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

2014 Contests, 1
Tags: digit , modular arithmetic , number theory
Consider the number $\left(101^2-100^2\right)\cdot\left(102^2-101^2\right)\cdot\left(103^2-102^2\right)\cdot...\cdot\left(200^2-199^2\right)$. [list=a] [*] Determine its units digit. [*] Determine its tens digit. [/list]

2022 Moscow Mathematical Olympiad, 1
Tags: algebra , inequalities
$a,b,c$ are nonnegative and $a+b+c=2\sqrt{abc}$. Prove $bc \geq b+c$

2001 India Regional Mathematical Olympiad, 6
Tags:
If $x,y,z$ are sides of a triangle, prove that \[ | x^2(y-z) + y^2(z-x) + z^2(x-y) | < xyz. \]

2019 Pan-African, 4
Tags: circumcircle , geometry
The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ meet at $D$. The circumcircle of $\triangle BCD$ meets sides $AC$ and $AB$ again at $E$ and $F$ respectively. Let $O$ be the circumcentre of $\triangle ABC$. Show that $AO$ is perpendicular to $EF$.

2017 ASDAN Math Tournament, 18
Tags:
Find the sum of all integers $0\le a \le124$ so that $a^3-2$ is a multiple of $125$.

2025 Macedonian TST, Problem 4
Tags: functional equation , algebra
Find all functions $f:\mathbb{N}_0\to\mathbb{N}$ such that [b]1)[/b] \(f(a)\) divides \(a\) for every \(a\in\mathbb{N}_0\), and [b]2)[/b] for all \(a,b,k\in\mathbb{N}_0\) we have \[ f\bigl(f(a)+kb\bigr)\;=\;f\bigl(a + k\,f(b)\bigr). \]

2010 Stanford Mathematics Tournament, 25
Tags:
There are balls that look identical, but their weights all di er by a little. We have a balance that can compare only two balls at a time. What is the minimum number of times, in the worst case, we have to use to balance to rank all balls by weight?

2012 Indonesia TST, 4
Tags: floor function , number theory
The sequence $a_i$ is defined as $a_1 = 1$ and \[a_n = a_{\left\lfloor \dfrac{n}{2} \right\rfloor} + a_{\left\lfloor \dfrac{n}{3} \right\rfloor} + a_{\left\lfloor \dfrac{n}{4} \right\rfloor} + \cdots + a_{\left\lfloor \dfrac{n}{n} \right\rfloor} + 1\] for every positive integer $n > 1$. Prove that there are infinitely many values of $n$ such that $a_n \equiv n \mod 2012$.

1996 Spain Mathematical Olympiad, 2
Tags: centroid , geometry , isosceles
Let $G$ be the centroid of a triangle $ABC$. Prove that if $AB+GC = AC+GB$, then the triangle is isosceles

Estonia Open Junior - geometry, 2012.2.5
Tags: perimeter , geometry
Is it possible that the perimeter of a triangle whose side lengths are integers, is divisible by the double of the longest side length?

2023 Pan-African, 4
Tags: combinatorics
Manzi has $n$ stamps and an album with $10$ pages. He distributes the $n$ stamps in the album such that each page has a distinct number of stamps. He finds that, no matter how he does this, there is always a set of $4$ pages such that the total number of stamps in these $4$ pages is at least $\frac{n}{2}$. Determine the maximum possible value of $n$.

2014 Albania Round 2, 2
Tags: arithmetic progression , sequence , algebra
Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its sides.

1966 IMO Longlists, 15
Tags: euler , geometry , perpendicular , circles
Given four points $A,$ $B,$ $C,$ $D$ on a circle such that $AB$ is a diameter and $CD$ is not a diameter. Show that the line joining the point of intersection of the tangents to the circle at the points $C$ and $D$ with the point of intersection of the lines $AC$ and $BD$ is perpendicular to the line $AB.$

2020 USMCA, 19
Tags:
Let $x_1, x_2, x_3$ be the solutions to $(x - 13)(x - 33)(x - 37) = 1337$. Find the value of $$\sum_{i=1}^3 \left[(x_i - 13)^3 + (x_i - 33)^3 + (x_i - 37)^3\right].$$

2015 Switzerland Team Selection Test, 6
Tags: algebra , function , functional equation , polynomial
Find all polynomial function $P$ of real coefficients such that for all $x \in \mathbb{R}$ $$P(x)P(x+1)=P(x^2+2)$$

2015 Junior Balkan Team Selection Tests - Romania, 3
Tags: circles , orthocenter , geometry , circumcircle , parallel
Let $ABC$ be an acute triangle , with $AB \neq AC$ and denote its orthocenter by $H$ . The point $D$ is located on the side $BC$ and the circumcircles of the triangles $ABD$ and $ACD$ intersects for the second time the lines $AC$ , respectively $AB$ in the points $E$ respectively $F$. If we denote by $P$ the intersection point of $BE$ and $CF$ then show that $HP \parallel BC$ if and only if $AD$ passes through the circumcenter of the triangle $ABC$.

2015 Irish Math Olympiad, 7
Tags: subset , set , combinatorics
Let $n > 1$ be an integer and $\Omega=\{1,2,...,2n-1,2n\}$ the set of all positive integers that are not larger than $2n$. A nonempty subset $S$ of $\Omega$ is called [i]sum-free[/i] if, for all elements $x, y$ belonging to $S, x + y$ does not belong to $S$. We allow $x = y$ in this condition. Prove that $\Omega$ has more than $2^n$ distinct [i]sum-free[/i] subsets.

2009 Indonesia TST, 4
Tags: geometry
Let $ ABCD$ be a convex quadrilateral. Let $ M,N$ be the midpoints of $ AB,AD$ respectively. The foot of perpendicular from $ M$ to $ CD$ is $ K$, the foot of perpendicular from $ N$ to $ BC$ is $ L$. Show that if $ AC,BD,MK,NL$ are concurrent, then $ KLMN$ is a cyclic quadrilateral.

1997 Romania Team Selection Test, 1
Tags: polynomial , algebra , inequalities
Let $P(X),Q(X)$ be monic irreducible polynomials with rational coefficients. suppose that $P(X)$ and $Q(X)$ have roots $\alpha$ and $\beta$ respectively, such that $\alpha + \beta $ is rational. Prove that $P(X)^2-Q(X)^2$ has a rational root. [i]Bogdan Enescu[/i]

2016 CHMMC (Fall), 11
Tags: probability , algebra
Let $a,b \in [0,1], c \in [-1,1]$ be reals chosen independently and uniformly at random. What is the probability that $p(x) = ax^2+bx+c$ has a root in $[0,1]$?

2001 Tuymaada Olympiad, 3
Tags: concyclic , angle bisector , geometry , circumcircle
Let ABC be an acute isosceles triangle ($AB=BC$) inscribed in a circle with center $O$ . The line through the midpoint of the chord $AB$ and point $O$ intersects the line $AC$ at $L$ and the circle at the point $P$. Let the bisector of angle $BAC$ intersects the circle at point $K$. Lines $AB$ and $PK$ intersect at point $D$. Prove that the points $L,B,D$ and $P$ lie on the same circle.

IV Soros Olympiad 1997 - 98 (Russia), 11.7
Tags: circles , geometry
On straight line $\ell$ there are points $A$, $B$, $C$ and $D$, following in the indicated order: $AB = a$, $BC = b$, $CD = c$. Segments $AD$ and $BC$ serve as chords of two circles, and the sum of the angular values of the arcs of these circles located on one side of $\ell$ is equal to $360^o$. A third circle passes through $A$ and $B$, intersecting the first two at points $K$ and $M$. The straight line $KM$ intersects $\ell$ at point $E$. Find $AE$.

2021 MOAA, 3
Tags:
What is the last digit of $2021^{2021}$? [i]Proposed by Yifan Kang[/i]

2023 Ecuador NMO (OMEC), 4
Tags: combinatorics
A number is [i]additive[/i] if it has three digits, all of them are different and the sum of two of the digits is equal to the remaining one. (For example, $123 (1+2=3), 945 (4+5=9)$). Find the sum of all additive numbers.

V Soros Olympiad 1998 - 99 (Russia), 10.4
Tags: algebra
Solve the equation $$ x + \sqrt{x^2-9} = \frac{2(x+3)}{(x-3)^2}$$