{x}$ and $\frac{96}{y}$ would be equal to their product.

2024 Iranian Geometry Olympiad, 5
Tags: geometry
Points $Y,Z$ lie on the smaller arc $BC$ of the circumcircle of an acute triangle $\bigtriangleup ABC$ ($Y$ lies on the smaller arc $BZ$). Let $X$ be a point such that the triangles $\bigtriangleup ABC,\bigtriangleup XYZ$ are similar (in this exact order) with $A,X$ lying on the same side of $YZ$. Lines $XY,XZ$ intersect sides $AB,AC$ at points $E,F$ respectively. Let $K$ be the intersection of lines $BY,CZ$. Prove that one of the intersections of the circumcircles of triangles $\bigtriangleup AEF,\bigtriangleup KBC$ lie on the line $KX$. [i]Proposed by Amirparsa Hosseini Nayeri - Iran[/i]

III Soros Olympiad 1996 - 97 (Russia), 10.6
Tags: analytic geometry , algebra
Find $m$ and $n$ such that the set of points whose coordinates $x$ and $y$ satisfy the equation $|y-2x|=x$, coincides with the set of points specified by the equation $|mx + ny| = y$.

2013 Taiwan TST Round 1, 1
Tags: circumcircle , rhombus , geometry
Let $\Delta ABC$ be a triangle with $AB=AC$ and $\angle A = \alpha$, and let $O,H$ be its circumcenter and orthocenter, respectively. If $P,Q$ are points on $AB$ and $AC$, respectively, such that $APHQ$ forms a rhombus, determine $\angle POQ$ in terms of $\alpha$.

1958 Polish MO Finals, 5
Tags: 3d geometry , tetrahedron , geometry
Prove the theorem: In a tetrahedron, the plane bisector of any dihedral angle divides the opposite edge into segments proportional to the areas of the tetrahedron faces that form this dihedral angle.