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

Kvant 2023, M2745
Tags: combinatorics , binary sequences
Two 100-digit binary sequences are given. In one operation, one may insert (possibly at the beggining or end) or remove one or more identical digits from a sequence. What is the smallest $k{}$ for which we can transform the first sequence into the second one in no more than $k{}$ operations? [i]Proposed by V. Novikov[/i]

Oliforum Contest I 2008, 2
Tags: algebra
Let $ a_1,a_2,...,a_n$ with arithmetic mean equals zero; what is the value of: $ \sum_{j=1}^n{\frac{1}{a_j(a_j+a_{j+1})(a_j+a_{j+1}+a_{j+2})...(a_j+a_{j+1}+...+a_{j+n-2})}}$ , where $ a_{n+k}=a_k$ ?

2023 Novosibirsk Oral Olympiad in Geometry, 1
Tags: geometry , tiles , combinatorial geometry
Let's call a corner the figure that is obtained by removing one cell from a $2 \times 2$ square. Cut the $6 \times 6$ square into corners so that no two of them form a $2 \times 3$ or $3 \times 2$ rectangle together.

1994 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 5
Tags: geometry , 3d geometry , rotation , symmetry
In how many ways can you color the six sides of a cube in black or white? (Do note that the cube is unchanged when rotated?) A. 7 B. 10 C. 20 D. 30 E. 36

1992 Mexico National Olympiad, 4
Tags: number theory , divisor
Show that $1 + 11^{11} + 111^{111} + 1111^{1111} +...+ 1111111111^{1111111111}$ is divisible by $100$.

1986 National High School Mathematics League, 6
Tags: geometry , circumcircle
Area of $\triangle ABC$ is $\frac{1}{4}$, circumradius of $\triangle ABC$ is $1$. Let $s=\sqrt{a}+\sqrt{b}+\sqrt{c},t=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$, then $\text{(A)}s>t\qquad\text{(B)}s=t\qquad\text{(C)}s<t\qquad\text{(D)}s>t$

Russian TST 2016, P1
Tags: algebra , polynomial , inequality
For which even natural numbers $d{}$ does there exists a constant $\lambda>0$ such that any reduced polynomial $f(x)$ of degree $d{}$ with integer coefficients that does not have real roots satisfies the inequality $f(x) > \lambda$ for all real numbers?

2011 China Team Selection Test, 3
Tags: combinatorics
For a given integer $n\ge 2$, let $a_0,a_1,\ldots ,a_n$ be integers satisfying $0=a_0<a_1<\ldots <a_n=2n-1$. Find the smallest possible number of elements in the set $\{ a_i+a_j \mid 0\le i \le j \le n \}$.

2015 AMC 10, 1
Tags: modular arithmetic , function
What is the value of $2-(-2)^{-2}$? $ \textbf{(A) } -2 \qquad\textbf{(B) } \dfrac{1}{16} \qquad\textbf{(C) } \dfrac{7}{4} \qquad\textbf{(D) } \dfrac{9}{4} \qquad\textbf{(E) } 6 $

2017 Romania National Olympiad, 1
Tags: limit , function , real analysis , definite integral , calculus , integration , sequence
[b]a)[/b] Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{>0} . $ Show that there exists a natural number $ n_0 $ and a sequence of positive real numbers $ \left( x_n \right)_{n>n_0} $ that satisfy the following relation. $$ n\int_0^{x_n} f(t)dt=1,\quad n_0<\forall n\in\mathbb{N} $$ [b]b)[/b] Prove that the sequence $ \left( nx_n \right)_{n> n_0} $ is convergent and find its limit.

1989 Romania Team Selection Test, 1
Tags: functional , functional equation , algebra
Let $F$ be the set of all functions $f : N \to N$ which satisfy $f(f(x))-2 f(x)+x = 0$ for all $x \in N$. Determine the set $A =\{ f(1989) | f \in F\}$.

1996 USAMO, 5
Tags: isosceles triangle , trigonometry , law of cosines , law of sines , reflection , geometry
Let $ABC$ be a triangle, and $M$ an interior point such that $\angle MAB=10^\circ$, $\angle MBA=20^\circ$, $\angle MAC=40^\circ$ and $\angle MCA=30^\circ$. Prove that the triangle is isosceles.

1989 Brazil National Olympiad, 3
Tags: function , algebra
A function $f$, defined for the set of integers, is such that $f(x)=x-10$ if $x>100$ and $f(x)=f(f(x+11))$ if $x \leq 100$. Determine, justifying your answer, the set of all possible values for $f$.

1988 India National Olympiad, 4
Tags: function , inequalities
If $ a$ and $ b$ are positive and $ a \plus{} b \equal{} 1$, prove that \[ \left(a\plus{}\frac{1}{a}\right)^2\plus{}\left(b\plus{}\frac{1}{b}\right)^2 \geq \frac{25}{2}\]

2003 IMO, 4
Tags: geometry , angle bisector , cyclic quadrilateral , quadrilateral , concurrency
Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

2024 Sharygin Geometry Olympiad, 10.5
Tags: geo , geometry
The incircle of a right-angled triangle $ABC$ touches the hypothenuse $AB$ at point $T$. The squares $ATMP$ and $BTNQ$ lie outside the triangle. Prove that the areas of triangles $ABC$ and $TPQ$ are equal.

2017 IFYM, Sozopol, 8
Tags: geometry
$k$ is the circumscribed circle of $\Delta ABC$. $M$ and $N$ are arbitrary points on sides $CA$ and $CB$, and $MN$ intersects $k$ in points $U$ and $V$. Prove that the middle points of $BM$,$AN$,$MN$, and $UV$ lie on one circle.

2007 Baltic Way, 20
Tags: geometry , search , function , number theory
Let $a$ and $b$ be positive integers, $b<a$, such that $a^3+b^3+ab$ is divisible by $ab(a-b)$. Prove that $ab$ is a perfect cube.

KoMaL A Problems 2018/2019, A. 754
Tags: geometry
Let $P$ be a point inside the acute triangle $ABC,$ and let $Q$ be the isogonal conjugate of $P.$ Let $L,M$ and $N$ be the midpoints of the shorter arcs $BC,CA$ and $AB$ of the circumcircle of $ABC,$ respectively. Let $X_A$ be the intersection of ray $LQ$ and circle $(PBC),$ let $X_B$ be the intersection of ray $MQ$ and circle $PCA,$ and let $X_C$ be the intersection of ray $NQ$ and circle $(PAB).$ Prove that $P,X_A,X_B$ and $X_C$ are concyclic or coincide. [i]Proposed by Gustavo Cruz (São Paulo)[/i]

1971 IMO, 3
Tags: linear algebra , matrix , algebra , combinatorial inequality , combinatorics
Let $ A \equal{} (a_{ij})$, where $ i,j \equal{} 1,2,\ldots,n$, be a square matrix with all $ a_{ij}$ non-negative integers. For each $ i,j$ such that $ a_{ij} \equal{} 0$, the sum of the elements in the $ i$th row and the $ j$th column is at least $ n$. Prove that the sum of all the elements in the matrix is at least $ \frac {n^2}{2}$.

1981 Bulgaria National Olympiad, Problem 2
Tags: geometry , angle , triangle
Let $ABC$ be a triangle such that the altitude $CH$ and the sides $CA,CB$ are respectively equal to a side and two distinct diagonals of a regular heptagon. Prove that $\angle ACB<120^\circ$.

1992 French Mathematical Olympiad, Problem 5
Tags: number theory , floor function
Determine the number of digits $1$ in the integer part of $\frac{10^{1992}}{10^{83}+7}$.

2013 All-Russian Olympiad, 1
Tags: algebra
Given three distinct real numbers $a$, $b$, and $c$, show that at least two of the three following equations \[(x-a)(x-b)=x-c\] \[(x-c)(x-b)=x-a\] \[(x-c)(x-a)=x-b\] have real solutions.

2014 Harvard-MIT Mathematics Tournament, 1
Tags: hmmt , geometry , analytic geometry , symmetry , pythagorean theorem , power of a point
Let $O_1$ and $O_2$ be concentric circles with radii 4 and 6, respectively. A chord $AB$ is drawn in $O_1$ with length $2$. Extend $AB$ to intersect $O_2$ in points $C$ and $D$. Find $CD$.

2007 Hungary-Israel Binational, 2
Tags: ellipse , geometry , 3d geometry , sphere , geometric transformation , rotation , conic
Given is an ellipse $ e$ in the plane. Find the locus of all points $ P$ in space such that the cone of apex $ P$ and directrix $ e$ is a right circular cone.