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 China Girls Math Olympiad, 2
Tags: integer , algebra , number theory
Find integers $a_1,a_2,\cdots,a_{18}$, s.t. $a_1=1,a_2=2,a_{18}=2019$, and for all $3\le k\le 18$, there exists $1\le i<j<k$ with $a_k=a_i+a_j$.

2010 Junior Balkan MO, 2
Tags: induction , inequalities , prime factorization , number theory
Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.

2010 Benelux, 3
Tags: geometry
On a line $l$ there are three different points $A$, $B$ and $P$ in that order. Let $a$ be the line through $A$ perpendicular to $l$, and let $b$ be the line through $B$ perpendicular to $l$. A line through $P$, not coinciding with $l$, intersects $a$ in $Q$ and $b$ in $R$. The line through $A$ perpendicular to $BQ$ intersects $BQ$ in $L$ and $BR$ in $T$. The line through $B$ perpendicular to $AR$ intersects $AR$ in $K$ and $AQ$ in $S$. (a) Prove that $P$, $T$, $S$ are collinear. (b) Prove that $P$, $K$, $L$ are collinear. [i](2nd Benelux Mathematical Olympiad 2010, Problem 3)[/i]

2017 Romania Team Selection Test, P1
Tags: geometry , circumcircle
Let $ABC$ be a triangle with $AB<AC$, let $G,H$ be its centroid and otrhocenter. Let $D$ be the otrhogonal projection of $A$ on the line $BC$, and let $M$ be the midpoint of the side $BC$. The circumcircle of $ABC$ crosses the ray $HM$ emanating from $M$ at $P$ and the ray $DG$ emanating from $D$ at $Q$, outside the segment $DG$. Show that the lines $DP$ and $MQ$ meet on the circumcircle of $ABC$.

2023 HMNT, 7
Tags: system of equations , algebra
Compute all ordered triples $(x, y, z)$ of real numbers satisfying the following system of equations: $$xy + z = 40$$ $$xz + y = 51$$ $$x + y + z = 19.$$

2015 BMT Spring, P2
Tags: geometry
Suppose that fixed circle $C_1$ with radius $a > 0$ is tangent to the fixed line $\ell$ at $A$. Variable circle $C_2$, with center $X$, is externally tangent to $C_1$ at $B \ne A$ and $\ell$ at $C$. Prove that the set of all $X$ is a parabola minus a point

2006 Germany Team Selection Test, 2
Tags: algebra , functional equation
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that $ f(x+y)+f(x)f(y)=f(xy)+2xy+1$ for all real numbers $ x$ and $ y$. [i]Proposed by B.J. Venkatachala, India[/i]

2001 Finnish National High School Mathematics Competition, 2
Tags: parabola , geometry , perpendicular bisector , algebra , conic
Equations of non-intersecting curves are $y = ax^2 + bx + c$ and $y = dx^2 + ex + f$ where $ad < 0.$ Prove that there is a line of the plane which does not meet either of the curves.

2013 China Team Selection Test, 3
Tags: geometry , parallelogram , combinatorial geometry , combinatorics
Let $A$ be a set consisting of 6 points in the plane. denoted $n(A)$ as the number of the unit circles which meet at least three points of $A$. Find the maximum of $n(A)$

1999 National Olympiad First Round, 2
Tags:
How many ordered integer pairs $ \left(x,y\right)$ are there such that $ xy \equal{} 4\left(y^{2} \plus{} x\right)$? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ \text{None}$

2017 QEDMO 15th, 7
Tags: algebra , system of equations , system
Find all real solutions $x, y$ of the system of equations $$\begin{cases} x + \dfrac{3x-y}{x^2 + y^2} = 3 \\ \\ y-\dfrac{x + 3y}{x^2 + y^2} = 0 \end{cases}$$

2013 IMO Shortlist, A4
Tags: algebra , inequalities , sequence
Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]

1983 National High School Mathematics League, 9
Tags: trigonometry
In $\triangle ABC,\sin A=\frac{3}{5},\cos B=\frac{5}{13}$, then $\cos C=$________.

2016 Serbia National Math Olympiad, 4
Tags: geometry , incenter
Let $ABC $be a triangle, and $I $ the incenter, $M $ midpoint of $ BC $, $ D $ the touch point of incircle and $ BC $. Prove that perpendiculars from $M, D, A $ to $AI, IM, BC $ respectively are concurrent

2008 Turkey Junior National Olympiad, 3
Tags:
There are $24$ cups on a table. In the beginning, only three of them placed upside-down. At each step, we are turning four cups. Can we turn all the cups right-side up in at most $100$ steps?

2019 Tournament Of Towns, 6
Tags: combinatorics
Peter has several $100$ ruble notes and no other money. He starts buying books; each book costs a positive integer number of rubles, and he gets change in $1$ ruble coins. Whenever Peter is buying an expensive book for $100$ rubles or higher he uses only $100$ ruble notes in the minimum necessary number. Whenever he is buying a cheap one (for less than $100$ rubles) he uses his coins if he has enough, otherwise using a $100$ ruble note. When the $100$ ruble notes have come to the end, Peter has expended exactly a half of his money. Is it possible that he has expended $5000$ rubles or more? (Tatiana Kazitsina)

2023 LMT Fall, 13
Tags: number theory
Given that the base-$17$ integer $\overline{8323a02421_{17}}$ (where a is a base-$17$ digit) is divisible by $\overline{16_{10}}$, find $a$. Express your answer in base $10$. [i]Proposed by Jonathan Liu[/i]

2021/2022 Tournament of Towns, P5
Tags: combinatorics , process
Consider the segment $[0; 1]$. At each step we may split one of the available segments into two new segments and write the product of lengths of these two new segments onto a blackboard. Prove that the sum of the numbers on the blackboard never will exceed $1/2$. [i]Mikhail Lukin[/i]

Swiss NMO - geometry, 2016.5
Tags: equal angles , geometry , concurrency
Let $ABC$ be a right triangle with $\angle ACB = 90^o$ and M the center of $AB$. Let $G$ br any point on the line $MC$ and $P$ a point on the line $AG$, such that $\angle CPA = \angle BAC$ . Further let $Q$ be a point on the straight line $BG$, such that $\angle BQC = \angle CBA$ . Show that the circles of the triangles $AQG$ and $BPG$ intersect on the segment $AB$.

2011 Today's Calculation Of Integral, 724
Tags: calculus , integration , limit , calculus computations , logarithm
Find $\lim_{n\to\infty}\left\{\left(1+n\right)^{\frac{1}{n}}\left(1+\frac{n}{2}\right)^{\frac{2}{n}}\left(1+\frac{n}{3}\right)^{\frac{3}{n}}\cdots\cdots 2\right\}^{\frac{1}{n}}$.

KoMaL A Problems 2020/2021, A. 782
Tags: graph theory , combinatorics
Prove that the edges of a simple planar graph can always be oriented such that the outdegree of all vertices is at most three. [i]UK Competition Problem[/i]

2005 France Pre-TST, 1
Tags: geometry , incenter , circumcircle , geometric transformation , reflection , ratio , dilation
Let $I$ be the incenter of the triangle $ABC$, et let $A',B',C'$ be the symmetric of $I$ with respect to the lines $BC,CA,AB$ respectively. It is known that $B$ belongs to the circumcircle of $A'B'C'$. Find $\widehat {ABC}$. Pierre.

2013 Switzerland - Final Round, 3
Tags: geometry , projection , equal angles
Let $ABCD$ be a cyclic quadrilateral with $\angle ADC = \angle DBA$. Furthermore, let $E$ be the projection of $A$ on $BD$. Show that $BC = DE - BE$ .

2024 AIME, 8
Tags: geometry
Eight circles of radius $34$ can be placed tangent to side $\overline{BC}$ of $\triangle ABC$ such that the first circle is tangent to $\overline{AB}$, subsequent circles are externally tangent to each other, and the last is tangent to $\overline{AC}$. Similarly, $2024$ circles of radius $1$ can also be placed along $\overline{BC}$ in this manner. The inradius of $\triangle ABC$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2020 CMIMC Geometry, 5
Tags: geometry
For every positive integer $k$, let $\mathbf{T}_k = (k(k+1), 0)$, and define $\mathcal{H}_k$ as the homothety centered at $\mathbf{T}_k$ with ratio $\tfrac{1}{2}$ if $k$ is odd and $\tfrac{2}{3}$ is $k$ is even. Suppose $P = (x,y)$ is a point such that $$(\mathcal{H}_{4} \circ \mathcal{H}_{3} \circ \mathcal{H}_2 \circ \mathcal{H}_1)(P) = (20, 20).$$ What is $x+y$? (A [i]homothety[/i] $\mathcal{H}$ with nonzero ratio $r$ centered at a point $P$ maps each point $X$ to the point $Y$ on ray $\overrightarrow{PX}$ such that $PY = rPX$.)