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

2012 Turkey Junior National Olympiad, 3
Tags: inequalities
Let $a, b, c$ be positive real numbers satisfying $a^3+b^3+c^3=a^4+b^4+c^4$. Show that \[ \frac{a}{a^2+b^3+c^3}+\frac{b}{a^3+b^2+c^3}+\frac{c}{a^3+b^3+c^2} \geq 1 \]

1971 Spain Mathematical Olympiad, 5
Tags: complex number , limit , analysis
Prove that whatever the complex number $z$ is, it is true that $$(1 + z^{2^n})(1-z^{2^n})= 1- z^{2^{n+1}}.$$ Writing the equalities that result from giving $n$ the values $0, 1, 2, . . .$ and multiplying them, show that for $|z| < 1$ holds $$\frac{1}{1-z}= \lim_{k\to \infty}(1 + z)(1 + z^2)(1 + z^{2^2})...(1 + z^{2^k}).$$

2012 Canadian Mathematical Olympiad Qualification Repechage, 6
Tags: calculus , derivative , quadratic , algebra
Determine whether there exist two real numbers $a$ and $b$ such that both $(x-a)^3+ (x-b)^2+x$ and $(x-b)^3 + (x-a)^2 +x$ contain only real roots.

Brazil L2 Finals (OBM) - geometry, 2002.5
Tags: circumcircle , isosceles , inscribed , perpendicular , geometry
Let $ABC$ be a triangle inscribed in a circle of center $O$ and $P$ be a point on the arc $AB$, that does not contain $C$. The perpendicular drawn fom $P$ on line $BO$ intersects $AB$ at $S$ and $BC$ at $T$. The perpendicular drawn from $P$ on line $AO$ intersects $AB$ at $Q$ and $AC$ at $R$. Prove that: a) $PQS$ is an isosceles triangle b) $PQ^2=QR= ST$

2010 ELMO Shortlist, 4
Tags: algebra , polynomial , floor function , number theory
Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$. [i]Evan O' Dorney.[/i]

2019 HMNT, 3
Tags: combinatorics
Katie has a fair $2019$-sided die with sides labeled $1, 2,..., 2019$. After each roll, she replaces her $n$-sided die with an $(n+1)$-sided die having the $n$ sides of her previous die and an additional side with the number she just rolled. What is the probability that Katie's $2019^{th}$ roll is a $ 2019$?

2015 Estonia Team Selection Test, 7
Tags: number theory , digit , consecutive , power
Prove that for every prime number $p$ and positive integer $a$, there exists a natural number $n$ such that $p^n$ contains $a$ consecutive equal digits.

2005 Oral Moscow Geometry Olympiad, 3
Tags: geometry , geometric inequality , equal segments
In triangle $ABC$, points $K ,P$ are chosen on the side $AB$ so that $AK = BL$, and points $M,N$ are chosen on the side $BC$ so that $CN = BM$. Prove that $KN + LM \ge AC$. (I. Bogdanov)

2024 HMNT, 32
Tags: guts
Let $ABC$ be an acute triangle and $D$ be the foot of altitude from $A$ to $BC.$ Let $X$ and $Y$ be points on the segment $BC$ such that $\angle{BAX} = \angle{YAC}, BX = 2, XY = 6,$ and $YC = 3.$ Given that $AD = 12,$ compute $BD.$

2023 USAMO, 1
Tags: geometry
In an acute triangle $ABC$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $C$ to $AM$. Suppose that the circumcircle of triangle $ABP$ intersects line $BC$ at two distinct points $B$ and $Q$. Let $N$ be the midpoint of $\overline{AQ}$. Prove that $NB=NC$. [i]Proposed by Holden Mui[/i]

2017-2018 SDML (Middle School), 1
Tags:
Let $N = \frac{1}{3} + \frac{3}{5} + \frac{5}{7} + \frac{7}{9} + \frac{9}{11}$. What is the greatest integer which is less than $N$?

1962 AMC 12/AHSME, 17
Tags: logarithm
If $ a \equal{} \log_8 225$ and $ b \equal{} \log_2 15,$ then $ a$, in terms of $ b,$ is: $ \textbf{(A)}\ \frac{b}{2} \qquad \textbf{(B)}\ \frac{2b}{3}\qquad \textbf{(C)}\ b \qquad \textbf{(D)}\ \frac{3b}{2} \qquad \textbf{(E)}\ 2b$

2010 Switzerland - Final Round, 6
Tags: function , algebra
Find all functions $ f: \mathbb{R}\mapsto\mathbb{R}$ such that for all $ x$, $ y$ $ \in\mathbb{R}$, \[ f(f(x))\plus{}f(f(y))\equal{}2y\plus{}f(x\minus{}y)\] holds.

2009 Jozsef Wildt International Math Competition, W. 8
Tags: combinatorics , algebra
If $n,p,q \in \mathbb{N}, p<q $ then $${{(p+q)n}\choose{n}} \sum \limits_{k=0}^n (-1)^k {{n}\choose{k}} {{(p+q-1)n}\choose{pn-k}}= {{(p+q)n}\choose{pn}} \sum \limits_{k=0}^{\left [\frac{n}{2} \right ]} (-1)^k {{pn}\choose{k}} {{(q-p)n}\choose{n-2k}} $$

2013 Greece National Olympiad, 1
Tags: induction , algebra
Let the sequence of real numbers $(a_n),n=1,2,3...$ with $a_1=2$ and $a_n=\left(\frac{n+1}{n-1} \right)\left(a_1+a_2+...+a_{n-1} \right),n\geq 2$. Find the term $a_{2013}$.

Putnam 1938, B3
Tags:
A horizontal disk diameter $3$ inches rotates once every $15$ seconds. An insect starts at the southernmost point of the disk facing due north. Always facing due north, it crawls over the disk at $1$ inch per second. Where does it again reach the edge of the disk?

2018 USA TSTST, 2
Tags: combinatorics , graph theory , matching
In the nation of Onewaynia, certain pairs of cities are connected by one-way roads. Every road connects exactly two cities (roads are allowed to cross each other, e.g., via bridges), and each pair of cities has at most one road between them. Moreover, every city has exactly two roads leaving it and exactly two roads entering it. We wish to close half the roads of Onewaynia in such a way that every city has exactly one road leaving it and exactly one road entering it. Show that the number of ways to do so is a power of $2$ greater than $1$ (i.e.\ of the form $2^n$ for some integer $n \ge 1$). [i]Victor Wang[/i]

2022 JBMO Shortlist, G2
Tags: parallelogram , geometry , shortlist , concentric
Let $ABC$ be a triangle with circumcircle $k$. The points $A_1, B_1,$ and $C_1$ on $k$ are the midpoints of arcs $\widehat{BC}$ (not containing $A$), $\widehat{AC}$ (not containing $B$), and $\widehat{AB}$ (not containing $C$), respectively. The pairwise distinct points $A_2, B_2,$ and $C_2$ are chosen such that the quadrilaterals $AB_1A_2C_1, BA_1B_2C_1,$ and $CA_1C_2B_1$ are parallelograms. Prove that $k$ and the circumcircle of triangle $A_2B_2C_2$ have a common center. [b]Comment.[/b] Point $A_2$ can also be defined as the reflection of $A$ with respect to the midpoint of $B_1C_1$, and analogous definitions can be used for $B_2$ and $C_2$.

2014 Contests, 1
Tags: inequalities , induction , algebra
Prove that for $n\ge 2$ the following inequality holds: $$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$

2014 AMC 10, 3
Tags:
Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip? $ \textbf {(A) } 30 \qquad \textbf {(B) } \frac{400}{11} \qquad \textbf {(C) } \frac{75}{2} \qquad \textbf {(D) } 40 \qquad \textbf {(E) } \frac{300}{7} $

2009 Turkey Team Selection Test, 1
Tags: polynomial , abstract algebra , number theory , prime number , algebra
For which $ p$ prime numbers, there is an integer root of the polynominal $ 1 \plus{} p \plus{} Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p \minus{} 2})$ such that $ Q(x)$ is a polynominal with integer coefficients?

IV Soros Olympiad 1997 - 98 (Russia), 11.4
Tags: geometry , 3d geometry , geometric inequality , cylinder , projection
Find the largest value of the area of the projection of the cylinder onto the plane if its radius is $r$ and its height is $h$ (orthogonal projection).

1983 IMO Shortlist, 18
Tags: modular arithmetic , number theory , frobenius , additive number theory , additive combinatorics
Let $a,b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed in the form $xbc+yca+zab$, where $x,y,z$ are non-negative integers.

1987 Putnam, B1
Tags:
Evaluate \[ \int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}}. \]

2024 Princeton University Math Competition, B1
Tags: number theory
Let $f(n)$ be the sum of the factors of $2^n \cdot 31.$ Find $\sum_{n=0}^{4} f(n).$