problem stringlengths 0 45.3k | solution stringlengths 0 26.8k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 9
values | question_type stringclasses 5
values | problem_is_valid stringclasses 5
values | solution_is_valid stringclasses 5
values | source stringclasses 11
values | synthetic bool 2
classes |
|---|---|---|---|---|---|---|---|---|
2. Let $S$ - the distance between cities, $v_{1}=\frac{8 S}{5}$ - the speed of the cyclist, $v_{2}=\frac{3 S}{5}-$ the speed of the pedestrian. The graph below shows the dependence of the distances from city A to the cyclist and to the pedestrian on time.
 8 times; 2) $\frac{15}{11}$ hours | \frac{15}{11} | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
Task 1. Dad and Mom are cooking pancakes on the stove, while their children Petya and Vasya are eating them. Dad can cook 70 pancakes in an hour, and Mom can cook 100 pancakes. Petya, if he tries hard, can eat 10 pancakes in 15 minutes, and Vasya can eat twice as much. After how much time will there be no less than 20 ... | Answer: 24 min
Solution. Let the baking rate of pancakes be $p_{1}=\frac{170}{60}$ pancakes/min, and the eating rate of pancakes be $p_{2}=\frac{10+20}{15}=2$ pancakes/min.
In $k$ minutes, there will be no less than $q=k\left(p_{1}-p_{2}\right)$ pancakes left on the table: $k \cdot \frac{5}{6} \geq 20 \rightarrow k \... | 24 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Problem 2. In a square of grid paper containing an integer number of cells, a hole in the shape of a square, also consisting of an integer number of cells, was cut out. How many cells did the large square contain if 209 cells remained after the cutout? | Answer: 225 cells
Solution. The side of the larger square contains $n$ sides of a cell, and the side of the smaller square contains $m$ sides of a cell. Then $n^{2}-m^{2}=209 \rightarrow(n-m)(n+m)=209=11 \cdot 19$.
Case 1. $\left\{\begin{array}{c}n+m=209 \\ n-m=1\end{array} \rightarrow\left\{\begin{array}{c}n=105 \\ ... | 225 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
Problem 3. A line passing through any point $D$ on the median $B M$ of triangle $A B C$ and parallel to side $A B$, intersects the line passing through vertex $C$ and parallel to $B M$, at point $E$. Prove that $B E=A D$. | Solution. Additional construction: through point $M$ we draw line $MN$, parallel to $DE$. Figure $MDEN$ is a parallelogram (by construction).
Triangles $ABM$ and $MNC$ are equal (by two equa... | proof | Geometry | proof | Yes | Yes | olympiads | false |
1. For what values of $x$ does the expression $\cos ^{2}(\pi \cos x)+\sin ^{2}(2 \pi \sqrt{3} \sin x)$ take its minimum value | 1. Answer: $x= \pm \frac{\pi}{3}+\pi t, t \in Z$. Solution. Let's check if the sine and cosine in the given expression can simultaneously be zero for some values of $x$. If so, the expression takes its minimum value at these values of $x$.
$$
\left\{\begin{array} { c }
{ \operatorname { c o s } ( \pi \operatorname { ... | \\frac{\pi}{3}+\pi,\inZ | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. To fight against the mice, the cat Leopold must drink animalin daily. The cat has a bottle of animalin solution in water (a colorless transparent liquid) with a volume of $V=0.5$ l and a volumetric concentration of animalin $C_{0}=40 \%$. Every day, Leopold drinks $v=50$ ml of the solution, and to avoid being notice... | 2. Let's find the change in the concentration of the absinthe solution after the cat drinks one portion. By definition of volume concentration,
$$
C_{0}=\frac{v_{O}}{V}
$$
where $C_{0}$ is the volume concentration of the solution, $v_{o}$ is the initial volume of absinthe in the bottle, and $V$ is the total volume of... | 23.6 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. A cyclic process $1-2-3-4-1$ is carried out with an ideal gas, consisting of two isotherms (2-3 and 4-1) and two isochors ($1-2$ and $3-4$; see figure). It is known that the ratio of temperatures on the isotherms 2-3 and 4-1 is $T_{2-3} / T_{4-1}=3$, and that during the isothermal expansion, the gas received twice a... | 3. Let the gas receive an amount of heat $Q$ during the isothermal expansion segment. Then, according to the condition, the gas receives an amount of heat $Q / 2$ during the isochoric heating segment. Therefore, the total amount of heat received by the gas from the heater during the cycle is
$$
Q_{n}=\frac{3}{2} Q
$$
... | \frac{4}{9} | Other | math-word-problem | Yes | Yes | olympiads | false |
4. The electrical circuit shown in the figure contains a very large number of identical resistors and identical voltmeters. It is known that the readings of voltmeters $V_{1}$ and $V_{2}$ are $U=4$ V and $U^{\prime}=6$ V (although it is unknown which reading corresponds to which voltmeter). Find the readings of voltmet... | 4. Since the current flowing through the voltmeter $V_{1}$ is greater than the current flowing through the voltmeter $V_{2}$, the resistances of the voltmeters are the same, and each voltmeter shows the voltage across it, the value $U^{\prime}=6$ V is the reading of the voltmeter $V_{1}$, $U=4$ V is the reading of the ... | 15\mathrm{~V} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. An integer N is given as input. The task is to write a program that finds a decomposition of the number N into a sum of squares of natural numbers, such that the number of terms in this decomposition is minimal. For example, N=11:
\[
\begin{aligned}
& 11=3^{2}+1^{2}+1^{2} \\
& 11=2^{2}+2^{2}+1^{2}+1^{2}+1^{2} \\
& ... | # 5.
\{pascal $\}$
program task 11 ;
var
$\mathrm{N}:$ integer;
m_len:array[1..10000] of integer; //array for recording the length of the minimum decomposition
m_val:array[1..10000] of integer; //array for saving the addends of the minimum decomposition
//declare the recursive function for calculating the result... | notfound | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
1. (mathematics) The probabilities of finding Kuzia the flea at vertices $A, B, C$ and $D$ of the square $A B C D$ are $p_{1}, p_{2}, p_{3}$ and $p_{4}$ respectively $\left(p_{1}+p_{2}+p_{3}+p_{4}=1\right)$. The flea jumps to the nearest vertex of the square, and which of the two it jumps to is completely random and eq... | Solution. Let $P_{A}(n), P_{B}(n), P_{C}(n), P_{D}(n)$ be the probabilities of finding the flea at vertices $A, B, C$, and $D$ after $n$ jumps. Then
$$
P_{A}(1)=\frac{1}{2}\left(p_{2}+p_{4}\right)=P_{C}(1), P_{B}(1)=\frac{1}{2}\left(p_{1}+p_{3}\right)=P_{D}(1)
$$
Figure 1 shows the probabilities of events after the f... | \frac{1}{2}(p_{1}+p_{3}) | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
2. (mathematics) Thirteen circles of the same radius $r=2 \sqrt{2-\sqrt{3}}$ are cut out of colored paper and laid out on a plane so that their centers lie on one straight line. The distance between the centers of adjacent circles is the same and equals 2. Find the perimeter of the figure on the plane formed by these c... | Answer: $P=44 \pi \sqrt{2-\sqrt{3}}$
## Solution:
The circles intersect since $r>1$. In the figure, three of the thirteen circles are depicted.
$$
A C=2, A B=B C=r=2 \sqrt{2-\sqrt{3}}, \squ... | 44\pi\sqrt{2-\sqrt{3}} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. (physics) A wire is made into a right-angled triangle with legs of length $l=1$ m. From vertices A and B, two bugs start crawling along the legs simultaneously with speeds $v_{A}=5$ cm/s and $v_{B}=2 v_{A}=10$ cm/s (see figure). What is the minimum time after the start of their movement when the distance between the... | Solution. Let the time that has passed since the beetles started moving be $t$. Then beetle
A will travel a distance of $v_{A} t$, beetle B - a distance of $v_{B} t=2 v_{A} t$. Therefore, the distance $S$ between the beetles at this moment will be
$$
S=\sqrt{\left(l-v_{A} t\right)^{2}+4 v_{A}^{2} t^{2}}
$$
We will f... | 4\mathrm{~} | Calculus | math-word-problem | Yes | Yes | olympiads | false |
4. (Physics) Find the total resistance of the electrical circuit, the scheme of which is shown in the figure. The values of all resistances are given in the figure. $r=1$ Ohm. | Solution. Let an electric current $I$ flow into the considered section
$I$, which splits at the first branching into currents $I_{1}$ (in the lower leg of the circuit) and $I-I_{1}$ (in the u... | 1.4 | Other | math-word-problem | Incomplete | Yes | olympiads | false |
5. (informatics) In a square matrix of size $N \times N$, all cells are filled with numbers from 1 to 5. A connected component in the matrix is defined as a set of cells that are filled with the same number, and between any two cells in the set, a path can be constructed. Cells can only be connected in a path if they a... | # Solution.
program prog11;
var n,i,j,k,l:integer;
a,b:array[0..11,0..11] of integer;
m,mc,mmax:integer;
begin
{initialize the array}
for i:=0 to 11 do
for j:=0 to 11 do
```
a[i,j]:=0;
{read input data}
readln(N);
for i:=1 to N do
begin
for j:=1 to N do
read(a[i,j]);
readln;
end;
mmax:=0;{c... | 12 | Other | other | Yes | Yes | olympiads | false |
1. Along one side of the road, there are 15 houses numbered $1,2, \ldots, 15$ (see figure). The residents of these houses travel to work on the road by bus. At a certain moment, one person from each house came out and headed to the bus stop. Indicate the house number where the bus stop should be built to minimize the a... | 1. The correct idea for solving the problem is the difference between the force of Archimedes acting on the hemisphere and the force of the liquid column pressure on its upper surface - 0.5 points | 8 | Logic and Puzzles | math-word-problem | Yes | Problem not solved | olympiads | false |
2. Thirty linear functions $y=f_{n}(x), n=1,2, \ldots, 30$ define the equations of thirty lines on the plane, passing through one point $A$ with ordinate 1 and intersecting the $\mathrm{OX}$ axis at points with abscissas $x_{1}, x_{2}, \ldots, x_{30}$. Calculate the value of the expression
$$
B=\left(f_{2}\left(x_{1}\... | 2. Correct determination of the Archimedes' force acting on a hemisphere - 0.5 points | notfound | Algebra | math-word-problem | Yes | Problem not solved | olympiads | false |
3. In what ratio does a line parallel to the bases of a trapezoid and passing through the point of intersection of its diagonals divide the area of the trapezoid, if one base of the trapezoid is twice as large as the other? | 3. Correct determination of the force acting on the upper surface of the hemisphere - 0.5 points | notfound | Geometry | math-word-problem | Yes | Problem not solved | olympiads | false |
4. A cottage has a gas water heater in which water flowing through pipes is heated to the required temperature by gas. The heater has a useful power of $P=20$ kW and an efficiency of $\eta=0.8$. What is the mass flow rate of gas in such a heater (the mass of gas burned per unit time)? How long will the heater take to f... | 4. Correct answer - 0.5 points
The score for the task is the sum of the scores according to the criteria listed above
## Evaluation of the participant's work
The final score for the work is the sum of the scores for each task (maximum score - 12 points). Conversion to a 50-point scale was carried out according to th... | Algebra | math-word-problem | Yes | Incomplete | olympiads | false | |
5. The star "tau" in the constellation Cetus has a planetary system. On the third planet from the star, there is a very unusual gravity: the acceleration due to gravity is $g=10 \mathrm{~m} / \mathrm{c}^{2}$ up to a height of $h=10$ m from the surface, but above that, it is half as much. An astronaut throws a body stra... | 5. Since the height of the body's rise on Earth is $H$, the initial velocity of the thrown body is
$$
v_{0}=\sqrt{2 g H}
$$
On the planet from the "tau Ceti" system, the body will move to a height $h$ with acceleration $g$, and then with acceleration $g / 2$. Therefore, above this boundary, the body will rise to a he... | 30\mathrm{~} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. A sphere of radius $R$, made of a material with a density equal to that of water, floats in water, touching its surface (see figure). Find the force with which the water acts on the lower half of the sphere. The density of water $\rho$ is known.
 There are scales with two pans, 4 weights of 2 kg each, 3 weights of 3 kg each, and two weights of 5 kg each. In how many different ways can a 12 kg load be balanced on the scales, if the weights are allowed to be placed on both pans? | Answer: 7 ways
Solution: Let $x$ be the number of 2 kg weights used in weighing, $y$ be the number of 3 kg weights, and $z$ be the number of 5 kg weights. Then the equilibrium condition is g... | 7 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
2. (mathematics) Segments $A B$ and $C D$, with lengths 3 and 4 respectively, are chords of a circle with radius 5. Points $M$ and $N$ are located on them such that $A M: M B=1: 2, \quad C N: N D=1: 3$. Find the smallest and largest possible lengths of segment $M N$. | Answer: $d_{\min }=\sqrt{23}-\sqrt{22} \quad d_{\max }=\sqrt{23}+\sqrt{22}$
Solution. The lengths of segments $A B=a, C D=b, \quad A M: M B=p: q$ $C N: N D=m: n$. Points $P, Q$ are the midpoints of segments $A B$ and $D C$. We will prove that points $M$ and $N$ lie on two concentric circles for any position of segment... | d_{\}=\sqrt{23}-\sqrt{22}\quadd_{\max}=\sqrt{23}+\sqrt{22} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. (physics) In a vessel, there is a mixture of equal masses of nitrogen $N_{2}$ and helium He at a pressure $p$. The absolute temperature of the gas is doubled, during which all nitrogen molecules dissociate into atoms. Find the pressure of the gas mixture at this temperature. The molar masses of the gases are $\mu_{\... | Solution. Let the mass of helium and nitrogen in the vessel be $m$. Then Dalton's law for the initial gas mixture gives
$$
p=\frac{v_{\mathrm{He}} R T}{V}+\frac{v_{N_{2}} R T}{V}
$$
where
$$
v_{N_{2}}=\frac{m}{\mu_{N_{2}}}=v ; \quad v_{\mathrm{He}}=\frac{m}{\mu_{\mathrm{He}}}=7 v
$$
are the amounts of substance of ... | \frac{9}{4}p | Other | math-word-problem | Yes | Yes | olympiads | false |
4. (physics) Two bodies with masses $m$ and $2 m$ are connected by ropes. One rope is thrown over a pulley, the axis of which is attached to the ceiling. Another rope has the same pulley placed on it, to the axis of which a body with mass $4 m$ is attached (see figure). Find the tension force in the rope connecting the... | Solution. Since the blocks do not move (only rotate), the acceleration of the body with mass $4 m$ is zero. Therefore, the tension force in the lower rope connecting the bodies $m$ and $2 m$ is
$$
T_{u}=2 \mathrm{mg}
$$
Newton's second law for the bodies with masses $m$ and $2 m$ in projections on the axis directed v... | \frac{10}{3} | Other | math-word-problem | Yes | Yes | olympiads | false |
1. The function $f(x)$ is a quadratic trinomial, $g(x)$ is a linear function, for which the inequality $\left|f^{\prime}(x)\right| \cdot g^{2}(x) \geq|f(x)|+|g(x)|$ holds for all $x$. Prove that $\lim _{x \rightarrow a} \frac{f(x)}{g^{2}(x)}$ exists for any $a$ and its value does not depend on $a$. | 1. The correct method for finding the amount of heat received or given (through areas) or through average heat capacity (with justification) - 0.5 points | proof | Algebra | proof | Yes | Problem not solved | olympiads | false |
2. Petya suggested Vasya find all integer solutions to the equation:
$$
\operatorname{tg} \frac{\pi}{x} \cdot \operatorname{tg} \frac{2 \pi}{x} \cdot \operatorname{tg} \frac{3 \pi}{x} \cdot \ldots \cdot \operatorname{tg} \frac{2021 \pi}{x}=0
$$
Vasya crossed out the first factor $\operatorname{tg} \frac{\pi}{x}$ and ... | 2. Correct quantities of received and given heat, work of the gas for the cycle - 0.5 points | notfound | Geometry | math-word-problem | More than one problem | Problem not solved | olympiads | false |
4. A body located on a rough horizontal surface is attached to a thread, which is positioned horizontally. The body is pulled by applying a constant force to the thread. As a result, the body moves with a constant speed $v$ along a circle of radius $R$, and the thread makes a constant angle $\alpha$ with the velocity v... | 4. Correct efficiency of the process $-0.5$ points
The score for the task is the sum of the scores according to the criteria listed above
## Evaluation of the Participant's Work
The final score for the work is the sum of the scores for each task (maximum score - 12 points). Conversion to a 50-point scale was carried... | notfound | Other | math-word-problem | Incomplete | Problem not solved | olympiads | false |
5. A square is made from a wire. Two
opposite sides have the same resistance $r$, the other two have the same resistance $2 r$. Sliding contacts can move along the two opposite sides with res... | 5. Let's find the resistance of a square for some arrangement of contacts and investigate this value for maximum and minimum. Obviously, for any arrangement of contacts on the square, the sum of the resistances of its upper and lower branches is equal to the resistance of the wire from which it is made, i.e.,
. Find the thermodynamic efficiency coefficient of this
![](https://cdn.ma... | 6. Since the temperature of the gas increased in processes 1-2 and 2-3, and the heat capacity is positive, the gas received heat in these processes. In process 3-1, the temperature of the gas decreased, and the gas gave off heat. This means that the contact with the engine's heater was during the process $1-2-3$, and w... | \eta=\frac{5}{17}=0.29 | Other | math-word-problem | Incomplete | Yes | olympiads | false |
1. The sum of the absolute values of the first 18 terms of an increasing arithmetic progression is 270. This sum does not change if 2 is added to each term of the progression. If 3 is added to each term of the progression, the sum also does not change. Find the first term and the common difference of the progression. | 1. Correct formulas for adding resistances - 0.5 points | notfound | Algebra | math-word-problem | Yes | Problem not solved | olympiads | false |
2. For what values of $a$ does the equation $\arcsin ^{2} x+\arccos ^{2} x=a$ have a unique solution? | 2. Correct circuit resistance as a function of the resistance between contacts along the "upper" or "lower" conductor. Correct answer for the maximum and minimum circuit resistance - 0.5 points | notfound | Algebra | math-word-problem | Incomplete | Problem not solved | olympiads | false |
3. Point $C$ is located outside the circle constructed on segment $A B$ as its diameter. Lines $A C$ and $B C$ intersect the circle at points $P$ and $Q$, respectively. The length of segment $P Q$ is equal to the radius of the circle. Find the angle $C A Q$. | 3. Justification of the maximum or minimum resistance - 0.5 points | notfound | Geometry | math-word-problem | Yes | Problem not solved | olympiads | false |
4. Two diatomic gases $A_{2}$ and $B_{2}$, taken in equal amounts of moles, are in a vessel under pressure $p$. A chemical reaction occurs with the formation of gaseous compound $A_{2} B$. It is known that the maximum possible amount of this gas has been formed. What will be the pressure in the vessel at the same tempe... | 4. Correct placement of contacts to ensure maximum and minimum resistance. Enumeration of all cases of connection with maximum and minimum resistance - 0.5 points
The score for the task is the sum of the scores according to the above criteria
Score for the participant's work
The final score for the work is the sum o... | notfound | Algebra | math-word-problem | Yes | Problem not solved | olympiads | false |
5. A sphere of radius $R$, made of a material with a density equal to that of water, floats in water, touching its surface (see figure). Find the force with which the water acts on the lower half of the sphere. The density of water $\rho$ is known.
. By what factor does the maximum resistance of the square differ from the ... | 6. Let's find the resistance of a square for some arrangement of contacts and investigate this value for maximum and minimum. Obviously, for any arrangement of contacts on the square, the sum of the resistances of its upper and lower branches is equal to the resistance of the wire from which it is made, i.e.,
, that the velocities of the body and the ring at this moment are the same - 0.5 points | notfound | Algebra | math-word-problem | Yes | Problem not solved | olympiads | false |
2. How many different irreducible fractions of the form $\frac{n}{m}$ exist, whose decimal representation is a periodic fraction of the form $0, b 8\left(a_{1} a_{2}\right)$, where $a_{1}, a_{2}, b$ are digits? | 2. The velocity of the body and the ring at the moment of reaching the maximum height (found using the law of conservation of momentum) is correctly determined - 0.5 points | notfound | Number Theory | math-word-problem | Incomplete | Problem not solved | olympiads | false |
3. The lengths of sides $B C, A C$ and $A B$ of triangle $A B C$ are 2, 3, and 4, respectively. Points $M$ and $N$ are located on sides $A B$ and $A C$ at a distance of 2 from vertices $B$ and $C$, respectively. Points $P$ and $Q$ are located on the extensions of sides $A C$ and $B C$ beyond vertex $C$ at a distance of... | 3. The correct law of conservation of energy for the moment of raising the body to the maximum height - $\mathbf{0 , 5}$ points | notfound | Geometry | math-word-problem | Yes | Problem not solved | olympiads | false |
4. The own volume of a bulk material - gravel, sand, grain - can be measured as follows. The bulk material is placed in a cylindrical vessel and hermetically sealed with a movable piston. Then, the gas pressures $p_{1}$ and $p_{2}$ in the vessel are measured at two different volumes under the piston - $V_{1}$ and $V_{2... | 4. Correct answer $-0.5$ points
The score for the problem is the sum of the scores according to the criteria listed above
## Evaluation of the Participant's Work
The final score for the work is the sum of the scores for each problem (maximum score - 12 points). Conversion to a 50-point scale was carried out accordin... | notfound | Algebra | math-word-problem | Yes | Problem not solved | olympiads | false |
5. A resistor with an unknown resistance was connected to a voltage source $U$, and two amperemeters $\mathrm{A}_{1}$ and $\mathrm{A}_{2}$ were connected in parallel (see the left diagram). In this case, the amperemeters show currents of $I_{1}=2$ A and $I_{2}=3$ A. Then, these amperemeters together with the same resis... | 5. Obviously, ammeters have different resistances, since different currents flow through them when connected in parallel in the circuit. From Ohm's law for ammeters, it follows that the ratio of their resistances is
$$
\frac{R_{1}}{R_{2}}=\frac{I_{2}}{I_{1}}=\frac{3}{2}
$$
Since the current flowing through the source... | 12.7\mathrm{~A} | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
6. At the ends of a weightless, inextensible rod, a small body and a massive ring of the same mass are attached. The ring can move without friction along a horizontal rigid spoke in a gravitational field (see figure). At some moment, the ring and the body were given velocities $v$ and $2 v$, directed horizontally along... | 6. At the moment of rising to the maximum height, the body moves horizontally. Therefore, from the condition of the inextensibility of the rod, we conclude that the speed is the same (their projections on the direction of the rod should be the same, and the speeds of the body and the ring at the moment of the body's ri... | \frac{9v^{2}}{4(1-\sin\alpha)} | Other | math-word-problem | Yes | Yes | olympiads | false |
1. Subtract the second equation from the first:
$x^{2}-2 x+y^{4}-8 y^{3}+24 y^{2}-32 y=-17 \rightarrow(x-1)^{2}+(y-2)^{4}=0 \rightarrow\left\{\begin{array}{l}x=1 \\ y=2\end{array}\right.$
Then $z=x^{2}+y^{4}-8 y^{3}=1+16-64=-47$ | Answer: the only solution is $x=1, y=2, z=-47$. | -47 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. Find all integer values of $x$ that are solutions to the equation $\sin (\pi(2 x-1))=\cos (\pi x / 2)$. | 1. Answer: $x=4 t \pm 1, t \in Z$. Solution.
$\sin (\pi(2 x-1))=\sin (\pi(1-x) / 2) \rightarrow\left\{\begin{array}{l}2 x-1=(1-x) / 2+2 m \rightarrow 5 x / 2-2 m=3 / 2 \\ 2 x-1=(x+1) / 2+2 n \rightarrow 3 x / 2-2 n=3 / 2\end{array}\right.$
The equation $5 x-4 m=3$ in integers has the following solutions $\left\{\begi... | 4\1,\inZ | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. A sealed horizontal cylindrical vessel of length $l$ is divided into two parts by a movable partition. On one side of the partition, there is 1 mole of oxygen, and on the other side, there is 1 mole of helium and 1 mole of oxygen, with the partition being in equilibrium. At some point in time, the partition becomes ... | 3. From the condition of equilibrium of the partition (equality of pressures to the right and left of it), we find that initially it is located at a distance of $2 l / 3$ and $l/3$ from the ends of the vessel. After helium is uniformly distributed throughout the vessel, its partial pressures to the right and left of th... | \frac{}{6} | Other | math-word-problem | Yes | Yes | olympiads | false |
4. A small weightless ring is hinged to a light rod of length $2 l$. At the midpoint and end of the rod, point masses of $m$ and $2 m$ are fixed. The ring is placed on a smooth horizontal rod. Initially, the rod was held horizontally and then released. Find the speed of the ring at the moment when the rod passes the ve... | 4. Since the external forces on the system of bodies and the spoke act only in the vertical direction (the gravitational forces of the bodies and the reaction force of the rod), the projection of the momentum of the system of bodies on the horizontal direction is conserved. Therefore, if the lower body has a speed $v$ ... | \sqrt{\frac{125}{3}} | Other | math-word-problem | Yes | Yes | olympiads | false |
5. Given an array of $N$ integers $m_{i}$, where $N$ is odd. The median of a set of numbers is the element $\mathrm{m}_{\mathrm{i}}$ that, after sorting this set in ascending or descending order, would be exactly in the middle of the sequence. For example: 7386513 --> $133 \underline{5} 67$ 8. The task is to write a pr... | # 5.
{pascal
program task_10;
var
N :integer;
m:array[1..10001] of integer;
i,j:integer;
r,c1,c2:integer;
begin
// read input data
readln(N);
for i:=1 to N do
read(m[i]);
for i:=1 to N do // for each element, count the number of smaller and equal elements
begin
c1:=0;
c2:=0;
for j:=1 to N do
begin
if... | notfound | Other | math-word-problem | Yes | Incomplete | olympiads | false |
1. Express z from the first equation and substitute into the second:
$x^{2}-2 x+y^{2}-2 \sqrt{3} y=-4 \rightarrow(x-1)^{2}+(y-\sqrt{3})^{2}=0 \rightarrow\left\{\begin{array}{c}x=1 \\ y=\sqrt{3}\end{array} \rightarrow z=x^{2}+y^{2}+2 x=6\right.$ | Answer: $x=1, y=\sqrt{3}, z=6$ | 6 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Task 1. The administration divided the region into several districts based on the principle: the population of a large district exceeds $8 \%$ of the region's population and for any large district, there are two non-large districts with a combined population that is larger. Into what minimum number of districts was the... | Answer: 8 districts.
Solution. The number of "small" districts is no less than 2 according to the condition, and their population does not exceed $8 \%$ of the total population of the region. We will show that the number of districts in the region is no less than 8. If the number of districts in the region is no more ... | 8 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
Problem 2. The number $x$ satisfies the condition $\frac{\sin 3 x}{\sin x}=\frac{5}{3}$. Find the value of the expression $\frac{\cos 5 x}{\cos x}$ for such $x$ | Answer: $-\frac{11}{9}$.
Solution.
$$
\begin{aligned}
& \sin x \neq 0 \rightarrow \frac{\sin 3 x}{\sin x}=\frac{3 \sin x-4 \sin ^{3} x}{\sin x}=3-4 \sin ^{2} x \rightarrow 3-4 \sin ^{2} x=\frac{5}{3} \rightarrow \sin ^{2} x=\frac{1}{3} \rightarrow \\
& \rightarrow \cos ^{2} x=\frac{2}{3} \rightarrow \cos 2 x=\frac{1}... | -\frac{11}{9} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Problem 3. Point $M$ is located on side $CD$ of a square such that $CM: MD=1: 3$. Line $AM$ intersects the circumcircle of the square at point $E$. The area of triangle $ACE$ is 14. Find the side length of the square. | Answer: 10.
Solution.
Triangles $A M D$ and $C M E$ are similar with a similarity coefficient $k=5$. Then
$$
C E=\frac{4 x}{5}, M E=\frac{3 x}{5} \rightarrow A E=5 x+\frac{3 x}{5}=\frac{28... | 10 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Problem 1. Seven students in the class receive one two every two days of study, and nine other students receive one two every three days each. The rest of the students in the class never receive twos. From Monday to Friday, 30 new twos appeared in the journal. How many new twos will appear in the class journal on Satur... | Answer: 9
Solution. Over the period from Monday to Saturday (six days), in the journal, there will be 3 new twos from each student of the first group (seven people) and 2 new twos from each of the 9 students of the second group. The total number of new twos for the school week is $7 \cdot 3 + 9 \cdot 2 = 39$. Then, on... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
Problem 2. Prove that for any positive numbers $a$ and $b$ the following inequality holds
$$
\sqrt{\frac{a^{2}}{b}}+\sqrt{\frac{b^{2}}{a}} \geq \sqrt{a}+\sqrt{b}
$$ | Solution. Let $b \geq a$. Divide the inequality by $\sqrt{a}$:
$$
\begin{aligned}
& \sqrt{\frac{a}{b}}+\sqrt{\frac{b^{2}}{a^{2}}} \geq 1+\sqrt{\frac{b}{a}} \rightarrow \sqrt{\frac{b}{a}}\left(\sqrt{\frac{b}{a}}-1\right) \geq 1-\sqrt{\frac{a}{b}} \rightarrow \sqrt{\frac{b}{a}} \cdot \frac{1}{\sqrt{a}}(\sqrt{b}-\sqrt{a}... | proof | Inequalities | proof | Yes | Yes | olympiads | false |
Problem 3. Through the vertex $B$ of an equilateral triangle $ABC$, a line $L$ is drawn, intersecting the extension of side $AC$ beyond point $C$. On line $L$, segments $BM$ and $BN$ are laid out, each equal in length to the side of triangle $ABC$. The lines $MC$ and $NA$ intersect at a common point $D$ and intersect t... | # Answer: 1.
Solution. Triangles $M B C$ and $A B N$ are isosceles, therefore
$$
\begin{gathered}
\angle B M C = \angle B C M = \alpha \rightarrow \angle N B C = 2 \alpha, \angle B A N = \angle B N A = \beta \rightarrow \angle A B M = 2 \beta \\
2 \alpha + 2 \beta + 60^{\circ} = 180^{\circ} \rightarrow \alpha + \beta... | 1 | Geometry | proof | Yes | Yes | olympiads | false |
1. Answer: 125 cm, 75 cm
# | # Solution
Notations:
$$
\vec{c}=\overrightarrow{O Q} \quad \vec{a}=\overrightarrow{O A} \quad \vec{b}=\overrightarrow{Q B}
$$
Then $\vec{c}+(\vec{b}-\vec{a})=\overrightarrow{A B}$. The pro... | 125 | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
2.
Suppose that $\square A=75^{\circ}, \square C=45^{\circ}$. Then $\square B=60^{\circ}$. Points $M$ and $N$ lie on the circle constructed on side $A C$ as the diameter. Triangle $B M N$ i... | Answer: $S_{M N P}: S_{A B C}=(\sqrt{3}-1): 4$ | (\sqrt{3}-1):4 | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
3. Answer: 1) $p=7, x=1, y=4$; 2) $p=3, x=3, y=6$
# | # Solution
Transformation:
$$
p^{x}=(y-3)(y+3) \rightarrow y>3 \rightarrow y-3=p^{u}, u \geq 0, y+3=p^{v}, v \geq 0, u, v \in Z, v \geq u, u+v=x
$$
Case $1 . u_{0}$
$y=4 \rightarrow p^{v}=7 \rightarrow p=7, x=1$
Case $2 . u>0$
GCD $(y-3, y+3)=p^{v-u}=$ GCD $(y-3,(y+3)-(y-3))=$ GCD $(y-3,6)$
Possible values $p^{v... | p=7,1,4;p=3,3,6 | Number Theory | math-word-problem | More than one problem | Yes | olympiads | false |
1. (mathematics) Find the values of $a$ for which the coordinates ( $x ; y$ ) of any point in the square $2 \leq x \leq 3,3 \leq y \leq 4$ satisfy the inequality $(3 x-2 y-a)\left(3 x-2 y-a^{2}\right) \leq 0$. | Answer: $a \in(-\infty ;-4]$
Solution: Points with coordinates $(x ; y)$ of the square $2 \leq x \leq 3, 3 \leq y \leq 4 A B C D$ lie in the strip
$$
(3 x-2 y-a)\left(3 x-2 y-a^{2}\right) \leq 0
$$
between the lines $L_{1}: 3 x-2 y-a=0$ and $L_{2}: 3 x-2 y-a^{2}=0$ (see figure), if this strip contains all the vertic... | \in(-\infty;-4] | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
2. (mathematics) An arbitrary point $E$ inside the square $A B C D$ with side 1 is connected to its vertices by segments. The points $P, Q, F$ and $T$ are the points of intersection of the medians of triangles $B C E, C D E, D A E$ and $A B E$ respectively. Find the area of quadrilateral PQFT. | Answer: $S=\frac{2}{9}$
Solution: Let's introduce the following notations (see the figure): $K, L$ - midpoints of segments $A E$ and $E C, G$ - midpoint of segment $B E$. $K L$ is parallel to the diagonal $A C$ and is equal in length to half of it. Triangles $B P T$ and $B L K$ are similar with a similarity coefficien... | \frac{2}{9} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. (physics) Two weightless glasses with different amounts of water are balanced on a lever of unequal-arm scales with a length of $L$. A mass of water $m$ is transferred from one glass to the other, and the scales lose their balance. To restore balance, the fulcrum of the lever needs to be moved by a distance $l$ rela... | Solution. Let the mass of water in the left glass be $m_{1}$, in the right glass be $m_{2}$, and the lever arms of the glasses be $l_{1}$ and $L-l_{1}$, respectively. Then the condition
for e... | m_{1}+m_{2}=\frac{L}{} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. (physics) From a piece of metal with mass $M$, a rectangular parallelepiped is cast. Then, a source of constant voltage is connected to different pairs of mutually opposite faces of the parallelepiped. It turned out that the ratio of the thermal powers dissipated in the parallelepiped when the source is connected to... | Solution. Let the edge lengths of the parallelepiped be $a$ (the smallest), $b$ (the middle), and $c$ (the largest). Then the resistance of the parallelepiped when electrical contacts are connected to the faces with sides $a$ and $b$ is
$$
R_{a b}=\frac{\rho_{0} a b}{c}
$$
where $\rho_{0}$ is the resistivity of the m... | =(\frac{M}{4\rho})^{1/3},b=\sqrt{2}(\frac{M}{4\rho})^{1/3},=2\sqrt{2}(\frac{M}{4\rho})^{1/3} | Other | math-word-problem | Yes | Yes | olympiads | false |
5. (informatics) On a chessboard, a white queen and a black pawn are positioned, given by coordinate pairs ( $R_{1}, C_{1}$ ) and ( $R_{2}, C_{2}$ ) respectively, where $R$ is the row number, and $C$ is the column number (as marked in the figure below). The task is to check if the queen can attack the pawn. The example... | # Solution.
program prog9;
var r1,c1,r2,c2:integer;
begin
{read input data}
readln(r1,c1,r2,c2);
if $(r 1=\mathrm{r} 2)$ {check horizontal}
or ( $\mathbf{c} 1=\mathrm{c} 2)\{$ check vertical $\}$
or $((\mathrm{r} 1+\mathrm{c} 1)=(\mathrm{r} 2+\mathrm{c} 2))\{$ check diagonal\}
or $((\mathrm{r} 1-\mathrm{c} 1)=(... | notfound | Other | other | Yes | Yes | olympiads | false |
1.
Notations:
$C$ - the point on the highway where Kolya got off his bicycle, $A C=10 t$
$D$ - the point on the highway where Vasya was at time $t, A D=5 t$
$E$ - the point where Petya pic... | Answer: 1) $t=\frac{16}{13}$ hour; 2) $T_{\min }=\frac{38}{13}$ hour | \frac{38}{13} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. Answer: $( \pm 1 ; 0),( \pm 3 ; 1),( \pm 3 ; 2)$
# | # Solution
Note that $y \geq 0$.
Estimate $24 y+1=\left(4 y^{2}-x^{2}\right)^{2} \geq\left(4 y^{2}-(2 y-1)^{2}\right)^{2}=(4 y-1)^{2}=16 y^{2}-8 y+1 \rightarrow y(y-2) \leq 0$
The inequality is satisfied by three values $y=0, y=1, y=2$.
Case 1. $y=0$
Equation $x^{4}=1 \rightarrow x= \pm 1$
Case 2. $y=1$
Equation... | (\1;0),(\3;1),(\3;2) | Algebra | math-word-problem | Not a problem | Yes | olympiads | false |
2. The drawing shows the corresponding constructions and some relationships between the sides of the triangles
Triangle $B C D$ is isosceles, and $C O$ is the angle bisector, so $B O=O D=y$.... | Answer: $S_{\text {ADOE }}: S_{\text {ABC }}=5: 12$ | 5:12 | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
1. For what integers $a$ does the equation $x^{2}-(a+12) x+12 a+3=0$ have only integer roots? | 1. The time of ascent and the total time of motion are correctly found - 0.5 points | notfound | Algebra | math-word-problem | Yes | Problem not solved | olympiads | false |
2. Petya wrote down an integer in decimal form, starting with the digits 2022 and divisible by 2023. What is the smallest number Petya could have written? | 2. The horizontal component of the velocity and the vertical component of the initial velocity are correctly found - 0.5 points | notfound | Number Theory | math-word-problem | Yes | Problem not solved | olympiads | false |
3. In triangle $ABC$, the angle bisectors $AD$ and $BE$ intersect at point $O$. The length of segment $OE$ is 1. It is known that a circle can be circumscribed around quadrilateral $OEC D$. Find the radius of this circle. | 3. The maximum height of the body's ascent is correctly found - 0.5 points | notfound | Geometry | math-word-problem | Yes | Problem not solved | olympiads | false |
4. From six identical pieces of wire, a pyramid was made and connected to an electrical circuit at the midpoints of two opposite edges (see the left figure). It turned out that the resistance of the pyramid in this connection is $R$. What will be the
. Conversion to a 50-poi... | notfound | Logic and Puzzles | math-word-problem | Incomplete | Problem not solved | olympiads | false |
5. Two loads with masses $m$ and $2 m$ are connected by three threads as shown in the figure. The thread $\mathrm{AB}$ is horizontal, and the thread CD forms an angle $\alpha$ with the vertical. Find the tensions in the threads AB and CD. The thread AB is cut. Find the accelerations of the loads immediately after this.... | 5. First, let's find the forces acting in all threads in equilibrium. On the body of mass \( m \), the force of gravity \( m \vec{g} \) and three tension forces of the threads \( \vec{T}_{A B}, \vec{T}_{C D} \), and the tension force of the lower thread, equal to \( 2 m \vec{g} \) (see the figure). Therefore, the equil... | a_{}=3\sin\alpha,\quada_{2}=0 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. A body was thrown from the surface of the earth at some angle to the horizon. It is known that the velocity vector of the body is directed at an angle $\alpha=30^{\circ}$ to the horizon at times $t_{1}$ and $t_{2}$ after the throw. Find the maximum height of the body's rise above the ground and the distance from the... | 6. Since the trajectory of the body is symmetric with respect to the highest point, the time of ascent $t_{\text {pod }}$ of the body to the maximum height lies exactly halfway between the moments $t_{1}$ and $t_{2}$:
$$
t_{\text {pod }}=\frac{t_{2}+t_{1}}{2}
$$
and the total time of motion $t_{\text {poli }}$ is twi... | S=\frac{\sqrt{3}}{2}(t_{2}^{2}-t_{1}^{2}),\quad=\frac{(t_{2}+t_{1})^{2}}{8} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. Solve the equation $\log _{x} 2+\log _{x+2} 2+\log _{x+3} 2=\frac{1}{\log _{2} x+\log _{2}(x+2)+\log _{2}(x+3)}$. | 1. Statement (with justification from the condition of the inextensibility of the rod and the fact that the body's velocity at the moment of lifting the body to the maximum height is horizontal), that the velocities of the body and the ring at this moment are the same - 0.5 points | notfound | Algebra | math-word-problem | Yes | Problem not solved | olympiads | false |
2. Petya decided to ask for advice from his friends Misha and Masha on whether he should get a dog or not. The probability that Misha will say "yes" is 0.6, and Masha - 0.4. If the advice of his friends coincides, Petya will follow it. If not, Petya will trust Misha's advice with a probability of 0.8. What is the proba... | 2. The velocity of the body and the ring at the moment of reaching the maximum height (found using the law of conservation of momentum) is correctly determined - 0.5 points | notfound | Logic and Puzzles | math-word-problem | Yes | Problem not solved | olympiads | false |
3. At the base of the triangular pyramid $A B C D$ lies an isosceles triangle $A B C$ with base $A C$. All lateral faces of the pyramid are equally inclined to the base. A sphere with radius 1 and center on the base of the pyramid touches the sides of triangle $A B C$ and two lateral edges $D A$ and $D C$. Find the sma... | 3. The correct law of conservation of energy for the moment of raising the body to the maximum height - 0.5 points | notfound | Geometry | math-word-problem | Yes | Problem not solved | olympiads | false |
4. A monoatomic ideal gas undergoes the following cyclic process: 1-2 isochoric heating, $2-3$ - isothermal expansion, 3-1 - isobaric compression (see the process graph in pressure-volume coordinates). For this process, the following are known: the pressure of the gas in state 1 - $p_{1}$, the volume of the gas in stat... | 4. The correct answer is $-0.5$ points
The score for the problem is the sum of the scores according to the criteria listed above
Evaluation of the participant's work
The final score for the work is the sum of the scores for each problem (maximum score - 12 points). Conversion to a 50-point scale was carried out acco... | notfound | Other | math-word-problem | Incomplete | Problem not solved | olympiads | false |
5. A resistor with an unknown resistance was connected to an electrical voltage source $U$, and two amperemeters $\mathrm{A}_{1}$ and $\mathrm{A}_{2}$ were connected in parallel (see the left figure). In this case, the amperemeters show currents of $I_{1}=2$ A and $I_{2}=3$ A. Then, these amperemeters, together with th... | 5. Obviously, ammeters have different resistances, since different currents flow through them when connected in parallel in the circuit. From Ohm's law for ammeters, it follows that the ratio of their resistances is
$$
\frac{R_{1}}{R_{2}}=\frac{I_{2}}{I_{1}}=\frac{3}{2}
$$
Since the current flowing through the source... | 12.7\mathrm{~A} | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
6. At the ends of a weightless, inextensible rod, a small body and a massive ring of the same mass are attached. The ring can move without friction along a horizontal rigid spoke in a gravitational field (see figure). At some moment, the ring and the body were given velocities $v$ and $2 v$, directed horizontally along... | 6. At the moment of reaching the maximum height, the body moves horizontally. Therefore, from the condition of the inextensibility of the rod, we conclude that the speed is the same (their projections on the direction of the rod should be the same, and the speeds of the body and the ring at the moment the body reaches ... | \frac{9v^{2}}{4(1-\sin\alpha)} | Other | math-word-problem | Yes | Yes | olympiads | false |
1. Two circles touch each other and the sides of two adjacent angles, one of which is $60^{\circ}$. Find the ratio of the radii of the circles. | 1. Answer: $r: R=1: 3$ Solution. The centers of the circles lie on the bisectors $A K$ and $A N$ of the adjacent angles,
$\alpha=\square B A C=60^{\circ}$
$\square K A N=90^{\circ}, \square C A K=\square A Q O=30^{\circ}, A M=h$. $r=h \tan \frac{\alpha}{2}, R=h \cot \frac{\alpha}{2} \rightarrow r: R=\tan^2 \frac{\alp... | r:R=1:3 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
2. In a garden partnership, they decided to make a pond using water from a spring. All the water from the spring was directed into the pond using a pipeline, and the pond began to fill with water. As a result, after \( t = 16 \) days, the pond was filled to \( 2 / 3 \) of its volume. At this moment, it was noticed that... | 2. Let the volume of the pond be $V$, and the volume of water flowing out of the source per unit time be $w$. Since before the leak was sealed, three-quarters of the water from the spring entered the pond, we have
$$
\frac{3}{4} w t = \frac{2}{3} V \quad \Rightarrow \quad \frac{9}{8} t = \frac{V}{w}
$$
![](https://cd... | t_1=\frac{3}{8} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. Two people simultaneously step onto an escalator from opposite ends, which is moving downward at a speed of $u=1.5 \mathrm{~m} / \mathrm{s}$. The person moving downward has a speed of $v=3 \mathrm{~m} / \mathrm{s}$ relative to the escalator, while the person moving upward has a speed of $2 v / 3$ relative to the esc... | 3. The speeds of the people relative to the ground are
$$
\text { descending } v_{c n}=v+u \text {, ascending } v_{\text {nоо }}=\frac{2}{3} v-u \text {. }
$$
Therefore, the time it takes for the people to meet is
$$
t=\frac{l}{v_{c n}+v_{n o d}}=\frac{3 l}{5 v}
$$
And since the ascending person will travel a dista... | 10\mathrm{~} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. A body of mass $m$ is attached to one end of a weightless, inextensible string, the other end of which is attached to the ceiling. The string is passed over a pulley of mass $M$, which is concentrated on its axis. At the initial moment, the body is held and then released. Find the accelerations of the body and the p... | 4. It is clear that the bodies will not fall freely with the acceleration due to gravity, and the string will be taut. Indeed, if the pulley falls with an acceleration of $g$, then on the side where the body is located, such a length of the string is "stretched" that the acceleration of the body should equal $2g$, whic... | a_{1}=\frac{2(2+M)}{4+M},\quada_{2}=\frac{(2+M)}{4+M} | Calculus | math-word-problem | Yes | Yes | olympiads | false |
5. Two segments on the co-
ordinate line are defined by the coordinates of their two endpoints each. The task is to write a program that calculates the length of the intersection of these se... | 5.
{pascal}
program task_9;
var
x1, x2, x3, x4 : double;
t1, t2: double;
begin
read(x1,x2,x3,x4); // read input data
// determine the larger of the start coordinates
t1 := x1;
if x3 > t1 then
t1 := x3;
// determine the smaller of the end coordinates
t2 := x2;
if x4 < t2 then
t2 := x4;
// output the res... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false |
7. [6 points] Find the number of pairs of integers $(x ; y)$ that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
y \geqslant 90+x-6^{90} \\
y \leqslant \log _{6} x
\end{array}\right.
$$ | The answer should be presented in the form of an algebraic sum of no more than three terms.
## VARIANT 14 | notfound | Inequalities | math-word-problem | Yes | Incomplete | olympiads | false |
7. [6 points] Find the number of pairs of integers $(x ; y)$ that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
y \geqslant 80 + x - 7^{80} \\
y \leqslant \log _{7} x
\end{array}\right.
$$ | The answer should be presented in the form of an algebraic sum of no more than three terms.
## VARIANT 15 | notfound | Inequalities | math-word-problem | Yes | Incomplete | olympiads | false |
7. [6 points] Find the number of pairs of integers $(x ; y)$ that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
y \geqslant 110+x-5^{110} \\
y \leqslant \log _{5} x
\end{array}\right.
$$ | The answer should be presented in the form of an algebraic sum of no more than three terms.
## VARIANT 16 | notfound | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
7. [6 points] Find the number of pairs of integers $(x ; y)$ that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
y \geqslant 70+x-4^{70} \\
y \leqslant \log _{4} x
\end{array}\right.
$$ | The answer should be presented in the form of an algebraic sum of no more than three terms.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | Inequalities | math-word-problem | Yes | Yes | olympiads | false | |
11. Let $\omega$ be the circumcircle of an acute-angled triangle $ABC$, and $AL$ be the angle bisector of triangle $ABC$. Denote by $P$ the point of intersection of the extension of the altitude $BH$ of triangle $ABC$ with the circle $\omega$. It is known that $BP = CP$, and param1. Find $\angle ALC$. Provide your answ... | 11. Let $\omega$ be the circumcircle of an acute-angled triangle $ABC$, and $AL$ be the angle bisector of triangle $ABC$. Denote by $P$ the point of intersection of the extension of the altitude $BH$ of triangle $ABC$ with the circle $\omega$. It is known that $BP = CP$, and param1. Find $\angle ALC$. Give your answer ... | Geometry | math-word-problem | Yes | Yes | olympiads | false | |
12. It is known that param 1 ( $a, b, c, d$ - some numbers). Find $a+b+c+d$.
| param1 | Answer |
| :---: | :---: |
| $-26 x^{3}+24 x^{2}-6 x=(a x+b)^{3}+(c x+d)^{3}$ | |
| $\frac{x^{2}}{6}+6 x+56=(a x+b)^{3}+(c x+d)^{3}$ | |
| $-7 x^{3}+18 x^{2}-12 x=(a x+b)^{3}+(c x+d)^{3}$ | |
| $6 x^{2}+24 x+26=(a x+b)^{3}+(c x+... | 12. It is known that param1 ( $a, b, c, d$ - some numbers). Find $a+b+c+d$.
| param1 | Answer |
| :---: | :---: |
| $-26 x^{3}+24 x^{2}-6 x=(a x+b)^{3}+(c x+d)^{3}$ | -2 |
| $\frac{x^{2}}{6}+6 x+56=(a x+b)^{3}+(c x+d)^{3}$ | 2 |
| $-7 x^{3}+18 x^{2}-12 x=(a x+b)^{3}+(c x+d)^{3}$ | -1 |
| $6 x^{2}+24 x+26=(a x+b)^{3}+(... | Algebra | math-word-problem | Incomplete | Yes | olympiads | false | |
13. Given an arithmetic progression param1, where param2. Find the sum of param3.
| param1 | param2 | param3 | Answer |
| :---: | :---: | :---: | :---: |
| $a_{1}, a_{2}, \ldots, a_{443}$ | $a_{1}=2, a_{443}=17$ | $\frac{1}{a_{1} a_{2}}+\frac{1}{a_{2} a_{3}}+\frac{1}{a_{3} a_{4}}+\ldots+\frac{1}{a_{442} a_{443}}$ | |... | 13. Given an arithmetic progression param1, where param2. Find the sum of param3.
| param1 | param2 | param3 | Answer |
| :---: | :---: | :---: | :---: |
| $a_{1}, a_{2}, \ldots, a_{443}$ | $a_{1}=2, a_{443}=17$ | $\frac{1}{a_{1} a_{2}}+\frac{1}{a_{2} a_{3}}+\frac{1}{a_{3} a_{4}}+\ldots+\frac{1}{a_{422} a_{443}}$ | 13... | Algebra | math-word-problem | Yes | Problem not solved | olympiads | false | |
14. Points $E$ and $F$ are the midpoints of sides $B C$ and $C D$ of parallelogram $A B C D$, respectively. The circle $\Omega$, constructed with $A E$ as its diameter, passes through point $F$. Find the length of segment $A D$ if param $1 \Omega$ equals param 2.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
... | 14. Points $E$ and $F$ are the midpoints of sides $B C$ and $C D$ of parallelogram $A B C D$, respectively. The circle $\Omega$, constructed with $A E$ as its diameter, passes through point $F$. Find the length of segment $A D$ if param $1 \Omega$ equals param 2.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
... | Geometry | math-word-problem | Incomplete | Yes | olympiads | false | |
15. What is the smallest integer value that the sum of the roots of the equation param 1 can take?
| param1 | Answer |
| :---: | :---: |
| $x^{2}-\left(a^{2}+2\right) x+a^{2}+6=0$ | |
| $x^{2}-\left(a^{2}+2\right) x+a^{2}+8=0$ | |
| $x^{2}-\left(a^{2}+4\right) x+2 a^{2}+7=0$ | |
| $x^{2}-\left(a^{2}+4\right) x+2 a^... | 15. What is the smallest integer value that the sum of the roots of the equation param 1 can take?
| param1 | Answer |
| :---: | :---: |
| $x^{2}-\left(a^{2}+2\right) x+a^{2}+6=0$ | 7 |
| $x^{2}-\left(a^{2}+2\right) x+a^{2}+8=0$ | 8 |
| $x^{2}-\left(a^{2}+4\right) x+2 a^{2}+7=0$ | 8 |
| $x^{2}-\left(a^{2}+4\right) x+2... | Algebra | math-word-problem | Yes | Yes | olympiads | false | |
16. The last digit of a six-digit number was moved to the beginning (for example, $456789 \rightarrow$ 945678), and the resulting six-digit number was added to the original number. Which numbers from the interval param 1 could have resulted from the addition? In the answer, write the sum of the obtained numbers.
| par... | 16. The last digit of a six-digit number was moved to the beginning (for example, $456789 \rightarrow$ 945678), and the resulting six-digit number was added to the original number. Which numbers from the interval param 1 could have resulted from the addition? In the answer, write the sum of the obtained numbers.
| par... | 1279267 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
17. We took natural numbers param 1 and wrote these raparam 2 along the top side of the table param3 (one number above each column), and wrote these same raparam2 along the left side of the table (one number to the left of each row). In the cells of the table, we wrote the products of the corresponding numbers (a "mult... | 17. We took natural numbers param 1 and wrote these param 2 along the top side of the table param3 (one number above each column), and wrote these same param2 along the left side of the table (one number to the left of each row). In the cells of the table, we wrote the products of the corresponding numbers (a "multipli... | notfound | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
18. According to the results of a volleyball tournament held in a round-robin format (i.e., each team played one game against each other), it turned out that the top three teams won against each of the other teams, and the sum of points scored by the top three teams was param1 less than the sum of points scored by the ... | 18. According to the results of a volleyball tournament held in a round-robin format (i.e., each team played one game against each other), it turned out that the top three teams won against each of the other teams, and the sum of points scored by the top three teams was param1 less than the sum of points scored by the ... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
19. Find param1, if param2 for all real $x$, and param3.
| param1 | param2 | param3 | Answer |
| :--- | :--- | :--- | :--- |
| $f(1000)$ | $f(x+3)=f(x)+x-7$ | $f(1)=1$ | |
| :---: | :---: | :---: | :---: |
| $f(700)$ | $f(x+3)=f(x)+2 x+5$ | $f(1)=2$ | |
| $f(1000)$ | $f(x+3)=f(x)-x+10$ | $f(1)=15$ | |
| $f(850)$ ... | 19. Find param1, if param 2 for all real $x$, and param3.
| param1 | param2 | param3 | Answer |
| :--- | :--- | :--- | :--- |
| $f(1000)$ | $f(x+3)=f(x)+x-7$ | $f(1)=1$ | 163837 |
| :---: | :---: | :---: | :---: |
| $f(700)$ | $f(x+3)=f(x)+2 x+5$ | $f(1)=2$ | 163801 |
| $f(1000)$ | $f(x+3)=f(x)-x+10$ | $f(1)=15$ | -... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false |
20. In the cells of a square board param 1, chips were placed. It turned out that for any chip, either in the column or in the row where this chip is located, there are no more chips. What is the maximum number of chips that can be on the board?
| param1 | Answer |
| :---: | :---: |
| $30 \times 30$ | |
| $25 \times ... | 20. In the cells of a square board param 1, chips were placed. It turned out that for any chip, either in the column or in the row where this chip is located, there are no more chips. What is the maximum number of chips that can be on the board?
| param1 | Answer |
| :---: | :---: |
| $30 \times 30$ | 58 |
| $25 \time... | Combinatorics | math-word-problem | Yes | Problem not solved | olympiads | false | |
1. Find the number of points in the $x O y$ plane having natural coordinates $(x, y)$ and lying on the parabola $y=-\frac{x^{2}}{4}+11 x+23$. | Answer: 22.
Solution. Let's find those values of $x$ for which $y$ is positive: $-\frac{x^{2}}{4}+11 x+23>0 \Leftrightarrow-\frac{1}{4}(x+2)(x-46)>0$, from which $-2<x<46$. On this interval, there are 45 natural values of $x: x=1, x=2, \ldots, x=45$. In this interval, $y$ takes integer values only for even $x$ - a tot... | 22 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. Solve the inequality $\frac{\sqrt{x}-2}{1-\sqrt{x+1}} \geq 1+\sqrt{x+1}$. | Answer: $x \in(0 ; 1]$.
Solution. The domain of the inequality is determined by the conditions $x \geq 0,1-\sqrt{x+1} \neq 0$, from which we get that $x>0$. Note that on the domain, the denominator of the fraction is negative, so we can multiply both sides of the inequality by it, changing the inequality sign. Then
$... | x\in(0;1] | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
3. In the number $2 * 0 * 1 * 6 * 0 * 2 *$, each of the 6 asterisks needs to be replaced with any of the digits $0,1,2,3,4,5,6,7,8$ (digits can be repeated) so that the resulting 12-digit number is divisible by 45. In how many ways can this be done? | Answer: 13122.
Solution. For a number to be divisible by 45, it is necessary and sufficient that it is divisible by 5 and by 9. To ensure divisibility by 5, we can choose 0 or 5 as the last digit (2 ways).
To ensure divisibility by nine, we proceed as follows. We will choose four digits arbitrarily (this can be done ... | 13122 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
4. Find all values of the parameter $a$, for each of which the equation $a|x-1|+\frac{x^{2}-7 x+12}{3-x}=0$ has exactly one solution. | Answer: $a \in\left[-1 ;-\frac{1}{2}\right) \cup\left(-\frac{1}{2} ; 1\right)$.
Solution. Given the condition $x \neq 3$, the equation is equivalent to $a|x-1|=x-4$. The graph of the right side of the equation is the line $y=x-4$. The graph of the left side of the equation is a "V" shape with its vertex at the point $... | \in[-1;-\frac{1}{2})\cup(-\frac{1}{2};1) | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. Solve the system of equations $\left\{\begin{array}{l}x+\sqrt{x+2 y}-2 y=\frac{7}{2}, \\ x^{2}+x+2 y-4 y^{2}=\frac{27}{2}\end{array}\right.$. | Answer: $\left(\frac{19}{4} ; \frac{17}{8}\right)$.
Solution. Let $\sqrt{x+2 y}=u, x-2 y=v$. Then the system takes the form
$$
\left\{\begin{array} { l }
{ u + v = \frac { 7 } { 2 } } \\
{ u ^ { 2 } v + u ^ { 2 } = \frac { 2 7 } { 2 } }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
v=\frac{7}{2}-u \\
u^{2}\lef... | (\frac{19}{4};\frac{17}{8}) | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.