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

2015 District Olympiad, 4
Tags: real analysis , epsilon-delta , contest , sequence
Let $ \left( x_n\right)_{n\ge 1} $ be a sequence of real numbers of the interval $ [1,\infty) . $ Suppose that the sequence $ \left( \left[ x_n^k\right]\right)_{n\ge 1} $ is convergent for all natural numbers $ k. $ Prove that $ \left( x_n\right)_{n\ge 1} $ is convergent. Here, $ [\beta ] $ means the greatest integer smaller than $ \beta . $

1955 AMC 12/AHSME, 6
Tags:
A merchant buys a number of oranges at $ 3$ for $ 10$ cents and an equal number at $ 5$ for $ 20$ cents. To "break even" he must sell all at: $ \textbf{(A)}\ \text{8 for 30 cents} \qquad \textbf{(B)}\ \text{3 for 11 cents} \qquad \textbf{(C)}\ \text{5 for 18 cents} \\ \textbf{(D)}\ \text{11 for 40 cents} \qquad \textbf{(E)}\ \text{13 for 50 cents}$

2020 Harvard-MIT Mathematics Tournament, 5
Tags:
A positive integer $N$ is \emph{piquant} if there exists a positive integer $m$ such that if $n_i$ denotes the number of digits in $m^i$ (in base $10$), then $n_1+n_2+\cdots + n_{10}=N$. Let $p_M$ denote the fraction of the first $M$ positive integers that are piquant. Find $\lim\limits_{M\to \infty} p_M$. [i]Proposed by James Lin.[/i]

1997 Akdeniz University MO, 1
Tags: equation , number theory
Prove that, $$15x^2-7y^2=9$$ equation has any solutions in integers.

1997 National High School Mathematics League, 5
Tags: function , trigonometry
Let $f(x)=x^2-\pi x$, $\alpha=\arcsin\frac{1}{3},\beta=\arctan\frac{5}{4},\gamma=\arccos\left(-\frac{1}{3}\right),\delta=\text{arccot}\left(-\frac{5}{4}\right)$ $\text{(A)}f(\alpha)>f(\beta)>f(\delta)>f(\gamma)$ $\text{(B)}f(\alpha)>f(\delta)>f(\beta)>f(\gamma)$ $\text{(C)}f(\delta)>f(\alpha)>f(\beta)>f(\gamma)$ $\text{(D)}f(\delta)>f(\alpha)>f(\gamma)>f(\beta)$

2020 Online Math Open Problems, 22
Tags:
Three points $P_1, P_2,$ and $P_3$ and three lines $\ell_1, \ell_2,$ and $\ell_3$ lie in the plane such that none of the three points lie on any of the three lines. For (not necessarily distinct) integers $i$ and $j$ between 1 and 3 inclusive, we call a line $\ell$ $(i, j)$-[i]good[/i] if the reflection of $P_i$ across $\ell$ lies on $\ell_j$, and call it [i]excellent[/i] if there are two distinct pairs $(i_1, j_1)$ and $(i_2, j_2)$ for which it is good. Suppose that exactly $N$ excellent lines exist. Compute the largest possible value of $N$. [i]Proposed by Yannick Yao[/i]

2018 Latvia Baltic Way TST, P15
Tags: system of equations , number theory , algebra
Determine whether there exists a positive integer $n$ such that it is possible to find at least $2018$ different quadruples $(x,y,z,t)$ of positive integers that simultaneously satisfy equations $$\begin{cases} x+y+z=n\\ xyz = 2t^3. \end{cases}$$

1997 AIME Problems, 7
Tags: analytic geometry , graphing lines , slope , calculus , conic
A car travels due east at $\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is 51 miles, moves southeast at $\frac 12\sqrt{2}$ mile per minute. At time $t=0,$ the center of the storm is 110 miles due north of the car. At time $t=t_1$ minutes, the car enters the storm circle, and at time $t=t_2$ minutes, the car leaves the storm circle. Find $\frac 12(t_1+t_2).$

2020 Bulgaria EGMO TST, 1
Tags: inequalities
The positive integers $a$, $p$, $q$ and $r$ are greater than $1$ and are such that $p$ divides $aqr+1$, $q$ divides $apr+1$ and $r$ divides $apq+1$. Prove that: a) There are infinitely many such quadruples $(a,p,q,r)$. b) For each such quadruple we have $a\geq \frac{pqr-1}{pq+qr+rp}$.

2002 Italy TST, 3
Tags: function , algebra
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ which satisfy the following conditions: $(\text{i})$ $f(x+f(y))=f(x)f(y)$ for all $x,y>0;$ $(\text{ii})$ there are at most finitely many $x$ with $f(x)=1$.

2023 pOMA, 1
Tags: permutation , combinatorics
Let $n$ be a positive integer. Marc has $2n$ boxes, and in particular, he has one box filled with $k$ apples for each $k=1,2,3,\ldots,2n$. Every day, Marc opens a box and eats all the apples in it. However, if he eats strictly more than $2n+1$ apples in two consecutive days, he gets stomach ache. Prove that Marc has exactly $2^n$ distinct ways of choosing the boxes so that he eats all the apples but doesn't get stomach ache.

2008 South africa National Olympiad, 2
Tags: geometry
Let $ABCD$ be a convex quadrilateral with the property that $AB$ extended and $CD$ extended intersect at a right angle. Prove that $AC\cdot BD>AD\cdot BC$.

2020 AMC 10, 2
Tags: arithmetic mean
The numbers $3, 5, 7, a,$ and $b$ have an average (arithmetic mean) of $15$. What is the average of $a$ and $b$? $\textbf{(A) } 0 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 30 \qquad\textbf{(D) } 45 \qquad\textbf{(E) } 60$

2018 CHMMC (Fall), 4
Tags: algebra , polynomial
Find the sum of the real roots of $f(x) = x^4 + 9x^3 + 18x^2 + 18x + 4$.

2018 ASDAN Math Tournament, 3
Tags:
In a bag are all natural numbers less than or equal to $999$ whose digits sum to $6$. What is the probability of drawing a number from the bag that is divisible by $11$?

2013 Mediterranean Mathematics Olympiad, 2
Tags: induction , combinatorics
Determine the least integer $k$ for which the following story could hold true: In a chess tournament with $24$ players, every pair of players plays at least $2$ and at most $k$ games against each other. At the end of the tournament, it turns out that every player has played a different number of games.

2019 Dutch IMO TST, 1
Tags: trinomial , quadratic trinomial , algebra , polynomial
Let $P(x)$ be a quadratic polynomial with two distinct real roots. For all real numbers $a$ and $b$ satisfying $|a|,|b| \ge 2017$, we have $P(a^2+b^2) \ge P(2ab)$. Show that at least one of the roots of $P$ is negative.

2015 JBMO TST - Turkey, 2
Tags: geometry
Let $ABCD$ be a convex quadrilateral and let $\omega$ be a circle tangent to the lines $AB$ and $BC$ at points $A$ and $C$, respectively. $\omega$ intersects the line segments $AD$ and $CD$ again at $E$ and $F$, respectively, which are both different from $D$. Let $G$ be the point of intersection of the lines $AF$ and $CE$. Given $\angle ACB=\angle GDC+\angle ACE$, prove that the line $AD$ is tangent to th circumcircle of the triangle $AGB$.

1992 Turkey Team Selection Test, 3
Tags: inequalities
$x_1, x_2,\cdots,x_{n+1}$ are posive real numbers satisfying the equation $\frac{1}{1+x_1} + \frac{1}{1+x_2} + \cdots + \frac{1}{1+x_{n+1}} =1$ Prove that $x_1x_2 \cdots x_{n+1} \geq n^{n+1}$.

1961 AMC 12/AHSME, 27
Tags:
Given two equiangular polygons $P_1$ and $P_2$ with different numbers of sides; each angle of $P_1$ is $x$ degrees and each angle of $P_2$ is $kx$ degrees, where $k$ is an integer greater than $1$. The number of possibilities for the pair $(x, k)$ is: ${{ \textbf{(A)}\ \text{infinite} \qquad\textbf{(B)}\ \text{finite, but greater than 2} \qquad\textbf{(C)}\ \text{Two} \qquad\textbf{(D)}\ \text{One} }\qquad\textbf{(E)}\ \text{Zero} } $

2024 IFYM, Sozopol, 2
Tags: geometry
Let \( n \geq 3 \) be an integer. For every two adjacent vertices \( A \) and \( B \) of a convex \( n \)-gon, we find a vertex \( C \) such that the angle \( \angle ACB \) is the largest, and write down the measure in degrees. Find the smallest possible value of the sum of the written \( n \) numbers.

1993 China Team Selection Test, 3
Tags: geometry , circumcircle , incenter , trigonometry , geometric transformation , reflection , ring theory
Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.

2019 China Team Selection Test, 3
Tags: algebra
Find all positive integer $n$, such that there exists $n$ points $P_1,\ldots,P_n$ on the unit circle , satisfying the condition that for any point $M$ on the unit circle, $\sum_{i=1}^n MP_i^k$ is a fixed value for \\a) $k=2018$ \\b) $k=2019$.

2021 Kyiv Mathematical Festival, 2
Tags: inequalities , algebra
Let $a,b,c\ge0$ and $a+b+c=3.$ Prove that $(3a-bc)(3b-ac)(3c-ab)\le8.$ (O. Rudenko)

2016 China Northern MO, 6
Tags: geometry
Four points $B,E,A,F$ lie on line $AB$ in order, four points $C,G,D,H$ lie on line $CD$ in order, satisfying: $$\frac{AE}{EB}=\frac{AF}{FB}=\frac{DG}{GC}=\frac{DH}{HC}=\frac{AD}{BC}.$$ Prove that $FH\perp EG$.